Speaker: Horng-Tzer Yau (Courant Institute)

Title: Superdiffusivity of Two Dimensional Stochastic Dynamics

Date: 2/5/2002

Abstract: The asymmetric simple exclusion process (ASEP) is a system of asymmetric random walks with hard core condition so that no two particles can be on the same site. It has become the paradigm of stochastic dynamics modeling the transport phenomena or the interface growth. Furthermore, combining asymmetric simple exclusion and suitable collision kernels, one can construct lattice gas models converging to the incompressible Navier-Stokes equations in dimension $d=3$. In particular, the viscosity is finite. For $d \le 2$, the diffusive behavior no longer holds. It was conjectured that its diffusion coefficient diverges as $t^{1/3}$ in $d=1$ and $(\log t)^{2/3}$ in $d=2$ by Beijeren-Kutner-Spohn or the Kardar-Parisi-Zhang equation. Recent results based on integrable systems has indicated strongly the exponent $1/3$ in the totally asymmetric case in dimension $d=1$. We shall describe an approach based on Euclidean field theory proving this conjecture in dimension $d=2$.

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Speaker: Victor Dela Pena ( Columbia University)

Title: The LIL for self-normalised martingales

Date: 2/12/2002

Abstract: In this talk I introduce an extension of Stout’s LIL to the case of self-normalised martingales. The proof relies on the construction of an exponential supermartingale involving the martingale and the sum of its squared martingale differences. This represents joint work with T. L. Lai and M. J. Klass

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Speaker: Davar Khoshnevisan (University of Utah)

Title: Greedy Search on a Sparse Tree with Random Edge-Weights

Date: 2/19/2002

Abstract: I will discuss the asymptotic value of the greedy search on a sub-exponentially growing tree, when the edge-weights are i.i.d. random variables. Amongst other things, I will show the nonexistence of a “universal algorithm” that can match greedy search. this is joint work with T. M. Lewis.

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Speaker: Jan Rosi{‘n}ski (University of Tennessee, Knoxville)

Title: Radonification of cylindrical semimartingales by a single Hilbert-Schmidt operator

Date: 3/9/2002

Abstract: It is well known that semimartingales constitute the most general basis for nonanticipating stochastic integration. An essential problem in the infinite dimensional stochastic integration is to describe the class of radonifying operators for cylindrical semimartingales. This problem was investigated by Badrikian and Ustunel (1996) who showed that a composition of three Hilbert-Schmidt operators radonifies a cylindrical semimartingale on a Hilbert space to a strong semimartingale. Analogous results regarding compositions of three operators were obtained by L. Schwartz (1994-96) in the context of Banach space valued semimartingales. We prove that just a single Hilbert-Schmidt operator suffices to radonify a cylindrical semimartingale on a Hilbert space. Since a Hilbert-Schmidt operator is necessary to radonify a cylindrical Brownian motion, this resolves the problem for Hilbert spaces. The proof relies on inequalities employing a Gaussian randomization. The talk will be based on a joint work with A. Jakubowski, S. Kwapien, and P. Raynaud de Fitte.

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Speaker: Joseph Yukich (Lehigh University)

Title: Limit Theory for Random Packing

Date: 3/16/2002

Abstract: Consider sequential packing of unit balls in a large cube, as in the Renyi car-parking model, but in any dimension and with finite input. We prove a law of large numbers and central limit theorem for the number of packed balls in the thermodynamic limit. We prove analogous results for numerous related applied models, including cooperative sequential adsorption, ballistic deposition, and spatial birth-growth models. The proofs are based on a general law of large numbers and central limit theorem for “stabilizing” functionals of marked point processes of independent uniform points in a large cube, which are of independent interest. “Stabilization” means, loosely, that local modifications have only local effects. (based on joint work with Mathew Penrose).

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Speaker: Kavita Ramanan (Bell Labs, Lucent Technologies)

Title: Large Deviations of Stationary Reflected Brownian Motion

Date: 3/23/2002

Abstract: We analyze the tail probabilities of stationary reflected Brownian motion in the N-dimensional nonnegative orthant having drift b, covariance matrix A and constraint matrix D. Under suitable stability and regularity conditions, the exponential decay rate of the tail probabilities (i.e. the rate function) is known to have a variational representation V(x). We discuss why this representation is hard to analyze, and develop new techniques for analyzing this problem.

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Speaker: Richard Bass (University of Connecticut)

Title: Local times on curves for space-time Brownian motion

Date: 3/30/2002

Abstract: The local time for a curve $f$ with respect to space-time Brownian motion measures the amount of time the Brownian motion spends in a band about $f$. I’ll discuss some characterizations of these local times and then address the question of when the supremum of local times on curves over a large class of curves is finite or not.

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Speaker: Evarist Gin'{e} (University of Connecticut, Storrs)

Title: On kernel density estimators: uniform convergence over the wholespace of the deviations from their means

Date: 4/7/2002

Abstract: Stute’s law of the logarithm (rate of a.s. convergence) for the sup norm over compact sets of the deviation from the mean of a kernel density estimator, as well as the Bickel-Rosenblatt result on shift-convergence in distribution of the same quantities, are strengthened to the norm of the supremum over the whole space. Some of the tools used include Talagrand’s exponential inequality for empirical processes, extension of classical results on stationary Gaussian processes to slightly not stationary and, of course, KMT. These results were obtained in collaboration with A. Guillou, and with V.I. Koltchinskii and L. Sachanenko.

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Speaker: Haya Kaspi (Technion)

Title: Splitting/Coalescence Phenomena in Skew Brownian Motion

Date: 1/11/2003

Abstract: Skew Brownian motion is a process that satifies the stochastic differential equation $$X^x_t=x+B_t+\beta L_t^x,$$ where $B_t$ is a given Brownian motion, $\beta \in [-1,1]$ is a fixed constant and $L^x_t$ is the symmetric local time of $X$ at $0$, i.e. $$L^x_t=\lim_{\epsilon\rightarrow 0}\frac{1}{2\epsilon}\int_0^t 1_{(-\epsilon, \epsilon)}(X^x_s)ds .$$ The existence and uniqueness of strong solutions to the above equation, starting at each real $x$, was proved by Harrison and Shepp. In the special case $\beta=1$ the solution to the above equation is the reflected Brownian motion. For any $\beta\in(-1,1)$ the absolute value of $X$ is a reflected Brownian motion. The set of measure $0$ on which the solution is not unique depends on the starting point $x$, and it turns out that there is no strong uniqueness for all starting points simultaneously. In this talk we shall consider a stochastic flow in which individual particles follow skew Brownian motions, with each one of these processes driven by the same Brownian motion and starting at a different points in space and time. Due to the lack of the simultaneous strong uniqueness for the whole system of stochastic differential equations, the flow contains lenses, i.e., pairs of skew Brownian motions which start at the same time at the same point, bifurcate, and then coalesce in a finite time. We shall discuss both qualitative and quantitative (distributional) results on the geometry of the flow and lenses. \vspace{1cm} The talk is based on a joint work with Krzysztof Burdzy.

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Speaker: Jay Rosen (CUNY)

Title: The fractal nature of late points for two dimensional random walk

Date: 1/18/2003

Abstract: \newcommand{\Bbb}[1]{\mathbf{#1}} \def \ov{\overline} \def \un{\underline} \newcommand{\be}{{\begin{equation}}} \newcommand{\ee}{{\end{equation}}} \def \bt{\begin{theorem}} \def \et{\end{theorem}} \def \bea{\begin{eqnarray}} \def \eea{\end{eqnarray}} \def \bas{\begin{eqnarray*}} \def \eas{\end{eqnarray*}} % ***************************************** % Greek Letters \def \al{\alpha} \def \bb{\beta} \def \IJK{\mathcal I} \def \bga{\bar\gamma} %OOO \def \HHofer{{\bf A}} %AAAAA \def\wtn{{\widetilde{n}}} \def\wta{{\widetilde{a}}} \def\apr{{a’}} \def\aast{{a^{\ast}}} \def\oD{{\oo{D}}} \def\tast{{t^{\ast}_{n}}} \def\taup{{\oo{\tau}}} \def\sip{{\oo{\zeta}}} \def\sii{{\zeta}} %NA \def\mm{{\widehat{n}}} \def\wtW{{\widetilde{W}}} %———– \def \ga{\gamma} \def \Ga{\Gamma} \def \de{\delta} \def \De{\Delta} \def \ep{\epsilon} \def \epon{\ep_1} \def \epr{\ep_0} \newcommand{\eps}{\varepsilon} \newcommand{\bart}{{\bar{\theta}}} \newcommand{\won}{{\mbox{\bf 1}}} \def \vep{\varepsilon} \def \la{\lambda} \def \La{\Lambda} \def \ka{\kappa} \def \om{\omega} \def \Om{\Omega} \def \va{\varrho} \def \ffi{\Phi} \def \vf{\varphi} \def \si{\sigma} \def \Si{\Sigma} \def \vsi{\varsigma} \def \th{\theta} \def \Th{\Theta} \def \ups{\Upsilon} \def \ze{\zeta} \def \tr{\nabla} \def \Gm{\Gamma} \def \dGm{({\rm det} \Gm)} \def \dA{{\rm det} A} \def\wpi{\pi_\Gamma} % Math Symbols \def \ff{\infty} \def \wh{\widehat} \def \wt{\widetilde} \def \dar{\downarrow} \def \rar{\rightarrow} \def \uar{\uparrow} \def \sbs{\subseteq} \def \mpt{\mapsto} \def \cd{\,\cdot\,} \def \cds{\cdots} \def \lds{\ldots} \newcommand{\ls}[1] {\dimen0=\fontdimen6\the\font \lineskip=#1\dimen0 \advance\lineskip.5\fontdimen5\the\font \advance\lineskip-\dimen0 \lineskiplimit=.9\lineskip \baselineskip=\lineskip \advance\baselineskip\dimen0 \normallineskip\lineskip \normallineskiplimit\lineskiplimit \normalbaselineskip\baselineskip \ignorespaces } \newcommand{\req}[1]{(\ref{#1})} \def \R{{\Bbb{R}}} \def \G{{\bf G}} %\def \H{{\bf H}} \def \Z{{\Bbb{Z}}} %\def \S{{\bf S}} \def \sfB{{\sf B}} \def \sfS{{\sf S}} \def \T{{\Bbb{T}}} \def \C{{\bf C}} \def \AA{{\mathcal A}} \def \BB{{\mathcal B}} \def \CC{{\mathcal C}} \def \DD{{\mathcal D}} \def \EE{{\mathcal E}} \def \FF{{\mathcal F}} \def \GG{{\mathcal G}} \def \HH{{\mathcal H}} \def \II{{\mathcal I}} \def \JJ{{\mathcal J}} \def \KK{{\mathcal K}} \def \LL{{\mathcal L}} \def \MM{{\mathcal M}} \def \NN{{\mathcal N}} \def \OO{{\mathcal O}} \def \PP{{\mathcal P}} \def \QQ{{\mathcal Q}} \def \RR{{\mathcal R}} %\def \SS{{\mathcal S}} \def \TT{{\mathcal T}} \def \UU{{\mathcal U}} \def \VV{{\mathcal V}} \def \YY{{\mathcal Y}} \def \ZZ{{\mathcal Z}} \def \Pxh{P^{x/h}} \def \Exh{E^{x/h}} \def \Px{P^{x}} \def \Ex{E^{x}} \def \Prh{P^{\rho/h}} \def \Erh{E^{\rho/h}} \def \p{p_{t}(x,y)} \def \({\left(} \def \){\right)} \def \lk{\left[} \def \rk{\right]} \def \lc{\left\{} \def \rc{\right\}} \def \bsq{\hfil $\Box$} \def \nn{\nonumber} \def \Proof{{\bf Proof: }} \def \Bo{\bigotimes} \def \bo{\times} \def \ot{\times} \def \bc{\begin{center} } \def \ec{\end{center} } \newcommand{\Ini}{{I_{n,i}}} %\newcommand{\reals}{{I\!\!R}} \newcommand{\reals}{{\Bbb{R}}} \newcommand{\F}{{\mathcal F}} \newcommand{\D}{{\mathcal D}} \newcommand{\Fn}{{{\mathcal F}_n}} \newcommand{\Gn}{{{\mathcal G}_n}} \newcommand{\Hn}{{{\mathcal H}_n}} \newcommand{\Fp}{{{\mathcal F}^p}} \newcommand{\Gp}{{{\mathcal G}^p}} \newcommand{\PPP}{{\mathbf P}} \newcommand{\Pop}{{P\otimes \PPP}} %\newcommand{\hm}{h_{\gamma,b}{\mbox{-meas}}} \newcommand{\hm}{\HH^\varphi} \newcommand{\nuw}{{\nu^W}} \newcommand{\ths}{{\theta^*}} \newcommand{\beq}[1]{\begin{equation}\label{#1}} \newcommand{\eeq}{\end{equation}} \newcommand{\integers}{{\rm I\!N}} %\newcommand{\E}{{\mathrm I\!E}} \newcommand{\E}{{\Bbb E}} \newcommand{\te}{{\tilde{\delta}}} \newcommand{\tI}{{\tilde{I}}} \newcommand{\loge}{{\log(1/\ep)}} \newcommand{\logen}{{\log(1/\ep_n)}} \newcommand{\epn}{{\ep_n}} \def\var{{\rm Var}} \def\cov{{\rm Cov}} \def\one{{\bf 1}} \def\leb{{\mathcal L}eb} \def\Ho{{\mbox{\sf H”older}}} %% This is H”older exponent in formulas. \def\thi{{\mbox{\sf Thick}}} \def\cthi{{\mbox{\sf CThick}}} \def\late{{\mbox{\sf Late}}} \def\clate{{\mbox{\sf CLate}}} %fraction with round brackets \newcommand{\ffrac}[2] {\left( \frac{#1}{#2} \right)} \newcommand{\calF}{{\mathcal F}} \newcommand{\dfn}{\stackrel{\triangle}{=}} \newcommand{\beqn}[1]{\begin{eqnarray}\label{#1}} \newcommand{\eeqn}{\end{eqnarray}} \newcommand{\oo}{\overline} \newcommand{\uu}{\underline} \newcommand{\bfcdot}{{\mbox{\boldmath$\cdot$}}} \newcommand{\Var}{{\rm \,Var\,}} %qed \def\squarebox#1{\hbox to #1{\hfill\vbox to #1{\vfill}}} \newcommand{\half}{\frac{1}{2}\:} \newcommand{\beaa}{\begin{eqnarray*}} \newcommand{\eeaa}{\end{eqnarray*}} \newcommand{\calK}{{\mathcal K}} \def\dimm{{\overline{{\rm dim}}_{_{\rm M}}}} %\def\dimm{{\overline{\rm {dim}}_M \def\dimp{\dim_{_{\rm P}}} \def\htaum{{\hat\tau}_m} \def\htaumk{{\hat\tau}_{m,k}} \def\htaumkj{{\hat\tau}_{m,k,j}} \noindent Let $\TT_n(x)$ denote the time of first visit of a point $x$ on the lattice torus $\Z_n^2=\Z^2/n\Z^2$ by the simple random walk. The size of the set of $\alpha,n$-late points $\LL_n(\al)=\{x\in \Z_n^2: \TT_n(x)\geq \al\frac{4}{\pi}(n\log n)^2\}$ is approximately $n^{2(1-\al)}$, for $\al\in (0,1)$ ($\LL_n(\al)$ is empty if $\al>1$ and $n$ is large enough). These sets have interesting clustering and fractal properties: we show that for $\bb \in (0,1)$ a disc of radius $n^\bb$ centered at non-random $x$ typically contains about $n^{2 \bb(1-\al/\bb^2)}$ points from $\LL_n(\al)$ (and is empty if $\bb >1$ vertices with each edge chosen independently with probability $0<p2$ in large random graphs, the derivation of which is yet an open problem. This talk is based on joint works with Wlodek Bryc, Francis Comets, and Tiefeng Jiang.

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Speaker: Jay Rosen (CUNY/CSI)

Title: Frequently visited sets for random walks

Date: 9/12/2004

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Speaker: Irene Hueter (Baruch College)

Title: Random convex hulls and the Stein method

Date: 9/19/2004

Abstract: I will survey a few of the results and many open problems of the wide literature on the convex hull of a random point set in $R^d$. Random convex sets have a long mathematical history and commonly arise in linear programming algorithms, imaging and in multivariate statistics. I will explain how the Stein method can be applied to prove asymptotic normality of the number of vertices of a random convex hull and bound the normal approximation error.

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Speaker: Olivier Daviaud (Stanford, Morgan Stanley)

Title: Extremes of the discrete Gaussian free field in 2-d

Date: 9/26/2004

Abstract: The two dimensional Gaussian free field (on the square lattice $\{1,…,N\}^2)$ is a well-known Gaussian random surface. Its asymptotic maximum (for large $N$) was first computed by Bolthausen et al. (2001). Building on this work and drawing inspiration from Dembo et al. (2004), we exhibit an intricate multi-fractal structure for the sets where the field is unusually high (or low). Next, we show that this structure remains unchanged when the field is conditioned on being everywhere positive (phenomenon of entropic repulsion). Finally, in light of these results we propose a suggestive analogy between the square of the free field and the two-dimensional simple random walk on the discrete torus.

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Speaker: David Asher Levin (University of Utah)

Title: Dynamical Random Walks

Date: 10/9/2004

Abstract: A dynamical random walk is a stationary, path-valued process which at any fixed time has the same law on paths as a random walk. Benjamini, Haggstrom, Peres, and Steif (2003) first studied which almost-sure properties of random walk may fail to hold at (random) times for the dynamical version. I will describe some atypical behavior which occurs for dynamical random walks, and discuss the role played by the Kolmogorov entropy in analyzing exceptional sets. This is joint work with Davar Khoshnevisan and Pedro Mendez.

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Speaker: Peter M”{o}rters (University of Bath)

Title: Large deviations for Markov chains on random trees

Date: 10/16/2004

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Speaker: Roger Mansuy (University of Paris)

Title: A Tanaka formula for the symmetric Levy processes

Date: 10/23/2004

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Speaker: Wenbo Li (University of Delaware)

Title: Small deviation probabilities for stable processes

Date: 1/1/2005

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Speaker: Dimitri Gioev (Courant, NYU)

Title: Random Matrix Theory: Applications and Universality Questions

Date: 1/8/2005

Abstract: Random Matrix Theory (RMT) is currently of considerable interest in physics as well as in mathematics. An important aspect of RMT is that its results are believed to be applicable to a wide variety of physical, mathematical and applied mathematical situations (universal). Applications of RMT include: – statistical properties of many-body quantum systems – statistics of eigenvalues of classically chaotic quantum systems – elastomechanic resonances – random particle systems with particular focus on percolation problems – random growth models – transport problems – number theory (distribution of zeros of the Riemann zeta-function) – combinatorial problems (longest increasing subsequences, hard-drive disk scheduling problem). A central mathematical issue in RMT which arose very early is universality within random matrix models themselves, and this is currently my main research focus. The talk will start with an intoduction to RMT and its applications. After that I will describe our recent work with Percy Deift (Courant Institute) on the proof of the Universality Conjecture in RMT for orthogonal and symplectic ensembles. Finally, several open problems and ongoing projects will be described.

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Speaker: Peter Bank (Columbia University)

Title: Optimal Control under a Dynamic Fuel Constraint

Date: 1/15/2005

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Speaker: Elena Kosygina (Baruch College, CUNY)

Title: Homogenization of Stochastic Hamilton-Jacobi-Bellman equations

Date: 1/22/2005

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Speaker: Leonid Koralov (Princeton University)

Title: Inverse Problem for Gibbs Fields

Date: 2/8/2005

Abstract: Given a potential of pair interaction and a value of activity, one can construct the corresponding Gibbs distribution in a finite domain $\Lambda \subset Z^d$. It is well known that for small values of activity there exist the infinite volume ($\Lambda \rightarrow {Z}^d$) limiting Gibbs distribution and the infinite volume correlation functions. We prove the converse of this classical result – we show that given $\rho_1$ and $\rho_2(x)$, where $\rho_1$ is a constant and $\rho_2(x)$ is a function on ${Z}^d$, which are sufficiently small, there exist a pair potential and a value of activity, for which $\rho_1$ is the density and $\rho_2(x)$ is the pair correlation function

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Speaker: Xia Chen (University of Tennessee)

Title: Moment asymptotics associated with large permutation groups

Date: 2/15/2005

Abstract: In estimating the tail probability, the moment estimation is one of the efective approaches, especially when the logarithmic generating function is hard to compute. This is often the case as we study the upper tail behaviors of the intersection local times or, the local times of the multi-parameter processes. In these situations, the moment can usually be written in the form of permutation sums. In this talk we discuss two examples, in which the moment asymptotics will be established and some new approaches of applying these moment asymptotics will be introduced.

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Speaker: Gerardo Hernandez-del Valle (Columbia Univ.)

Title: The density of the first crossing time of Brownian motion over a non-decreasing right continuous barrier

Date: 3/5/2005

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Speaker: Haya Kaspi (Technion and Cornell University)

Title: Infinitely Divisible Gaussian Squares

Date: 3/12/2005

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Speaker: Victor de la Pena (Columbia University)

Title: Copulas, Information, Dependence and Decoupling

Date: 4/3/2005

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Speaker: Krzysztof Burdzy (University of Washington)

Title: ON THE ROBIN PROBLEM IN FRACTAL DOMAINS

Date: 5/7/2005

Abstract: The “Robin problem” or the “third boundary problem” is a mathematical model for the flow of a substance (or heat) out of a domain through a semipermeable membrane. I will address the question of when the concentration of the substance (or the temperature) is bounded below by a constant over the whole domain. The problem is analytic, the techniques used in proofs are largely probabilistic. Joint work with Rich Bass and Zhenqing Chen.

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Speaker: Martin Zerner (University of Tuebingen)

Title: On random ranchers and cookie monsters: some self-interacting random walks with bias.

Date: 8/20/2005

Abstract: We consider two models of self-interacting random walks:\\ 1. Excited Random Walks:\\ We put two cookies on each integer and start a random walker at 0. Whenever there is at least one cookie at the walker’s present location, the walker eats one of these cookies and then jumps to the right with probability $p$ and to the left with probability $1-p$, where $p$ is a fixed parameter greater than 1/2. At sites without any cookies left over the walker jumps with probability 1/2 to the right and 1/2 to the left. We consider recurrence, transience and the speed of this and similar walks. Such models have also been investigated e.g. by Benjamini, Wilson, Kozma and Volkov.\\ 2. Random Rancher: \\ We consider a model due to Angel, Benjamini and Virag, in which a random walker in the plane takes steps of length one but avoids the convex hull of its past positions. We show that this walk has positive lim inf speed

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Speaker: Max von Renesse (TU Berlin, CIMS)

Title: A preliminary talk for the geometers introducing the necessary probability techniques.

Date: 8/27/2005

Abstract: This will be a preliminary talk for the speaker’s 4pm talk

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Speaker: Max von Renesse (TU Berlin, CIMS)

Title: Mass Transportation and Synthetic Ricci Curvature Bounds

Date: 8/27/2005

Abstract: The problem of optimal mass transportation has appeared first in the 18th century in economic models. In recent years the theory has undergone a remarkable development with applications in PDE, probability and geometry. The talk will give a short review of some basic concepts involved. The focus of the second part will be geometric. Mass transportation is used for the defintion and analysis of generalized lower Ricci curvature bounds for metric measure spaces with no or almost no regularity. (This talk is given in Differential Geometry and Lie Theory seminar. The talk is at 4pm, but in room 4419 not our usual 5417.)

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Speaker: Gady Kozma, (IAS)

Title: Isoperimetric inequalities in probability

Date: 10/1/2005

Abstract: This talk will survey connection between isoperimetric inequalities on infinite graphs and random walk and percolation on these graphs.

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Speaker: Anotonia F”{o}ldes (CUNY, )

Title: Joint asymptotic behavior of local and occupation times

Date: 10/8/2005

Abstract: Considering a simple symmetric random walk in dimension 3 or higher, we study the almost sure joint asymptotic behavior of two objects: first the local times of a pair of neighboring points, then the local time of a point and the occupation time of the surface of the unit ball around it.

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Speaker: Natella O’Bryant (College of Staten Island, CUNY)

Title: Ballistic points of finite-mode Kolmogorov flows

Date: 10/29/2005

Abstract: This talk is of purely mathematical nature, but some connections to problems dealing with turbulence of incompressible fluids will be drawn. We consider planar flows driven by velocity fields with prescribed spectrum of Kolmogorov’s type. Knowing that a finite-mode approximation of such a flow expands diameters of certain sets linearly in time, we show that a point moving with linear speed is guaranteed to be found almost surely. A similar result is known to hold for isotropic Brownian flows with strictly positive Lyapunov exponent, and also for more general martingale flows, and has been conjectured for the ‘full’ Kolmogorov flow as well.

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Speaker: Greg Markowsky (CUNY)

Title: The derivative of intersection local time in two dimensions

Date: 11/6/2005

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Speaker: Mark Brown (The City College, CUNY)

Title: The Waiting Time for Patterns

Date: 1/28/2006

Abstract: The following are examples of pattern problems.. 1) How long will it take for an 80% free-throw shooter to make 20 consecutive foul shots?. This is an example of the problem of “success runs.” 2) In rolls of a fair die how long will it take to get m consecutive identical outcomes, of any kind?. This is the minimum of 6 dependent success run times. We refer to it as multi-type success runs. 3) How long will it take for a random number generator to produce a consecutive string of 5 digits where the digits 1,2,3,4,5 appear once each in any order?. We refer to that as the waiting time for permutations. We can also consider cyclic permutations. 4) How long will it take for the oft-mentioned monkey randomly banging on a typewriter to type out Hamlet? The waiting time for patterns has interested many authors, going back at least to Von-Mises in 1912. The mean waiting time is a favorite of probability texts. For example many people are surprised to learn that in flips of a fair coin the expected waiting time for HHH equals 14, for HTH equals 10, and for HHT equals 8. We derive approximations for the waiting time distribution, along with bounds on the total variation distance between the approximate and true distribution. Our approach is based on generating functions and hazard rate inequalities. The results improve upon those obtained by Chen-Stein methodology, in several respects.

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Speaker: Michael Marcus (CUNY/City College)

Title: Strong Laws for Gaussian Differences

Date: 2/7/2006

Abstract: Using the Borell, Sudakov-Tsirelson Theorem we obtain a surprising, robust, strong law of large numbers for the $L^P$ norm of integrated Gaussian differences. This is a preliminary report on joint work with Professor Jay Rosen

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Speaker: Scott Sheffield (Courant)

Title: Tug of war and the infinity Laplacian

Date: 2/28/2006

Abstract: The infinity Laplacian (informally, the “second derivative in the gradient direction”) is a simple yet mysterious operator with many applications. “Tug of war” is a two player random turn game played as follows: SETUP: Assign each player one of two disjoint target sets $T_1$ and $T_2$ in the plane, and fix a starting position $x$ and a constant $\epsilon$. Place the game token at $x$. GAME PLAY: Toss a fair coin and allow the player who wins the coin toss to move the game token up to epsilon units in the direction of his or her choice. Repeat the above until the token reaches a target set $T_i$. The $i$th player is then declared the winner. Given parameters $\epsilon$ and $x$, write $u_\epsilon(x)$ for the probability that player one wins when both players play optimally. We show that as epsilon tends to zero, the functions $u_\epsilon(x)$ converge to the infinity harmonic function with boundary conditions 1 on $T_1$ and 0 on $T_2$. Our strategic analysis of tug of war enables us to solve the “optimal Lipschitz extension problem” and leads to new formulations and significant generalizations of several classical results about infinity laplacians. The game theoretic arguments are simpler and more elementary than the original proofs. This talk is based on joint work with Yuval Peres, Oded Schramm, and David Wilson.

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Speaker: Pavel Hitzcenko (Drexel University)

Title: Probabilistic analysis of a class of WHT algorithms.

Date: 3/4/2006

Abstract: The Walsh-Hadamard transform (WHT) is one of the frequently used transforms in signal processing. Since its computation typically involves large data, a significant effort was put into developing efficient algorithms for its computation and into construction of theoretical models that would explain empirical observations. In this talk, after briefly describing a family of algorithms based on a factorization of a WHT matrix, I will discuss probabilistic aspects of one of the important measures of their complexity, namely the instruction count. \ The talk is based on a joint work with J.~Johnson(CS, Drexel) and his (former) student H-J. Huang.

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Speaker: Makram Talih (Hunter College of the City University of New York)

Title: Strategies for Online Inference in Dynamic Graphical Models

Date: 4/2/2006

Abstract: We present different strategies for online learning of Dynamic Graphs (DG’s) via Sequential Monte Carlo (SMC). SMC algorithms are based on maintaining in parameter space an ensemble of particles, each of which tracks a particular realization of the process under study. In our framework, the process is governed by the posterior distribution of the parameters (eg. the precision matrix) and hidden variables (eg. the underlying undirected graph) given the data sequence.

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Speaker: Soumik Pal (Columbia University)

Title: Capital Requirement to Achieve Acceptability

Date: 4/9/2006

Abstract: Consider an agent who wishes to trade in a financial market between day $t=0$ and day $t=T$. The agent starts with a specific objective in mind, which defines a class of acceptable or riskless (random) financial positions at the end of day T. It has been shown in the theory of convex measures of risk that any definition of acceptability will stipulate a random variable to be acceptable if its expectation under a variety of probability measures (called scenarios) dominate a corresponding real number (called floors). By suitable choices of scenarios and floors, this simple structure becomes flexible to encompass several major problems in mathematical finance, including superhedging, efficient hedging, and robust trading. In this talk we find, under general modeling assumptions, the minimum initial capital, the agent would require to drive his portfolio towards acceptability. The solution leads to a beautiful mathematical picture, which ties its analytical side (Frechet derivatives in Hilbert spaces), geometric side (nearest point projections in uniformly convex Banach spaces), and how such analytic and geometric properties become amenable under probabilistic structures of the Brownian filtration. In specific cases, such minimum initial capital becomes sellers’ price for efficient hedging on the one hand, and no-good-deals valuation on the other hand. And we shall see examples in this direction.

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Speaker: Jay Rosen (City University of New York)

Title: Frequent Points and Harnack Inequalities for Random Walks in Two Dimensions

Date: 8/12/2006

Abstract: For a random walk in $\Z^2$ which does not necessarily have bounded range we study those points which are visited an unusually large number of times. We prove the analogue of the Erd\H{o}s-Taylor conjecture and obtain the asymptotics for the number of visits to the most visited site. We also obtain the asymptotics for the number of points which are visited very frequently by time $n$. The key to these results are good Harnack Inequalities for the interior and exterior of a disc. Joint work with Rich Bass.

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Speaker: Michael Marcus (CUNY)

Title: Central limit theorems for moduli of continuity of Gaussian processes

Date: 9/17/2006

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Speaker: Van Vu (Rutgers)

Title: Central limit theorems for random polytopes.

Date: 9/24/2006

Abstract: Let K be a convex body in $R^d$, (where $d$ is fixed, say 4) with unit volume. Sample $n$ points in $K$, randomly and independently with respect to the uniform distribution. The convex hull of these points (called $K_n$) is a classicial model for random polytopes. The study of random polytopes was started systematically by Renyi and Sulanke in the 1960s. One of the major questions in this field is to prove central limit theorems (as n tends to infinity) for the key parameters (such as volume or number of vertices) of $K_n$. I will survey recent developments that lead to the solution of this problem.

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Speaker: Richard Gundy (Rutgers Univ)

Title: Ergodic theory of low-pass filters

Date: 11/12/2006

Abstract: I will talk about a problem that originated in the theory of wavelets: the characterization of functions that generate multi- resolution analyses. The initial results, with Dobric and Hitczenko, were significant but the proofs were complicated. With the benefit of hindsight, the ideas can be made very simple. It turns out that we are led to the study of a basic “historical Markov process” of the type encountered in statistical mechanics, and the set of invariant measures associated with this process. The wavelet setting presents some new features of these processes. If time permits, I will show how to obtain the basic class of lowpass polynomial filters, first discovered by Ingrid Daubechies in 1986 from elementary probability considerations, going back to 1634,( the correspondence between Pascal and Fermat.)

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Speaker: Dan Romik (Bell Labs)

Title: Random square Young tableaux

Date: 1/6/2007

Abstract: An $NxN$ square Young tableaux is an NxN matrix containing the numbers 1 through $N^2$ such that each row and column are increasing. It can be thought of as a sequence of instructions for building an $N$-by-$N$-shaped wall made of $N^2$ unit square bricks by laying one brick at a time so that at each point during the construction the wall is stable and will not collapse. In this talk, I will describe the problem of choosing a uniformly random square Young tableau when $N$ is large, and how the typical shape profile of the wall can be found using a large deviations analysis. I will also describe some applications of this result, and connections to other important problems such as finding the length of the longest increasing subsequence in a random permutation.

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Speaker: Robin Pemantle (University of Pennsylvania)

Title: Quantum random walks in one and two dimensions

Date: 1/13/2007

Abstract: I start by defining and motivating a quantum version of the simple random walk. This object has been understood in one dimension for some time, but not in higher dimensions. I will give a complete analysis in one dimension. In two dimensions, I will describe results that are currently being written down.

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Speaker: Bo’az Klartag (Princeton University)

Title: A central limit theorem for convex sets.

Date: 2/6/2007

Abstract: Suppose $X$ is a random vector, that is distributed uniformly in some $n$-dimensional convex set. It was conjectured that when the dimension n is very large, there exists a non-zero vector $u$, such that the distribution of the real random variable $$ is close to the gaussian distribution. A well-understood situation, is when $X$ is distributed uniformly over the $n$-dimensional cube. In this case, $$ is approximately gaussian for, say, the vector $u = (1,…,1) / sqrt(n)$, as follows from the classical central limit theorem. We prove the conjecture for a general convex set. Moreover, when the expectation of $X$ is zero, and the covariance of $X$ is the identity matrix, we show that for ‘most’ unit vectors $u$, the random variable $$ is distributed approximately according to the gaussian law. We argue that convexity – and perhaps geometry in general – may replace the role of independence in certain aspects of the phenomenon represented by the central limit theorem.

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Speaker: Vladimir Dobric (Lehigh University)

Title: Fractional Brownian motion,its martingales and natural wavelets

Date: 2/20/2007

Abstract: We have constructed the whole family of fractional Brownian motions as a single Gaussian field indexed by time and the Hurst index simultaneously. That field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. In this Gaussian field the pairs (H,H’) of Hurst indices with the property H+H’=1, which we call the dual pairs, are essential tools for constructing ”natural” martingales associated with fractional Brownian motions. The existence of those martingales, via their stochastic representations, leads to ”the natural wavelet expansions” of those processes in the spirit of our earlier work on construction of natural wavelets associated to Gaussian-Markov processes.

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Speaker: Joseph Yukich (Lehigh University)

Title: Limit theorems for convex hulls.

Date: 3/17/2007

Abstract: We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap (joint with T. Schreiber).

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Speaker: Wenbo V. Li (University of Delaware)

Title: Spectral Analysis of Brownian Motion with Jump Boundary

Date: 3/24/2007

Abstract: Consider a family of probability measures $\{\mu_y : y \in \partial D\}$ on a bounded open domain $D\subset R^d$ with smooth boundary. For any starting point $x \in D$, we run a a standard $d$-dimensional Brownian motion $B(t)$ until it first exits $D$ at time $\tau$, at which time it jumps to a point in the domain $D$ according to the measure $\mu_{B(\tau)}$ and starts the Brownian motion afresh. The same evolution is repeated independently each time the process reaches the boundary. The resulting diffusion process is called Brownian motion with jump boundary (BMJ). The spectral gap of non-self-adjoin generator of BMJ, which describes the exponential rate of convergence to the invariant measure, is studied. The main analytic tool is Fourier transforms with only real zeros.

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Speaker: Antonia F”{o}ldes (College of Staten Island, CUNY)

Title: TRANSIENT NEAREST NEIGHBOR RANDOM WALK ON THE LINE

Date: 4/1/2007

Abstract: \noindent Let $X_0=0,\, X_1,X_2,…$ be a Markov chain with \begin{eqnarray*} P\{X_{n+1}=i+1\mid X_n=i\}&=&1- P\{X_{n+1}=i-1\mid X_n=i\}=\\ &=&\left\{\begin{array}{ll} 1/2\quad & {\rm if\quad } i=0,\\ 1/2+p_i\quad &{\rm if\quad} i=1,2,…,\\ 1/2-p_i\quad &{\rm if\quad} i=-1,-2,… \end{array}\right. \end{eqnarray*} This sequence $\{X_i\}$ describes the motion of a particle which is going away from 0 with a larger probability than to the direction of 0. That is to say 0 has a repelling power which becomes small if the particle is far away from 0. We intend to characterize the motion $\{X_i\}$ and its local time.

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Speaker: Elena Kosygina (CUNY, Baruch)

Title: On the positive speed for one-dimensional “cookie” random walks

Date: 4/8/2007

Abstract: We consider exited random walks in one dimension: put $M$ “cookies” at each site of the one dimensional integer lattice and let a random walker start from the origin. Whenever there is a “cookie” at the walker’s present site he will “eat” one cookie and make one step to the right or one step to the left with probabilities $p>1/2$ or $1-p1$. Very recently A.-L. Basdevant and A. Singh have shown that the walk has positive speed if and only if $M (2p-1)>2$. We shall discuss the method and give a sketch of the proof of this result.

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Speaker: Kavita Ramanan (Carnegie Mellon University)

Title: ON SOLUTIONS TO A CLASS OF STOCHASTIC DIFFERENTIAL INCLUSIONS

Date: 4/15/2007

Abstract: We establish sufficient conditions for existence and pathwise uniqueness of strong solutions to a class of possibly degenerate stochastic differential equations with discontinuous and possibly singular drift (interpreted as stochastic differential inclusions). We also allow for possibility of reflection in a polyhedral domain with piecewise constant reflection field. As motivation for this work, we show how our results may be applied to obtain limit theorems for several classes of stochastic networks. This includes joint work with Rami Atar and Amarjit Budhiraja.

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Speaker: Jay Rosen (CUNY)

Title: Large Deviations for Riesz Potentials of Additive Processes

Date: 9/9/2007

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Speaker: Michael B. Marcus (CUNY)

Title: Infinitely divisible squares of non associated Gaussian vectors

Date: 9/16/2007

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Speaker: Jan Rosinski (University of Tennessee and Cornell University)

Title: Simulation of Levy Processes: an Overview and Current Issues

Date: 9/23/2007

Abstract: Levy processes arise in many areas of applied probability and stochastic finance. They are continuous time random walks comprised of independent diffusion and jump parts (Brownian motion and a Levy process of the Poissonian-type). Simulation of a Brownian motion and/or of a compound Poisson process can be found in many textbooks and will not be discussed here. We will concentrate on a situation when a Levy process has infinitely many jumps in each finite interval. Exact simulation of such processes is obviously impossible and we must use approximate methods. A choice of the method depends on the Levy measure of a process and other characteristics. We will discuss simulation methods based on (a) random walk approximation; (b) series representations of Levy processes; (c) Poisson and Gaussian approximation; together with their ramifications, and pros and cons. Contrary to the one-dimensional case, closed formulae for simulation of increments of multidimensional Levy processes are rarely available. This essentially rules out an approximation by a random walk from a discrete skeleton. If time permits, we will discuss current issues of simulation in the multidimensional case.

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Speaker: Christian Benes (Brooklyn College, CUNY)

Title: Some Properties of the Complement of Planar Random Walk

Date: 9/30/2007

Abstract: Consider simple random walk $S$ in the plane and the continuous curve obtained from it by linear interpolation between integer times. If one defines the set of ‘holes’ to be the set of connected components of $C \ S[0,2n]$ and the set of ‘lattice holes’ of $S$ to be the set of connected components of $Z^2 \ {S_j}_{0<= j<= 2n}$, one can assign to each hole an “area”, its Lebesgue measure, and to each lattice hole a “lattice area”, its cardinality. In this talk, we will show that the number of holes (resp. lattice holes) of area (resp. lattice area) greater than $A(n)*n$ is, up to a logarithmic correction term, asymptotic to $A(n)^{-1}$, if $n^{-d_0} < A(n) 0$. This confirms an observation by Mandelbrot. A consequence is that the largest hole has an area which is logarithmically asymptotic to n. We will also mention the different and mysterious exponent of $5/3$ observed by Mandelbrot for ‘small’ lattice holes

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Speaker: Kevin O’Bryant, CUNY (CUNY, College of Staten Island)

Title: The Central Limit Theorem without the limit

Date: 10/6/2007

Abstract: A $B*[g]$ set is a set $S$ of integers with the property that the set does not have more than $g$ solutions to $x=a_1 + a_2 = b_1 + b_2$ for any $x$. The foundational problem here is to bound the size of a $B*[g]$ set contained in $\{1,2,…,n\}$. Recent progress on this problem has been made through proving statements of the form: If $X_1, X_2$ are $i.i.d.$ with diameter less than $L$, then the pdf of $X_1+X_2$ has infinity norm at least $Q/L$. The “trivial” value of $Q$ is $1/2$. I will present recent progress on this problem and its generalization from 2 summands to h summands.

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Speaker: Alan Michael Hammond (Courant Institute, NYU)

Title: The scaling limit of a biased random walk on a supercritical Galton Watson tree

Date: 10/27/2007

Abstract: We study the long-term behaviour of a random walk on a super- critical Galton Watson tree, where the walk is biased away from the root. The walk is liable to be trapped in finite trees that are attached to the backbone of the tree, and this trapping makes the walk sub- ballistic, if the bias is strong enough. I will explain how, if the bias is taken to be a constant at each vertex, a discrete inhomogeneity is present on all time scales, so that a scaling limit (at least in a conventional sense) does not exist for the process. If we randomize the bias at each vertex so that it is independently sampled according to a smooth law, on the other hand, we show convergence of the scaled distance from the root to a stable subordinator. Joint work with G. Ben Arous.

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Speaker: Ross Pinsky (Technion, Israel Institute of Technology)

Title: Increasing subsequences in random permuations

Date: 10/27/2007

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Speaker: Olympia Hadjiliadis (Brooklyn College, CUNY)

Title: Optimal quickest detection of two-sided alternatives and connections to drawdown and rally processes.

Date: 11/4/2007

Abstract: In this talk we present the problem of quickest detection of two-sided alternatives in the Brownian motion model. In particular, we consider this problem in the min-max setting where the change-point is assumed to be an unknown constant. We formulate a stochastic optimization problem that arises as the trade-off between minimizing quick detection and keeping the mean time between false alarma above a certain threshold. We present properties of the optimal stopping time. We then proceed to find the best 2-CUSUM stopping time by means of evaluating an expression for its first moment. We describe a connection between the 2-CUSUM process and the drawdown and rally processes. We present the probability that a drawdown of a given level precedes a rally of an equal or unequal level in a special class time homogeneous diffusion processes. We provide closed form expressions of this probability in the case of an Ornstein Uhlenbeck process, a special case of which includes the Vasicek model for short-term interest rates.

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Speaker: Davar Khoshnevisan (University of Utah)

Title: The packing dimension of the range of a Levy process

Date: 11/6/2007

Abstract: Let $\{X(t)\}_{t\ge 0}$ denote a L\’evy process in $\R^d$ with exponent $\Psi$. Taylor (1986) proved that the packing dimension of the range $X([0\,,1])$ is described by the index \begin{equation} \gamma’ = \sup\left\{\alpha\ge 0: \liminf_{r \to 0^+}\, \, \int_0^1 \frac{\P \left\{|X(t)| \le r\right\}}{r^\alpha} \, dt =0\right\}. \end{equation} We provide an alternative formulation of $\gamma’$ in terms of the L\’evy exponent $\Psi$. Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Cauchy transform. We show, through examples, some applications of our formula. \\ Time permitting, we introduce also the resolution to a question of J. D. Howroyd (1997) in geometric measure theory. \\ This is based on joint work with Yimin Xiao.

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Speaker: Robert Adler (Technion)

Title: INTEGRAL GEOMETRY IN GAUSS SPACE

Date: 1/19/2008

Abstract: The three basic results of classical, Euclidean, Integral Geometry are the the Kinematic Fundamental Formula, Crofton’s Formula, and Steiner’s (Weyl’s) Formula. After describing these results and their importance, I will describe new versions of them in Gauss space and in Gaussian function space, as well as touching briefly on some of the applications of the new results. This is joint work with Jonathan Taylor.

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Speaker: Peter Carr (NYU)

Title: Options on Maxima, Drawdown, Trading Gains, and Local Time

Date: 1/26/2008

Abstract: We show how to replicate the payoff from a hypothetical option written on the drawdown and/or maximum of an asset price. In general, the hedge uses static positions in both standard and barrier options. Since barrier options are not yet liquid in many markets, we also impose some structure on the underlying price dynamics under which hedging involves occasional trading in just standard options. This structure further permits options on local time and options on the gains \ from binary trading strategies to be semi-statically hedged using standard options.

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Speaker: Gerardo Hernandez-del-Valle ( Columbia)

Title: On Schrodinger’s equation, 3-dimensional Bessel bridges and passage time problems

Date: 2/4/2008

Abstract: $See \url{http://www.math.csi.cuny.edu/probability/Notebook/abstract-Hernandez-del-Valle.pdf}$

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Speaker: Rama Cont (Columbia University)

Title: Levy copulas

Date: 2/11/2008

Abstract: We will discuss the notion of “Levy copula” which, just as the notion of ‘copula’ characterize the dependence structure of a random vector, characterizes the dependence among components of multidimensional Lévy processes (Cont & Tankov 2003, Kallsen & Tankov 2006). We discuss the dynamic analogue of Sklar’s theorem for Levy porcesses and the relation between the Levy copula and the copula of the law of the Levy process. We give parametric examples of Levy copulas, and illustrate how they can be used for constructing multidimensional infinitely divisible distribution with given marginals.

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Speaker: Xia Chen (University of Tennessee)

Title: LARGE DEVIATIONS FOR LOCAL AND INTERSECTION LOCAL TIMES OF FRACTIONAL BROWNIAN MOTIONS

Date: 2/18/2008

Abstract: It is well known that the local and intersection local times of Gaussian process can be constructed by a method known as local non-determinism. In this talk, I will show how this method can be used to establish the large deviations for the local and intersection local times of fractional Brownian motions. In addition, I will post some related conjectures and remaining problems. Part of of the talk comes is based a collaborative project with Qiman Shao.

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Speaker: Omri Sarig (Pennsylvania State University)

Title: Generalized laws of large numbers for horocycle flows

Date: 2/25/2008

Abstract: (joint with F. Ledrappier) Suppose T:X–>X is a map, and m is a T-invariant ergodic measure. We say that m has a “generalized law of large numbers” is there is a procedure which (1) accepts as input the times n such that T^n(x) is in a set E (but not x or E) (2) gives as output the measure of E and so that the procedure works for all E measurable, for almost all x in X. If m(X)=1, then the strong law of large numbers gives us such a procedure (take the limit of (1/n)[1_E(x)+1_E(Tx)+…+1_E(T^n x)]. But if m(X) is infinite, this fails. I will discuss alternative “generalized laws of large numbers” for a natural class of dynamical systems arising from hyperbolic geometry. No background in hyperbolic geometry is required.

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Speaker: Maria Cristina Mariani (New Mexico State University)

Title: Extreme events in financial markets

Date: 3/8/2008

Abstract: This presentation is devoted to the development and analysis of mathematical models to enhance understanding of extreme events in financial markets. This will be undertaken through two specific problems in the mathematics of risk management: * The analysis of asset-price dynamics in models that capture the possibility of sudden, large changes in prices — i.e., “jumps”; * The development and application of tools from mathematical physics to analyze market dynamics leading to a “crash”, and the corresponding matching with tools from the Mathematical Finance. Solutions to the equations arising in the corresponding mathematical models will be analized.

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Speaker: William Cuckler (University of Delaware)

Title: Entropy bounds for perfect matchings and Hamiltonian cycles

Date: 3/15/2008

Abstract: For a graph $G=(V,E)$ and ${\bf x}:E\rightarrow \Re^+$ satisfying $\sum_{e\ni v}{\bf x}_e =1$ for each $v\in V$, set $h({\bf x})= \sum_e {\bf x}_e\log (1/{\bf x}_e)$ (with $\log =\log_2$). We show that for any $n$-vertex $G$, random (not necessarily uniform) perfect matching ${\bf f}$ satisfying a mild technical condition, and ${\bf x}_e=\Pr(e\in{\bf f})$, $$H({\bf f}) 1)>0$ was analysed by Kesten who showed that $R$ is always heavy–tailed. The complementary case $0\le| M|\le 1$ is much less understood. Goldie and Gr”ubel showed that in that case, the tails are never heavier than exponential and that if $|M|$ behaves near 1 as a uniform random variable then the tails of $R$ are Poissonian. In this talk we will present further results about the tails of $R$ and their connection to the behavior of $|M|$ near 1. This is a based on a joint work with Jacek Weso{\l}owski, Technical University of Warsaw.

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Speaker: Julien Dubedat

Title: Dimers and analytic torsion

Date: 11/9/2008

Abstract: We discuss Gaussian invariance principles for dimer models in relation with variational formulae for zeta-determinants of Cauchy-Riemann operators. an introductory text: http://arxiv.org/pdf/math/0310326v1

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Speaker: Emmanuel Shertzer (Columbia University)

Title: The Voter Model and the Potts Model in one dimension.

Date: 1/10/2009

Abstract: The voter model can be seen as a simple model for describing the propagation of opinions in a population where neighbors influence each other. More precisely, every integer is assigned with an original opinion at time t=0 and then updates its opinion by taking on the opinion of one of its neighbors chosen uniformly at random with rate 1. In the first part of the talk, I will show that such a model can easily be described in terms of a system of coalescing random walks. In the second part of the talk, I will introduce a variation of the preceding model where the voters do not only change their mind under the influence of their environment, but where they are also able to come up with an opinion differing from their neighbors. This model is closely related to a classical model in statistical physics called the one dimensional stochastic Potts model. I will show that under the appropriate scaling, this model converges to a continuum object which can be constructed by a marking procedure of a family of coalescing Brownian motions. \ This is joint work with C. Newman and K. Ravishankar.

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Speaker: Anja Sturm ( University of Delaware)

Title: Coexistence and survival in some cancellative spin systems

Date: 1/17/2009

Abstract: We consider variations of the usual voter model, which favor the type that is locally less common. These voter models are dual to systems of branching annihilating random walks that preserve the parity of the number of particles where we interpret sites occupied by a 1 as a particle. Both sets of models into the category of cancellative spin systems. We consider coexistence of types in the voter models which is related to the survival of particles in the branching annihilating random walk. We find conditions for the uniqueness of a homogeneous coexisting invariant law as well as for convergence to this law from homogeneous and coexisting initial laws. For a particular one dimensional model we also show a complete convergence result for any initial condition. This is based on comparison with oriented percolation of the associated branching annihilating random walk. \ This is joint work with Jan Swart (UTIA Prague).

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Speaker: Dmitry Dolgopyat (University of Maryland)

Title: Central Limit Theorem for Random walk in Markovian environment.

Date: 1/24/2009

Abstract: We prove the central limit theorem for a random walk at Z^d where the transition probabilities at different sites are governed by independent finite state mixing Markov chains. This is a joint work with Carlangelo Liverani.

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Speaker: Janos Englander (UC Santa Barbara)

Title: Strong Law of Large Numbers for Branching Diffusions

Date: 2/3/2009

Abstract: Let X be the branching particle diffusion corresponding to a second order semilinear elliptic operator on a Euclidean domain. Under appropriate spectral theoretical assumptions on the operator, we prove that the exponentially discounted random measures X(t) converge almost surely in the vague topology as t tends to infinity. The exponential rate is the generalized principal eigenvalue of the linear part of the operator, which is assumed to be finite and positive. \ This result was motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions by Englander, Turaev and Winter. We extend significantly the results obtained by Asmussen and Hering in the seventies, and by Chen and Shiozawa very recently. We also include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and `spine’ decompositions or `immortal particle pictures’. \ This is joint work with Andreas Kyprianou and Simon Harris (Bath, UK).

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Speaker: Olympia Hadjiliadis (Brooklyn College, CUNY)

Title: Formulas for Stopped Diffusion Processes with Stopping Times based on Drawdowns and Drawups

Date: 2/17/2009

Abstract: This paper studies drawdown and drawup processes in a general diffusion model. The main result is a formula for the joint distribution of the running minimum and the running maximum of the process stopped at the time of the first drop of size $a$. As a consequence, we obtain the probabilities that a drawdown of size $a$ precedes a drawup of size $b$ and vice versa. The results are applied to several examples of diffusion processes, such as drifted Brownian motion, Ornstein-Uhlenbeck process, and Cox-Ingersoll-Ross process. We also discuss applications of the results to the problem of mis-identification of a two-sided change in the drift of a diffusion process. \\ This is joint work with Libor Pospisil and Jan Vecer.

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Speaker: Jose Blanchet (Columbia University)

Title: Algorithms and Large Deviations

Date: 2/28/2009

Abstract: This talk concentrates on the interplay between large deviations theory and the design of efficient stochastic simulation algorithms that are aimed at both estimating rare-event probabilities and sampling stochastic processes conditional on a rare event. The point is designing simulation estimators that can be easily implemented and whose coefficient of variation remains uniformly bounded as the event becomes rarer and rarer. Typically, a large deviations result is helpful to guide the construction of the estimator, but, as we shall see, the complexity analysis of the algorithm often demands a refinement of the underlying large deviations argument behind the rare-event probability of interest. In this talk we illustrate the techniques both in light and heavy-tailed stochastic processes. Applications to stochastic networks will be given as motivation.

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Speaker: Victor de la Pena (Columbia University)

Title: New Exponential Inequalities for Self-Normalized Martingales

Date: 3/21/2009

Abstract: In this talk I will introduce a new class of exponential inequalities for self-normalized martingales. I will show their usefulness by an application to hypothesis testing of the variance.

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Speaker: Souvik Ghosh (Columbia University)

Title: LARGE DEVIATION PRINCIPLE FOR A CLASS OF LONG RANGE DEPENDENT INFINITELY DIVISIBLE PROCESS

Date: 3/28/2009

Abstract: We make an attempt at understanding the effect of long range dependence on the large deviation principle for the partial sums of an infinitely divisible process. It has been observed in certain short memory processes that the large deviation principle is very similar to that of an i.i.d sequence. Whereas, if the process is long range dependent the large deviations change dramatically. We want to see if such a phenomenon holds for infinitely divisible processes. We consider a stationary, mean zero infinitely divisible process $ (X_{ n},n\in \mathbb{Z})$ without a Gaussian component but with exponentially light tails. The process is characterized by its L\’evy measure $ \Pi$ on $ \mathbb{R}^{ \mathbb{Z}}$ which is shift invariant. With the aim of modeling long range dependence for such processes, we consider the situation where the L\’evy measure is the law of the paths of an irreducible null recurrent Markov Chain with the marginals being the invariant measure $ \pi$ of the chain, i.e., for any $ n\ge 1$ and $ A_{ 0},\ldots,A_{ n}\in\mathcal{B}(\mathbb{R})$, \[ \Pi\big( z\in \mathbb{R}^{ \mathbb{Z}}:(z_{ 0},\ldots,z_{ n})\in A_{ 0}\times\cdots\times A_{ n} \big) = \int_{ A_{ 0}}\cdots \int_{ A_{ n}} \pi(dz_{ 0})P(z_{ 0},dz_{ 1})\cdots P(z_{ n-1},dz_{ n-1}), \] where $ P(\cdot,\cdot)$ is the transition kernel of the Markov chain. We study how the structure of this Markov chain affects the large deviation principle for the partial sums of the process $ (X_{ n})$. References: Alparslan, U.T., Samorodnitsky, G. (2007) {\sl Ruin probability with certain stationary stable claims generated by conservative flows\/}, Advances in Applied Probability. 39, 360–384. Ghosh, S. (2008){\sl The effect of memory on large deviations of moving average processes and infinitely divisible processes \/}, Thesis, Cornell University. Mikosch, T., Samorodnitsky, G. (2000) {\sl The supremum of a negative drift random walk with dependent heavy-tailed steps\/}, The Annals of Probability. 10, 1025–1064. Rosinski, J., Samorodnitsky, G. (1996) {\sl Classes of mixing stable processes \/}, Bernoulli. 2, 365–377.

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Speaker: Michael Kozdron (University of Regina)

Title: TBA

Date: 4/5/2009

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Speaker: David Nualart (University of Kansas)

Title: Central limit theorems for functionals of Gaussian processes.

Date: 4/12/2009

Abstract: In this talk we first establish a central limit theorem for a normalized sequence of random variables which are functionals of an underlying Gaussian stochastic process, and they belong to a fixed Wiener chaos. The convergence in law to the normal distribution is equivalent to the convergence of the moments of order 4, and also to the convergence of the square norm of the derivatives in the sense of Malliavin calculus. In a second part of the talk we introduce the fractional Brownian motion with Hurst parameter H, and discuss some of its basic properties. We also present several asymptotic properties of functionals of the fractional Brownian related to power variations and to the Levy area.

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Speaker: Jean Bertoin ( Universite Paris VI)

Title: A limit theorem for the tree of alleles in branching processes with rare neutral mutations

Date: 8/15/2009

Abstract: We are interested in the genealogical structure of alleles for a Bienayme-Galton-Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process in discrete time. Ito’s excursion theory and the L'{e}vy-It^{o} decomposition of subordinators provide fundamental insights for the results.

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Speaker: Fredrik Johansson ( KTH)

Title: Optimal Holder exponent for the SLE path

Date: 8/22/2009

Abstract: The Schramm-Loewner evolution (SLE) is a family of random fractal curves obtained by solving the Loewner equation with a Brownian motion input. SLE has attracted much attention in recent years since, for example, it can be used to rigorously understand scaling limits of several discrete models from statistical physics. In the talk we give a very brief introduction to SLE and discuss our proof of J. Lind’s conjecture about the optimal Holder exponent for the SLE path parametrized by half-plane capacity. \ This is joint work with G. F. Lawler (University of Chicago).

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Speaker: David Mason (University of Delaware)

Title: Large Self-normalized Levy process distributional behavior at small time and large time

Date: 9/13/2009

Abstract: \noindent Let $X_{t}$ be a L\'{e}vy process and \[ V_{t}\] be its quadratic variation process, where $\Delta X_{t}=X_{t}-X_{t-}$ denotes the jump process of $X$. We give stability and compactness results at small time, i.e, as $t\downarrow0$, and large time, i.e, as $t\rightarrow\infty$ for the \textquotedblleft self-normalized\textquotedblright\ process $X_{t}% /\sqrt{V_{t}}$. Special cases of our results characterize the possible limit laws of $X_{t}/\sqrt{V_{t}}$ at both small and large times. One such result says that \[ X_{t}/\sqrt{V_{t}}\overset{\mathrm{D}}{\longrightarrow}N(0,1),\text{ as }t\downarrow0, \] a standard normal random variable if and only if, for some nonstochastic function $b(t)>0$, \[ X_{t}/b(t)\overset{\mathrm{D}}{\longrightarrow}N(0,1),\text{ as }t\downarrow0, \] with the same statement holding as $t\rightarrow\infty$. Our asymptotic normality results are the small time and large time self-normalized L\'{e}vy process analogs of what is known for self-normalized sums of i.i.d. random variables. It turns out that roughly speaking for small time behavior everything is controlled by the tails of the L\'{e}vy measure of the process near zero, and for large time behavior, it is determined by the tails of the L\'{e}vy measure at infinity. Finally we show how stochastically compact L\'{e}vy processes, both at zero and infinity, arise via subsequential distributional limits of partial sums of i.i.d. random variables in the Feller Class. This talk is based on ongoing joint work with Ross Maller of the Australian National University.

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Speaker: Ron Peled (Courant Institute, NYU)

Title: Gravitational Allocation to Poisson Points

Date: 9/20/2009

Abstract: One way to quantify how uniformly spread a point process is, is to allocate cells of equal volume to each of its points and measure the regularity of the resulting partition of space. Such allocations (also known as transportations, matchings or marriages) with an additional equivariance constraint, have been the subject of many investigations in recent years. I will survey results in the field, with special focus on a natural allocation rule – the Gravitational Allocation, and its quantitative geometry when used for the Poisson point process.

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Speaker: Elena Kosygina ( Baruch College, CUNY)

Title: Limit laws of excited random walks on integers

Date: 9/27/2009

Abstract: We consider excited random walks on integers with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the “cookies”. E. Kosygina and M.P.W. Zerner have shown that when the total expected drift per site, $\delta$, is larger than 4 then the walks under the averaged measure obey the Central Limit Theorem. We show that when $\delta\in(2,4]$ the limiting behavior of an appropriately centered and scaled excited random walk is described by a strictly stable law with parameter $\delta/2$. Our method also extends the results obtained by A.-L. Basdevant and A. Singh for $\delta\in(1,2]$ under the non-negativity assumption to the setting, which allows both positive and negative cookies. \\ (joint with T. Mountford)

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Speaker: Dick Gundy (Rutgers University)

Title: Tilings of R1, scaling functions, and a Markov process

Date: 10/10/2009

Abstract: TBA

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Speaker: Hana Kogan (CUNY)

Title: On infinite divisibility of Gaussian squares with non-zero mean

Date: 11/1/2009

Abstract: I will present Feller’s coefficient positivity condition for infinite divisibility and outline Griffith and Bapat’s theorem on infinite divisibility of zero-mean Gaussian squares. \ I will present the series expansion for the Laplace Transform in the non-zero mean case (content of my thesis) and give applications (in particular to the upper bound on their critical point). I will present the theorem on non-zero mean Gaussian squares divisibility, describe the significance of Ray-Knight Isomorphism theorem in the original proof and give an (elementary) alternative proof. \ I will mention the unsolved questions connected to this topic.

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Speaker: Kay Kirkpatrick (Courant, NYU)

Title: Bose-Einstein condensation: from many quantum particles to one quantum “super-particle.”

Date: 11/8/2009

Abstract: I will discuss work with Benjamin Schlein and Gigliola Staffilani on the low-temperature phenomenon of Bose-Einstein condensation on the plane and the torus. Starting from the microscopic quantum dynamics, we prove that the cubic nonlinear Schrodinger equation (NLS) provides the macroscopic description of Bose-Einstein condensation. Time permitting I’ll also mention work in progress: with Gerard Ben Arous on quantum probability and large deviations for quantum many-body systems, and with Sourav Chatterjee on a phase transition for the invariant measures of the NLS.

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Speaker: Ivan Corwin ( Courant, NYU)

Title: Fluctuations in traffic flow, crystal growth and random matrices

Date: 1/9/2010

Abstract: Analogous to the central limit theorem, we study the asymptotic behavior of the fluctuations of traffic flow, crystal growth and eigenvalues of random matrices around their expectations. These three categories of random processes are closely related. Drawing on this relationship we give a classification theory for the fluctuations in these models.

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Speaker: Antonia F”{o}ldes (CUNY)

Title: Strong limit theorems for simple random walk on the 2-dimensional comb

Date: 1/16/2010

Abstract: We study the path behaviour of a simple random walk on the 2-dimensional comb lattice ${\mathbb C}^2$ that is obtained from ${\mathbb Z}^2$ by removing all horizontal edges off the $x$-axis. In particular, we prove a strong approximation result for such a random walk which, in turn, enables us to establish strong limit theorems, like the joint Strassen type law of the iterated logarithm of its two components, as well as their marginal Hirsch type behaviour. joint work with Endre Cs\'{a}ki, Mikl\’os Cs”org\H{o} and P\’al R\’ev\’esz

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Speaker: Sourav Chaterjee (Courant, NYU)

Title: Superconentration

Date: 1/23/2010

Abstract: We introduce the term `superconcentration’ to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens, the field becomes chaotic under small perturbations and a `multiple valley picture’ emerges. Conversely, chaos implies superconcentration. While a few notable examples of superconcentrated functions already exist, e.g. the largest eigenvalue of a GUE matrix, we show that the phenomenon is widespread in physical models; for example, superconcentration is present in the Sherrington-Kirkpatrick model of spin glasses, directed polymers in random environment, the Gaussian free field and the Kauffman-Levin model of evolutionary biology. As a consequence we resolve the long-standing physics conjectures of disorder-chaos and multiple valleys in the Sherrington-Kirkpatrick model, which is one of the focal points of this talk.

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Speaker: Balint Toth

Title: Brownian Random Polymers

Date: 2/2/2010

Abstract: I will give a survey of recent results about the asymptotics of the Brownian Random Polymer process introduced by Durrett and Rogers in 1992. The process is pushed by some signed average of its own occupation time measure. In the most interesting self-repelling cases it is driven by a smeared out negative gradient of its local time. The process in the continuous space-time analogue of the so-called “myopic (or true) self-avoiding random walk”. I will present two classes of results: (1) In 1d I will show various diffusive and superdiffusive bounds, depending on the infrared asymptotics of the driving function. (2) In three and more dimensions we prove a diffusive limit (CLT) for the self-repelling case. The talk will be based on joint work with P. Tarres and B. Valko, respectively, with I. Horvath and B. Veto.

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Speaker: Hongzhong Zhang (CUNY)

Title: Drawdowns, Drawups, and Their Applications

Date: 2/16/2010

Abstract: Drawdown processes and their counterpart, drawup processes, arise frequently in the problem of quickest detection of abrupt changes and in the area of finance. The drawdown of a given process is defined as the drop of the present value from the running maximum, and the drawup of this process is defined as the increase of the present value over the running minimum. The drawup of a log-likelihood ratio process is usually referred to as the cumulative sum (CUSUM) process in statistics, which has been used as a means of detecting an abrupt change in the dynamics of processes. In finance, drawdowns provide a dynamic measure of risk in that they measure the drop of a stock price from its running maximum. Similarly, drawups can be perceived as measures of return. In this talk, we will discuss some optimality results of CUSUM stopping rule, and a new risk measure related to drawdowns and drawups in finance.

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Speaker: Patrick Cheridito (Princeton)

Title: Processes of class Sigma, last passage times and drawdowns

Date: 3/13/2010

Abstract: $We propose a general framework to study last passage times, suprema and drawdowns of a large class of stochastic processes. A central role in our approach is played by processes of class Sigma. After investigating convergence properties and a family of transformations that leave processes of class Sigma invariant, we provide three general representation results. The first one allows one to recover a process of class Sigma from its final value and the last time it visited the origin. In many situations this gives access to the distribution of the last time a stochastic process hit a certain level or was equal to its running maximum. It also leads to a formula recently discovered by Madan, Roynette and Yor expressing put option prices in terms of last passage times. Our second representation result is a stochastic integral representation of certain functionals of processes of class Sigma, and the third one gives a formula for their conditional expectations. From the latter one can deduce the laws of a variety of interesting random variables such as running maxima, drawdowns and maximum drawdowns of suitably stopped processes. As an application we discuss the pricing and hedging of options that depend on the running maximum of an underlying price process and are triggered when the underlying drops to a given level or alternatively, when the drawdown or relative drawdown of the underlying attains a given height. \\ The paper is on Math Arxiv \url{http://arxiv.org/abs/0910.5493}$

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Speaker: Richard Bass (University of Connecticut)

Title: Stable-like processes

Date: 3/20/2010

Abstract: The class of stable-like processes is a subset of the class of multidimensional jump processes. They stand in the same relationship to stable processes as multidimensional diffusions do to Brownian motion. I’ll describe several models of stable-like processes, and then talk about relatively recent results, such as uniqueness of martingale problems and Harnack inequalities. Then I’ll talk about very recent results on the regularity of potentials of stable-like processes.

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Speaker: Peter Carr (Morgan Stanley)

Title: Local Time and Option Pricing

Date: 3/27/2010

Abstract: We survey how local time arises naturally in option pricing. We also consider multivariate analogs of the Tanaka Meyer formula.

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Speaker: Christian Benes (CUNY)

Title: A Rate of Convergence for SLE

Date: 9/5/2010

Abstract: Among the open problems for SLE suggested by Oded Schramm in his 2006 ICM talk is that of obtaining “reasonable estimates for the speed of convergence of the discrete processes which are known to converge to SLE.” In this talk, we give a rate of convergence of the Loewner driving process of loop-erased random walk to Brownian motion with variance 2, the driving process of SLE(2). If time permits, we will also discuss a rate of convergence for the paths with respect to the Hausdorff distance. This is joint work with F. Johansson Viklund (Columbia U.) and M. Kozdron (U. of Regina).

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Speaker: Pierre Nolin (Courant Institute)

Title: Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model

Date: 9/12/2010

Abstract: For two-dimensional independent percolation, Russo-Seymour-Welsh (RSW) bounds on crossing probabilities are an important a-priori indication of scale invariance, and they turned out to be instrumental to describe the phase transition. They are in particular a key tool to derive the so-called scaling relations, that link the critical exponents associated with the main macroscopic functions. \ In this talk, we prove RSW-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov’s fermionic observable for the FK Ising model, that makes some harmonicity appear on the discrete level, providing precise estimates on boundary connection probabilities. We also prove several related results – including some new ones – among which the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the half-plane one-arm exponent. \ This is joint work with H. Duminil-Copin and C. Hongler

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Speaker: David M. Mason (University of Delaware)

Title: A Tale of Two Inequalities

Date: 9/19/2010

Abstract: n the first part of my talk I will show how an extension of a quantile inequality due to Komlos, Major and Tusnday (1975) yields a number of interesting couplings of a statistic and a standard normal random variable. These statistics include standardized sums of dependent random variables under various mixing conditions. Associated with these couplings are certain generalized Bernstein-type inequalities. In order to apply these couplings it is often helpful to have a maximal Bernstein-type inequality. This need led to a new and unexpected maximal Bernstein-type inequality, which will be described in the second part of my talk, along with applications. It is especially useful to bound the tails of the maximum of sums of dependent random variable. This part of my talk is based upon joint work with Peter Kevei.

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Speaker: Clement Hongler (Columbia University )

Title: Ising interfaces with free boundary conditions

Date: 9/26/2010

Abstract: We study the Ising model at criticality from an SLE point of view.\ The interfaces between + and – spins of the Ising model with Dobrushin +/- boundary conditions have been shown to converge to SLE(3) by Smirnov (on the square lattice) and Chelkak and Smirnov (on more general lattices), thanks to the introduction and proof of convergence of a discrete holomorphic martingale observable in this setup. \ We show conformal invariance of the Ising interfaces in presence of free boundary conditions. In particular we prove the conjecture of Bauer, Bernard and Houdayer about the scaling limit of interfaces arising in a so-called dipolar setup. The limiting process is a Loewner chain guided by a drifted Brownian motion, known as dipolar SLE or SLE(3,-3/2) in the literature. \ This case is made harder by the absence of natural discrete holomorphic martingales, requiring us to introduce “exotic” martingale observables. The study of these observables is allowed by Kramers-Wannier duality and Edwards-Sokal coupling, and the computation of the scaling limit is made by appealing to discrete complex analysis methods, to three existing convergence results about discrete fermions, to the scaling limit of critical Fortuin-Kasteleyn model interfaces and to the introduction of Coulomb-gas integrals. \ Our result allows to show early predictions by Langlands, Lewis and Saint-Aubin about conformal invariance of crossing probabilities for the Ising model. \ This is based on joint work with Kalle Kyt”{o}l”{a} and work in progress with Hugo Duminil-Copin

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Speaker: Haya Kaspi (Technion)

Title: STOCHASTIC PDE LIMITS OF OF MANY SERVERS QUEUES

Date: 10/2/2010

Abstract: In this talk we shall consider a many servers queueing system in which customers with i.i.d service times enter service in their order of arrival. Motivating examples of such systems are large call centers and computer networks but the results apply to other service systems with many servers and high arrival rates. The state of the system is represented by a process that describes the number of customers in the system and a measure valued process that keeps track of the ages (amount of time in service) of the customers in service. This two component process is a Markov process with dynamics that satisfy a stochastic evolution equation. In this talk I’ll discuss a functional strong law of large numbers(a fluid limit), as the number of servers and the arrival rates go to infinity, and a functional central limit theorem (FCLT, a diffusion limit), for the above pair of processes. The diffusion limit process describing the total number of customers in the system (properly centered and scaled) obtained by the above FCLT is shown to be an Ito diffusion whose diffusion coefficient is insensitive to the service distribution and its drift is described by the limiting measure valued process and the hazard rate function of the service distribution. The corresponding limit of the measure valued process is a distribution valued diffusion and, applied to a family of test functions, it is characterized as the unique solution of a set of stochastic PDE’s. \ Joint work with Kavita Ramanan from Brown University

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Speaker: Vladimir Dobric ( Lehigh University)

Title: Fractional Brownian motions canonical martingales

Date: 10/9/2010

Abstract: In order to study invariances of fractional Brownian motions for all Hurst parameters at once, we have embedded those processes into the fractional Brownian Field. It turns out that the covariance structure of that Field contains additional information about each fractional Brownian motion $(Z_H(t))_{t \geq 0}$ which can not be accessed by studying it as a single process. In particular, in that Field pairs $(Z_H(t);Z_{H0} (t))_{t \geq 0}$ of fractional Brownian motions contain information sufficient to transform each of them to a new process with independent increments. This transformation is canonical, that is, it follows directly from covariance function of the Field. The new processes are martingales respect to the natural Filtrations. When $H+H0 = 1$ the martingales, up to a multiplicative constant, are the fundamental martingales of Molchan. Their canonical nature leads to their natural integral representations and to the canonical Gaussian-Markov processes.

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Speaker: Jay Rosen (CUNY)

Title: Continuity of permanental processes

Date: 1/8/2011

Abstract: We present a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the sufficient condition for continuity which is also known to be necessary. Using an isomorphism theorem of Eisenbaum and Kaspi which relates Markov local times and permanental processes we obtain a general sufficient condition for the joint continuity of Markov local times. We show that for certain Markov processes the associated permanental process is equal in distribution to the loop soup local time. This is joint work with Michael Marcus.

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Speaker: Louis-Pierre Arguin (Courant Institute)

Title: Statistics of Branching Brownian Motion at the edge

Date: 1/22/2011

Abstract: Branching Brownian motion (BBM) is a Markov process where particles perform Brownian motion and independently split into two independent Brownian particles after an exponential holding time. The extreme value statistics of BBM in the limit of large time is of interest since BBM constitutes a borderline case, among Gaussian processes, where correlations start to affect the statistics. The law of the maximum of BBM has been understood since the works of Bramson and McKean. But little is known about the distribution of the particles close to the maximum. In this talk, I will present results on the correlation structure of these particles. This is used to unravel a Poissonian structure underlying the point process of particles at the edge. This is joint work with A. Bovier and N. Kistler.

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Speaker: Murad Taqqu (Boston and Columbia University)

Title: Properties and numerical evaluation of the Rosenblatt distribution

Date: 2/1/2011

Abstract: The Rosenblatt process is the next simplest self-similar process which belongs to the Wiener Chaos after fractional Brownian motion. While fractional Brownian motion is Gaussian, the Rosenblatt porcess is not Gaussian. Thus obtaining the Rosenblatt distribution which is the distribution of the Rosenblatt process at time 1 is not easy. Our goal is to derive various properties of the Rosenblatt distribution and to describe a technique for computing it numerically with a high degree of precision. This is joint work with Mark Veillette.

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Speaker: Philip Protter (Columbia University)

Title: Recent Results on Filtration Shrinkage

Date: 2/8/2011

Abstract: The expansion of filtrations was a popular subject in the 1980s, as it (among other things) allowed one to increase the space of possible integrands for stochastic integration. However filtration shrinkage, its mirror image sister, went largely unstudied, possibly due to the lack of potential applications. Current progress in the theory of credit risk, however, has made filtration shrinkage seem much more important. We study the issue of the behavior of local martingales under filtration shrinkage (this is based on joint work with Hans F”{o}llmer), and also the behavior of compensators of stopping times (based on joint work with with Svante Janson and Sokhne M’Baye).

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Speaker: Tom LaGatta (Courant Institute)

Title: Continuous Disintegrations for Gaussian Processes

Date: 2/15/2011

Abstract: What is the conditional law of a stochastic process once it has been observed over an interval? To make this question precise, we introduce the notion of a continuous disintegration: a regular conditional probability measure which varies continuously in the conditioned parameter. The conditioning is infinite-dimensional in character, which leads us to consider the general case of probability measures in Banach spaces. Our main result is that for a certain quantity M based on the covariance structure, the finiteness of M is a necessary and sufficient condition for a Gaussian measure to have a continuous disintegration. The condition that M is finite is quite reasonable: for the familiar case of stationary processes, M = 1. In this talk, I will present some basics on Radon probability measures on Banach spaces. This is essentially just linear algebra; no functional analytic background is required.

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Speaker: Olympia Hadjiliadis (CUNY)

Title: Drawdowns and the last passage times preceding them

Date: 2/28/2011

Abstract: In this work we derive analytical formulas for the joint distribution of the drawdown, the last visit time of the maximum of a process preceding the drawdown and the maximum of the process under general diffusion dynamics. The motivation for this work arises in the financial risk management of drawdowns.\ Drawdowns measure the first time the current drop of an investor’s wealth from its historical maximum reaches a pre-specified level. Therefore a quantity related to the drawdown is the duration of time between the drawdown and the last time at which the maximum was achieved. This quantity is studied here and it is called the speed of market crash. The derivation of the joint Laplace transform is achieved by decomposing the path to before and after the last reset of the maximum preceding the drawdown. A key observation in our derivations is that the path before the last reset of the maximum and the path after conditional on the level of the process at the last rest of the maximum before the drawdown are in fact independent. An important tool used in our derivations is the progressive enlargement of filtrations, which is used in order to turn the last reset of the maximum preceding the drawdown (an honest time) into a stopping time. We finally develop explicit formulas in the special cases of a drifted Brownian motion and the constant elasticity of variance model.\ This is joint work with Hongzhong Zhang

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Speaker: Hana Kogan (CUNY)

Title: Permanental vectors: The question of existence.

Date: 3/5/2011

Abstract: The Laplace transform of a real n-dimensional random vector with its entries the squares of Gaussian random variables is of the form: |I+aG|^(-1/2), where G is an nxn covariance matrix of the Gaussian vector. It follows that G is symmetric and positive definite. In his 1997 paper D.Vere-Jones described the generalization to general positive random vectors with the Laplace transform of the form |I+ aB|^(-q), where B is a square matrix, q any positive number. These vectors are called Permanental vectors with index q and kernel B. The matrix B in this case is, a priori, any square matrix. The question posed is to identify the pairs of square matrices and indices q for which the Permanental vector exists. (It is obvious that the Gaussian squares vector is a permanental vector with index 1/2 and the kernel a symmetric positive definite matrix.) We give a complete description of permanental vectors in dimension 3.We also give some results and conjecture for higher dimensions. References: D. Vere-Jones, Alpha-permanents, New Zealand J. of Math., 26, (1997), 125–149. N. Eisenbaum and H. Kaspi, On permanental processes, Stochastic Processes and their Applications, 119, (2009), 1401-1415.

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Speaker: Mark Brown (CUNY)

Title: Gambler’s Ruin

Date: 4/3/2011

Abstract: TBA

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Speaker: Pawel Hitczenko (Drexel University)

Title: Two-sided tail bounds for thin-tailed perpetuities

Date: 4/10/2011

Abstract: We discuss a tail behavior of perpetuities, i.e. random variables satisfying distributional equation: R=MR+Q, where (M,Q) on the right-hand side is independent of R. The case when |M|>1 with positive probability results in R being heavy-tailed and the asymptotic behavior of the tails of R has been given by Kesten in 1973 with subsequent contributions from Goldie, among others. The complementary case when |M| is bounded by 1 turns out to be more difficult and no asymptotic results are known. In this talk, complementing earlier work by Goldie and Gruebel (1991) we provide a two-sided bounds on the tails of R when |M| is bounded by 1, good up to absolute multiplicative constants.

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Speaker: Martin Zerner (University of Tuebingen)

Title: Interpolation percolation

Date: 8/27/2011

Abstract: We consider the following two-dimensional infinitesimal continuum percolation model: Let $X$ be a countably infinite set of real numbers and let $Y_x$, $x$ in $X$, be an independent family of stationary random closed subsets of the real numbers, e.g. homogeneous Poisson point processes. We give criteria for the almost sure existence of various “regular” functions f with the property that $f(x)$ is an element of $Y_x$ for all $x$ in $X$. This model is related to several other models in probability including Lipschitz percolation, oriented percolation, first-passage percolation, Poisson matchings, coverings of the circle by random arcs, and Brownian motion. Several open questions are posed.

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Speaker: Dmitry Ioffe (Technion)

Title: Stochastic representation of ground states for a class of mean field models.

Date: 9/4/2011

Abstract: A family of mean field quantum models with a transverse component can be analyzed in terms of classical Markov chains in killing potentials. The low lying spectrum of such quantum Hamiltonians is asymptotically described in terms of an appropriate generalization of Wentzell-Freidlin theory via (Fathi’s) weak KAM approach to Hamilton-Jacobi equations. The main example to be considered is Curie-Weiss model in transverse field. \ Joint work with Anna Levit.

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Speaker: Ionut Florescu (Stevens Institute of Technology)

Title: Estimation for regime switching stochastic volatility models

Date: 9/18/2011

Abstract: In this talk we will discuss a methodology to estimate the regimes and transition probabilities associated with a Hidden Markov chain using only a discrete set of observations. Specifically we consider a stochastic volatility model where the stochastic volatility is modeled as a continuous time Markov chain. The problem is to estimate the parameters of this Markov chain. The chain is unobservable and furthermore the main process can only be observed at discrete times. We consider a filter and we estimate the HMC in a two-step procedure. The procedure is applied to finance, climate data and earthquake data. The frequency at which data is sampled seems to be crucial for the problem and the methodology only works with data sampled with high enough frequency to detect the changes in regimen. \ This is joint work with F. Levin.

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Speaker: Rama CONT (Columbia University)

Title: FUNCTIONAL ITO CALCULUS

Date: 10/1/2011

Abstract: FUNCTIONAL ITO CALCULUS We develop a non-anticipative functional calculus which extends the Ito calculus to path-dependent functionals of right- continuous semimartingales [1, 3], using a notion of non-anticipative functional derivative introduced by B. Dupire [6]. This calculus is shown to be, in a precise sense, a non-anticipative analogue of the Malliavin calculus; however, our construction holds for a large class of semimartingales and makes no use of the Gaussian properties of the Wiener space. Our framework, which is suffi ciently general to cover functionals depending on quadratic variation and involving exit times of a process, is used to obtain several new results. First, we obtain a martingale representation formula for square integrable functionals of a semimartingale [2]. Second, we characterize local martingales which satisfy a regularity property as solutions of a functional di fferential equation, for which existence and uniqueness results are given [5]. These results have natural applications in stochastic control and mathematical finance: they allow to derive a universal pricing equation and a general hedging formula for path-dependent options, and reformulate Backward Stochastic Differential Equations (BSDEs) as PDEs on path space. Based on joint work with David FOURNIE (Columbia University). References [1] R Cont and D Fournie (2010) A functional extension of the Ito formula, Comptes Rendus de l’Academie des Sciences, Volume 348, Issues 1-2, January 2010, Pages 57-61. [2] R Cont and D Fournie (2009) Functional Ito calculus and stochastic integral representation of martingales, http: //arxiv.org/abs/1002.2446. To appear in: Annals of Probability. [3] R Cont and Fournie (2010) Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, Volume 259, No 4, Pages 1043-1072. [4] R Cont (2010) Numerical computation of martingale representations, Working Paper. [6] R Cont, D Fournie (2010) Martingales and functional differential equations, Working Paper. [5] B Dupire (2009) Functional Ito calculus, www.ssrn.com.

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Speaker: Amir Dembo (Stanford University )

Title: TBA

Date: 10/8/2011

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Speaker: NEPS

Title:

Date: 10/17/2011

Abstract: Northeast Probability Seminar

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Speaker: NEPS

Title:

Date: 10/18/2011

Abstract: Northeast Probability Seminar

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Speaker: Christian Benes (CUNY)

Title: Tutorial on SLE, I

Date: 10/22/2011

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Speaker: Christian Benes (CUNY )

Title: Tutorial on SLE, II

Date: 10/29/2011

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Speaker: Frederi Viens (Purdue University)

Title: TBA

Date: 11/6/2011

Abstract:

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Speaker: Peter Carr (Morgan Stanley)

Title: Some Applications of Azema Yor Martingales to Finance

Date: 1/14/2012

Abstract: In 1979, Azema and Yor introduced a family of local martingales which they exploited to solve the Skorokhod embedding problem. In the same year, Harrison and Kreps related the financial concept of no arbitrage to local martingales. In this talk, I suppose that the underlying of an option is a positive Azema Yor martingale driven by a standard Brownian motion and its running infimum. I show that a continuum of co-terminal one touches can be used to identify the martingale, and that a subset of this continuum can be used to statically replicate the payoff of a co-terminal down-and-in call.

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Speaker: Edward Waymire (Oregon State University)

Title: Dispersion Phenomena in the Presence of Interfacial Discontinuities

Date: 1/28/2012

Abstract: This talk will focus on questions arising in the geophysical and biological sciences concerning dispersion in highly heterogeneous environments, as characterized by abrupt changes (discontinuities) in the diffusion coefficient. Some specific laboratory and field experiments involving breakthrough curves (first passage times), occupation times, and local times will be addressed within a probabilistic framework based on variants of the It^{o}-McKean-Feller classic skew Brownian motion. \ This is based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, and Brian Wood at Oregon State University and partially supported by a grant from the National Science Foundation.

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Speaker: GERARDO HERNANDEZ-DEL-VALLE (Columbia University)

Title: ON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER

Date: 2/6/2012

Abstract: This talk deals with rst hitting time densities of Ito processes whose local drift can be modeled in terms of a solution to Burgers’ equation. In particular, we derive the densities of the first time that these processes reach a moving boundary. We distin- guish two cases: (a) the case in which the process has unbounded domain before absorption, and (b) the case in which the process has bounded domain before absorption. The reason as to why this distinction has to be made will be claried. Next, we classify processes whose local drift can be expressed as a linear combination to solutions of Burgers’ equation. For example the local drift of a Bessel process of order 5 can be modeled as the sum of two solutions to Burgers’ equation and thus will be classified as of class B2. Alternatively, the Bessel process of order 3 has a local drift that can be modeled as a solution to Burgers’ equation and thus will be classied as of class B1. Examples of diffusions within class B1, and hence those to which the results described within apply, are: Brownian motion with linear drt, the 3D Bessel process, the 3D Bessel bridge, and the Brownian bridge.

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Speaker: Milan Bradonjic (Bell Labs, Alcatel-Lucent)

Title: Structural and Dynamical Properties of Random Generative Models for Real Structured Networks

Date: 2/13/2012

Abstract: The existing probabilistic methods have been most successful in the analysis of unstructured random graph instances that express a large degree of statistical independence. Real-world networks do not always respond well to such existing methods: they possess multiscale structure imposed by geometry and other intrinsic correlations among nodes and edges. We developed and analyzed a new structured random generative model, the Geographical Threshold Graph (GTG) model, which is a node-weighted generalization of Random Geometric Graphs (RGGs). Nodes are distributed in space, and edges are assigned according to a threshold function involving the distance between nodes and randomly chosen node weights. We show how the structural properties, such as connectedness, diameter, existence and absence of the giant component, clustering coefficient, and chromatic number, as well as mixing time, relate to the threshold value and node weight distribution. These values play an important role not only in theory, but also in applications, such as latency in wireless communications, epidemic spread, or job scheduling. We also analyzed bootstrap percolation on RGGs. Bootstrap percolation is a process that starts by random and independent activation of nodes with a fixed probability $p$. Consequently, every inactive node becomes and remains active if at least $\theta$ of its neighbors are activate. The process is repeated until no more nodes become active. We derive tight bounds on the critical threshold $p_c(\theta)$ on the initial probability $p$ above which full activation of nodes in a connected RGG takes place.

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Speaker: Krishnamurthi Ravishankar (SUNY New Paltz)

Title: EULER HYDRODYNAMICS FOR ATTRACTIVE PARTICLE SYSTEMS IN RANDOM ENVIRONMENT

Date: 2/27/2012

Abstract: We Prove quenched hydrodynamic limit under hyperbolic scaling for bounded attractive particle systems on $Z$ in random ergodic environment. Our result is a strong law of large numbers and applies to a large class of asymmetric models such as exclusion process, misanthrope process, $k$-step exclusion process and various trac ow models.

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Speaker: Ido Ben Ari (University of Connecticut )

Title: On an exit problem for a jump-diffusion model.

Date: 3/3/2012

Abstract: We consider a model of a pure-jump process on a bounded open interval perturbed by Brownian Motion. The jumps occur according to a continuous and spatially dependent rate. We study the model under the assumption that the jump rate vanishes at the boundary. We provide sharp asymptotic bounds on the principal eigenvalue for the generator of the process as the diffusion coefficient of the Brownian Motion tends to zero. Probabilistically, the principal eigenvalue gives the exponential rate of decay of the probability of not exiting the interval for a long time. Our results show non-trivial dependence of the principal eigenvalue on the behavior of the jump rate near the boundary, including a phase transition. This work answers a question recently posed by Arcusin and Pinsky who studied the multi-dimensional setting with jump rate that is bounded below by a positive constant

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Speaker: Leonid Koralov (University of Maryland )

Title: Random, Deterministic and Nonlinear Perturbations of Dynamical Systems.

Date: 3/24/2012

Abstract: In the first part of the talk we discuss deterministic and stochastic perturbations of incompressible flows. Even in the case of purely deterministic perturbations, the long-time behavior of such systems can be stochastic, in a certain sense. The stochasticity is caused by the instabilities near the saddle points of the non-perturbed system as well as by the ergodic components of the flow. In the second part of the talk we describe the asymptotic behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of corresponding diffusion processes. In particular, we discuss the exit problem and metastability for the processes corresponding to quasi-linear initial-boundary value problems.

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Speaker: Xia Chen (University of Tennessee)

Title: Quenched asymptotics for Brownian motion in generalized Gaussian potential

Date: 4/1/2012

Abstract: Recall that the notion of generalized function is introduced for the functions that can not be defined pointwise, and is given as a linear functional over the test functions. The same idea applies to random fields. In this talk, we discuss the quenched long term asymptotics for Brownian motion in generalized Gaussian field. The major progress made in this direction includes: Solution to an open problem posted by Carmona and Molchanov (1995) with an answer different from what was conjectured; the quenched laws for Brownian motions in Newtonian-type potentials, and in the potentials driven by white noise or by fractional white noise.

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Speaker: Felisa Vazquez-Abad (Hunter College, CUNY )

Title: GRADIENT ESTIMATION FOR MARKOV CHAINS, with application to threshold control problems.

Date: 9/9/2012

Abstract: When dealing with stochastic processes and simulation-based optimization, it is necessary to build estimators of the gradients with respect to the various control variables. In this talk we present a brief survey of unbiased gradient estimation methods for Markov chains. In particular, we will introduce the weak differentiation method for derivative estimation for general state Markov chains with random horizons and discuss the proof of unbiasedness, which is established in the weak topology (thus the name of the method). We will be using some examples for on-line optimization of dynamic processes.

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Speaker: Dmitry Dolgopyat (University of Maryland )

Title: Traps for one-dimensional random walks in random environment.

Date: 9/16/2012

Abstract: It is well known that random walks in one dimensional random environment can exhibit slow behavior due to the presence of traps. We show that traps have asymptotically Poisson distribution. Then we review the consequences of this fact for quenched and annealed behavior of the walker, aging, localization, intersection local times etc.

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Speaker: Ron Blei (University of Connecticut)

Title: MEASUREMENTS OF RANDOMNESS AND INTERDEPENDENCE

Date: 9/23/2012

Abstract: Our motif is that interdependence and randomness can be calibrated by indices based separately on combinatorial measurements, p-variations, and tail- probability estimates. These notions had naturally originated in a context of harmonic analysis, and appeared later in stochastic settings. I intend in this talk to survey and explain these ideas, and (hopefully) also shed some new light on them. No formal proofs will be given. I will speak heuristically, but will try to be precise.

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Speaker: Marcel Nutz (Columbia University )

Title: Pathwise Construction of Stochastic Integrals

Date: 9/30/2012

Abstract: We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Path-by-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that the integrator is a semimartingale. This method applies to any predictable integrand.

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Speaker: Jay Rosen (CUNY)

Title: Markovian loop soups, permanental processes and isomorphism theorems

Date: 10/6/2012

Abstract: We show how to construct loop soups for general Markov processes and explain how loop soups offer a deep understanding of Dynkin’s isomorphism theorem, and allow us to go beyond.

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Speaker: NEPS 11 (Northeast )

Title: 11th Northeast Probability Seminar

Date: 10/15/2012

Abstract: Main speakers of the day are: Sandra Cerrai and Amir Dembo

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Speaker: NEPS 11 (Northeast)

Title: Northeast Probability Seminar, day 2

Date: 10/16/2012

Abstract: The main speakers for Friday are Maury Bramson and Alexander Soshnikov.

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Speaker: Dan Jenkins (Courant, NYU)

Title: Exceptional Times for an Evolving System of Coalescing Random Walks

Date: 10/20/2012

Abstract: The dynamical discrete web is a system of one-dimensional coalescing random walks that evolves in an extra dynamical time parameter. At any deterministic dynamical time, the paths behave as coalescing simple symmetric random walks. In this talk we will discuss the existence of (random) exceptional dynamical times at which the paths violate certain almost sure properties of random walks. It was shown in 2009 by Fontes, Newman, Ravishankar and Schertzer that there exist exceptional times at which the path starting from the origin violates the law of the iterated logarithm. Their results gave exceptional times at which the path is slightly subdiffusive in one direction. We will discuss recent extensions of this result, showing the existence of exceptional times at which the path is superdiffusive and times at which it is subdiffusive in both directions.

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Speaker: Fredrik Viklund (Columbia University)

Title: On the continuity of SLE(k) in k and related topics

Date: 11/4/2012

Abstract: The Schramm-Loewner evolution with parameter $k > 0$, $SLE(k)$, is a family of random fractal curves constructed using the Loewner differential equation driven by a standard Brownian motion times the square-root of $k$. These curves arise as scaling limits of cluster interfaces in certain planar critical lattice models. A natural question that has been asked is whether the (parameterized) $SLE(k)$ curves almost surely change continuously if the Brownian motion sample is kept fixed while k is varied. \\ In the talk I will further motivate the study of this question and present recent work giving a positive answer, at least for an interval of k. I will also give some background on geometric properties of SLE and the (deterministic) Loewner equation, and describe other applications of the basic tools we use for the proof. \\ The talk is based on joint work with Steffen Rohde and Carto Wong.

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Speaker: Mike Ludkovski (University of California, Santa Barbara )

Title: Sequential Detection in Stochastic Models of Epidemics

Date: 1/5/2013

Abstract: I will discuss several recent projects on applying stochastic filtering techniques for online inference problems of detection and optimal response to infectious disease epidemics. Working in the framework of continuous-time compartmental models, I will review the theory of filtering doubly stochastic point processes and the resulting detection problems. This setup provides a completely explicit description of the filter and subsequent links to control of piecewise deterministic processes. Moreover, a recent extension has allowed us to consider joint inference of parameters and states in generic stochastic kinetic models. We also construct novel sequential Monte Carlo algorithms that exploit the underlying structure. Several examples illustrating this approach will be provided, including (i) detection of seasonal flu outbreaks; (ii) detection of co-dependent epidemics at several sites; (iii) joint inference with discrete-time observations.

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Speaker: Alexander Drewitz (Columbia University )

Title: A new rearrangement inequality around infinity and applications to L’evy processes

Date: 1/19/2013

Abstract: We start with showing how rearrangement inequalities may be used in probabilistic contexts such as e.g.for obtaining bounds on survival probabilities in trapping models. This naturally motivates the need for a new rearrangement inequality which can be interpreted as involving symmetric rearrangements around infinity. After outlining the proof of this inequality we proceed to give some further applications to the volume of L’evy sausages as well as to capacities for L’evy processes. \ (Joint work with P. Sousi and R. Sun)

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Speaker: Marcel Nutz (Columbia University )

Title: A Stochastic Game of Control and Stopping

Date: 1/26/2013

Abstract: We study the existence of optimal actions in a zero-sum game $\inf_\tau \sup_P E^P[X_\tau]$ between a stopper and a controller choosing the probability measure. We define a nonlinear Snell envelope $Y$ via the theory of sublinear expectations and show that the first hitting time $\inf\{t:\, Y_t=X_t\}$ is an optimal stopping time. The existence of a saddle point is obtained under a compactness condition. (Joint work with Jianfeng Zhang.)

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Speaker: Georgios Fellouris (USC )

Title: Parameter estimation in a semimartingale model under communication constraints

Date: 2/5/2013

Abstract: We will consider the parameter estimation problem for a linear semimartingale model when there are many sources of observations, but only partial information is available from all these sources. This problem is motivated by application areas, such as wireless sensor networks, where communication constraints prohibit the transmission of the complete information to the decision maker. We will propose an estimating scheme which requires communication of only one-bit messages at stopping times of the local filtrations. The proposed estimator is strongly consistent and — for a large class of processes — asymptotically optimal; that is, after a sufficiently long horizon, it behaves as the optimal estimator that has full access to the local observations. These properties remain valid even under an asymptotically low rate of communication and an asymptotically large number of sources, which is important for the control of the communication load in large sensor networks.

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Speaker: Michael Carlisle (CUNY)

Title: Sequential Decision Making in Two-Dimensional Hypothesis Testing

Date: 2/19/2013

Abstract: We consider the problem of sequential decision making on the state of a two-sensor system. Each of the sensors is either receiving or not receiving a signal (drift) obstructed by Brownian noise (diffusion). We set up the problem as a min-max optimization in which we devise a decision rule that minimizes the length of continuous observation time required to make a decision about the state of the system subject to error probabilities. We discuss the differences in the cases where the two-dimensional noise is uncorrelated vs correlated, as well as the degeneracy of the perfect correlation cases. Finally, we examine the proposed rule applied to the problem of a decentralized sensor system versus one in constant communication with a fusion center. (Joint work with Olympia Hadjiliadis.)

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Speaker: Jan Rosinski (University of Tennessee)

Title: CLT for Gaussian and non Gaussian chaos via asymptotic independence

Date: 3/9/2013

Abstract: The celebrated fourth moment theorem of Nualart and Peccati (2005) states that, for homogeneous Wiener chaoses of a fixed order, the convergence of the second moments to 1 and of the fourth moments to 3 implies the convergence of chaoses in distribution to the standard normal law. On the other hand, Rosinski and Samorodnitsky (1999) observed that homogeneous Wiener chaoses are independent if and only if their squares are uncorrelated. In this talk we relate both results and show that the fourth moment theorem follows from an analogous criterion for the asymptotic independence of Wiener chaoses. Furthermore, we derive a multidimensional version of the fourth moment theorem, applicable in the study of stochastic processes, give new bounds on the rate of convergence, and show other related results involving Gaussian and non Gaussian limits. Applications to the limit theory of short and long range dependent stationary Gaussian time series will also be discussed. If time permits, an extension to a non-Gaussian discrete chaos will be mentioned. This talk is based on a joint work with Ivan Nourdin.

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Speaker: Daniel Conus (Lehigh University)

Title: Intermittency and chaotic properties for a family of Stochastic PDEs.

Date: 3/16/2013

Abstract: We study a family of non-linear stochastic heat equations under di fferent assumptions on the noise, the non-linearity and the initial condition. Our purpose is to show that the supremum (and, hence the solution to the equation) exhibits drastically di fferent behavior for diff erent initial conditions and non-linearities, thereby illustrating a \emph{chaotic} behavior of the equation. This chaotic behavior is related to the intermittency of the solution. Quantitative estimates are given, which will illustrate differences between the Gaussian universality class and the KPZ universality class in the case of the Parabolic Anderson Model. Similar results are also valid for the wave equation, but exhibit different quantitative behavior. Time permitting, we will say a few words on the techniques behind the results. \\ This presentation is based on joint works with M. Joseph (Utah), D. Khoshnevisan (Utah) and S.-Y. Shiu (Academica Sinica).

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Speaker: Clement Hongler (Columbia University)

Title: Ising model, discrete complex analysis, SLE, CFT, etc.

Date: 3/23/2013

Abstract: Two-dimensional lattice models at continuous phase transitions are expected (and sometimes proven) to exhibit conformal symmetry in the scaling limit. As a result, their scaling limits can be described (rigorously or conjecturally) by Conformal Field Theory and Schramm-Loewner Evolution. Very interesting algebraic, geometric and probabilistic structures emerge, that yield in particular exact formulae. I will mostly discuss the case of the Ising model, which is exactly solvable on the lattice level, a feature that allows one to pass to the scaling limit, to prove conformal symmetry and to connect the model with continuous theories. I will in particular discuss how the scaling limits of the fields and curves of the model can be identified, and how one can understand some of the algebraic structures behind those. The results rely mostly on complex analytic and probabilistic techniques. Based on joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, K. Izyurov, F. Johansson Viklund, A. Kemppainen, K. Kytölä, D.H. Phong and S. Smirnov

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Speaker: Michael Kozdron (University of Regina)

Title: The Green’s function for the radial Schramm-Loewner evolution

Date: 3/30/2013

Abstract: The Schramm-Loewner evolution (SLE), a one-parameter family of random two-dimensional growth processes introduced in 1999 by the late Oded Schramm, has proved to be very useful for studying the scaling limits of discrete models from statistical mechanics. One tool for analyzing SLE itself is the Green’s function. An exact formula for the Green’s function for chordal SLE was used by Rohde and Schramm (2005) and Beffara (2008) for determining the Hausdorff dimension of the SLE trace. In the present talk, we will discuss the Green’s function for radial SLE. Unlike the chordal case, an exact formula is known only when the SLE parameter value is 4. For other values, a formula is available in terms of an expectation with respect to SLE conditioned to go through a point. This talk is based on joint work with Tom Alberts and Greg Lawler.

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Speaker: Nathalie Eisenbaum (University of Paris)

Title: Characterization of the positively correlated squared Gaussian processes

Date: 4/7/2013

Abstract: When does a centered Gaussian vector have the property of positive correlation (also called association or positive association) ? The answer is found by Loren Pitt in 1982. In 1991 Steve Evans raises the problem of the characterization of the centered Gaussian vectors $(G_1,\ldots, G_d)$ such that $(G^2_1,\ldots, G^2_d)$ is positively correlated. This talk will present a solution to that problem.

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Speaker: Isaac Meilijson (Tel Aviv University)

Title: The Azema Martingale and the Azema-Yor stopping time

Date: 4/9/2013

Abstract: [NOTE DIFFERENT TIME] First question: For a non-decreasing, concave utility function U, is E[U(X)] >= E[U(Y)] if Y is a Martingale dilation of X? The answer is certainly positive, as U(X) is a supermartingale. Second question: For a non-decreasing, concave utility function U, is max_{0 <= a = max_{0 <= a <= 1}E[U(aY+(1-a)*b)] for all b, if Y is a Martingale dilation of X? The answer is positive as well, as can be ascertained after a little thinking: the best “a” for Y works better for X because of the first question, and the best “a” for X is even better. Third question: Is argmax_{0 <= a = argmax_{0 <= a 3$. This result follows from an explicit description of the tail behaviour of the return time as a function of delta, which is achieved by diffusion approximation of related branching processes by squared Bessel processes. (This is a joint work with Martin Zerner, University of Tuebingen)

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Speaker: Jonathon Peterson (Purdue)

Title: Extreme slowdowns for excited random walks

Date: 1/11/2014

Abstract: CUNY Probability Seminar Tuesday, February 11, 4:15 PM, Rm. 5417. Excited random walks (also called cookie random walks) are a model for self-interacting motion where the transition probabilities are governed by the local time of the random walk at the current site. If the excited random walk has a positive limiting speed $v$ ($v$ is the limit of $X_n/n$), then for $u$ between 0 and $v$ the events $\{X_n n/u\}$ are “slowdown events”. In a previous study of the large deviations for excited random walks, it was shown that the probabilities of these slowdown events decay at a polynomial rate that can be explicitly calculated. In this talk we consider the asymptotics of the probabilities of the extreme slowdown events $\{X_n n \$} with $\gamma<1$. We compute precise estimates on the polynomial rate of decay for these events and show that an interesting transition occurs at $\gamma = 1/2$.

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Speaker: Jingchen Li ( Columbia University )

Title: Extreme Analysis of Gaussian Random Fields

Date: 1/18/2014

Abstract: Gaussian random fields are employed to model spatially varying errors in various stochastic systems. In this talk, I present several recent results of the extreme analysis for such systems. In particular, the topic covers various nonlinear functionals of Gaussian random fields including the supremum norm, integrals of exponential functions, and differential equations admitting random coefficients driven by Gaussian random fields. We present the asymptotic approximations of certain tail events, their practical interpretations, and intuitions behind the technical developments. The results have applications to material science, financial risk analysis, statistical analysis of point processes, etc.

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Speaker: Jeremy Quastel (University of Toronto)

Title: What does it mean to solve KPZ?

Date: 1/25/2014

Abstract: The last few years have seen significant progress in understanding the KPZ equation and its universality class, from the meaning of the equation, to a big picture conjecture of asymptotic fluctuations, to several exact formulas. We will try to give a survey of some of this progress, and use recent results on flat exclusion to illustrate some of the exact solvability.

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Speaker: Louis-Pierre Arguin (Université de Montréal)

Title: Variance bounds for Ising spin glasses on $Z^d$

Date: 2/4/2014

Abstract: Good bounds on the variance of the maximum (or the log-partition function) of a collection of correlated random variables are generally very hard to obtain. However, if this is possible, these bounds can yield a lot of information on the general behavior of the process. A famous example is the proof of the absence of phase transition in 2D for the Ising model with random magnetic field by Aizenman-Wehr. In this talk, I will explain this approach to prove the absence of phase transition in disordered systems. I will prove variance bounds for Ising spin glasses on $Z^d$ and show how it restricts the possible states of the systems (though falling short of proving the absence of phase transition). This is joint work with D. Stein, C. Newman and J. Wehr.

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Speaker: Ivan Matic (Baruch College, CUNY)

Title: Deterministic processes in random environments

Date: 2/11/2014

Abstract: The solutions to ordinary differential equations of the form $dX_t/dt=b(X_t,\omega)$ are called deterministic processes in random environments. Their discrete analogs are deterministic walks in random environments. We establish large deviation properties for these two classes of processes. \\ We also study natural generalizations to excited random environment. Such environment can be understood as stacks of cookies on each site of the environment. Each cookie represents transition probabilities for the walk to move to one of the nearby sites. Once all cookies are consumed at a given site, every subsequent visit will result in a walk taking a step according to the direction prescribed by the last consumed cookie. \\ We will see how large deviations can be obtained in some cases for deterministic walks in excited random environments.

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Speaker: Mariya Bessonov (NY Tech)

Title: Phase transitions of a quadratic contact process

Date: 2/18/2014

Abstract: The contact process is a well-known model for the spread of a species for which an offspring has a single parent. In this talk, we consider the quadratic contact process that describes the situation when there are two parents for an offspring. We will discuss results that give bounds on the critical values for the process to survive from a finite set and for the existence of a nontrivial stationary distribution. This is joint work with Rick Durrett.

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Speaker: Patrik Ferrari (University of Bonn)

Title: Anomalous shock fluctuations in TASEP and last passage percolation models

Date: 2/25/2014

Abstract: We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time $t$ will have a width of order $t^{1/3}$. We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the Airy$_1$ process. The limiting distribution is a product of two distribution functions, which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics. We show that the result generalizes to generic last passage percolation models. Joint work with Peter Nejjar.

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Speaker: David Belius (Université de Montréal)

Title: Finer properties of torus cover times

Date: 3/8/2014

Abstract: The cover time is the time it takes for a Markov chain to visit all of its state space. Cover times have received a lot of attention in the last few decades. In my talk I will discuss the finer properties of cover times of torii (fluctuations, structure of late points, etc).

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Speaker: David Sivakoff ( Ohio State University)

Title: Nucleation scaling in jigsaw percolation.

Date: 3/29/2014

Abstract: Jigsaw percolation is a non-local process that iteratively merges elements of a partition of the vertices in a deterministic “puzzle graph” according to the connectivity properties of a random “collaboration graph”. We assume the collaboration graph is an Erdos–Renyi graph with edge probability $p$, and investigate the probability that the puzzle graph is solved, that is, that the process eventually produces the partition $\{V\}$. In some generality, for puzzle graphs with N vertices of degrees about $D$, this probability is close to 1 or 0 depending on whether $pD(log N)$ is large or small. We give more detailed results for the one dimensional ring and two dimensional torus puzzle graphs, where in many instances we can prove sharp phase transitions.

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Speaker: Konstantinos Spiliopoulos, ( Boston University)

Title: Large deviations and Monte Carlo methods for problems with multiple scales

Date: 4/6/2014

Abstract: Rare events, metastability and Monte Carlo methods for stochastic dynamical systems have been of central scientific interest for many years now. In this talk we focus on multiscale systems that can exhibit metastable behavior, such as rough energy landscapes. We discuss quenched large deviations in related random rough environments and design of provably efficient Monte Carlo methods, such as importance sampling, in order to estimate probabilities of rare events. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. We resolve this issue and demonstrate the theoretical results by examples and simulation studies.

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Speaker: Vladas (IMPA)

Title: Continuity of phase transitions and near critical geometry.

Date: 8/9/2014

Abstract: During my talk I will touch two interrelated topics: continuity of phase transition in lattice models of Statistical Mechanics, such as percolation and Ising model, report recent results and discuss geometry of critical and near critical clusters. This includes connection with self-destructive percolation and forest fires.

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Speaker: Ivan Corwin (Columbia University )

Title: Integrable probability: beyond the gaussian universality class

Date: 8/23/2014

Abstract: Methods originating in representation theory and integrable systems have led to detailed descriptions of new non-Gaussian statistical universality classes. This talk will focus on some of the probabilistic systems (ASEP, $q$-TASEP, the O’Connell-Yor polymer, and the KPZ equation) and methods (Schur / Macdonald processes and quantum integrable systems) which have played a prominent role in this story.

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Speaker: Olympia Hadjiliadis (Brooklyn College, CUNY )

Title: Quickest detection in coupled systems

Date: 8/30/2014

Abstract: We consider the problem of quickest detection of signals in a coupled system of $N$ sensors, which receive continuous sequential observations from the environment. It is assumed that the signals, which are modeled by general Ito processes, are coupled across sensors, but that their onset times may differ from sensor to sensor. Two main cases are considered; in the first one signal strengths are the same across sensors while in the second one they differ by a constant. The objective is the optimal detection of the first time at which any sensor in the system receives a signal. The problem is formulated as a stochastic optimization problem in which an extended minimal Kullback-Leibler divergence criterion is used as a measure of detection delay, with a constraint on the mean time to the first false alarm. The case in which the sensors employ cumulative sum (CUSUM) strategies is considered, and it is proved that the minimum of $N$ CUSUMs is asymptotically optimal as the mean time to the first false alarm increases without bound. In particular, in the case of equal signal strengths across sensors, it is seen that the difference in detection delay of the N-CUSUM stopping rule and the unknown optimal stopping scheme tends to a constant related to the number of sensors as the mean time to the first false alarm increases without bound. Alternatively, in the case of unequal signal strengths, it is seen that this difference tends to zero. This is joint work with Hongzhong Zhang, Tobias Schaefer and H.Vincent Poor

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Speaker: Yuri Bakhtin (NYU)

Title: Burgers equation with random forcing

Date: 9/21/2014

Abstract: The Burgers equation is one of the basic nonlinear evolutionary PDEs. The study of ergodic properties of the Burgers equation with random forcing began in 1990’s. The natural approach is based on the analysis of optimal paths in the random landscape generated by the random force potential. For a long time only compact cases of the Burgers dynamics on a circle or bounded interval were understood well. In this talk I will discuss the Burgers dynamics on the entire real line with no compactness or periodicity assumption on the random forcing. The main result is the description of the ergodic components and existence of a global attracting random solution in each component. The proof is based on ideas from the theory of first or last passage percolation. The kicked forcing case is an extension of the Poissonian forcing case considered in a joint work with Eric Cator and Kostya Khanin.

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Speaker: Sumit Mukherjee (Columbia University)

Title: Universal Limit Theorems in Graph Coloring Problems

Date: 9/28/2014

Abstract: In this talk the limiting distribution of the number of monochromatic edges in a uniform random coloring of any sequence of random graphs will be characterized. The origin of these problems can be traced back to the classical birthday paradox, and are often helpful in the study of coincidences. I will show that the number of monochromatic edges converge in distribution to either a Poisson mixture or a Normal when the number of colors grow to infinity. The results are universal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors, and work for any graph sequence, deterministic or random. The necessary and sufficient condition for asymptotic normality when the number of colors is fixed, will also be discussed. Finally, using results from the emerging theory of graph limits, the asymptotic distribution is also characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relates to results of Erd”{o}s and Alon on extremal subgraph counts. \ This is based on joint work with Bhaswar Bhattacharya and Persi Diaconis.

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Speaker: Louis-Pierre Arguin (Baruch College, CUNY)

Title: Probabilistic approach for the maxima of the Riemann Zeta function on the critical line

Date: 10/4/2014

Abstract: A recent conjecture of Fyodorov, Hiary & Keating states that the maxima of the Riemann Zeta function on a bounded interval of the critical line behave similarly to the maxima of a specific class of Gaussian fields, the so-called log-correlated Gaussian fields. These include important examples such as branching Brownian motion and the 2D Gaussian free field. In this talk, we will highlight the connections between the number theory problem and the probabilistic models. We will outline the proof of the conjecture in the case of a randomized model of the Zeta function. We will discuss possible approaches to the problem for the function itself. This is joint work with D. Belius (NYU) and A. Harper (Cambridge).

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Speaker: Michael Carlisle (Baruch College)

Title: Recent topics in the two-dimensional gambler’s ruin

Date: 10/11/2014

Abstract: The one-dimensional gambler’s ruin is one of the oldest problems in probability theory. While most problems using a one-dimensional gambler’s ruin reside on a line segment, what can be examined by the gambler’s ruin in two dimensions? We examine two recent problems that use the framework of the gambler’s ruin on planar regions (one discrete, one continuous), and see how the interpretation of a ruin problem can change based on the question asked and the shape of the boundary considered. (This is based on the speaker’s dissertation work and joint work with Olympia Hadjiliadis and Zhenyu Cui.)

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Speaker: Eyal Lubetzky (Courant)

Title: Harmonic pinnacles in the Discrete Gaussian model

Date: 11/2/2014

Abstract: The 2D Discrete Gaussian model gives each height function $\eta : Z^2 \to Z$ a probability proportional to $\exp[-\beta H(\eta)]$, where $\beta$ is the inverse-temperature and $H(\eta)=\sum (\eta_x-\eta_y)^2$ sums over nearest-neighbor bonds. We consider the model at large fixed $\beta$, where it is flat unlike its continuous analog (the Gaussian Free Field). We first establish that the maximum height in an $L\times L$ box with 0 boundary conditions concentrates on two integers $M,M+1$ with $M\sim [(2/\pi\beta)\log L \log\log L]^{1/2}$. The key is a large deviation estimate for the height at the origin in Z^2, dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on $\eta \geq 0$ (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels $H,H+1$ where $H\sim M/\sqrt{2}$. This in particular pins down the asymptotics, and corrects the order, in results of Bricmont, El-Mellouki and Fröhlich (1986). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to $p$-harmonic analysis and alternating sign matrices. \\ Joint work with Fabio Martinelli and Allan Sly.

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Speaker: Paul Bourgade (Courant, NYU )

Title: From the mesoscopic to microscopic scale in random matrix theory.

Date: 1/10/2015

Abstract: Eugene Wigner has envisioned that the distributions of the eigenvalues of large Gaussian random matrices are new paradigms for universal statistics of large correlated quantum systems. These random matrix eigenvalues statistics supposedly occur together with delocalized eigenstates. I will explain recent developments proving this paradigm, both for eigenvalues and eigenvectors of random matrices. This is achieved by bootstrap on scales, from mesoscopic to microscopic. Random walks in random environments, homogenization and the coupling method play a key role. This talk is based on joint works L. Erdos, J. Yin and H-T. Yau.

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Speaker: Daniel Fresen (Yale)

Title: Asymptotic structure of large random samples: global shape and local limiting Poisson process

Date: 1/24/2015

Abstract: We consider a random sample of 4n$ i.i.d. points in d dimensional Euclidean space, with common distribution $u$. The emphasis is on the limiting behavior as $n$ tends to infinity and $d$ remains fixed. Various properties are imposed on $u$ at different times that involve (for example) convexity, smoothness and rapid tail-decay. We study properties of the random polytope $p_n$ defined as the convex hull of the sample. These properties include the volume and number of vertices, and concentration properties/deviation inequalities (multivariate Gnedenko law of large numbers). We also study the local structure of the sample by ‘zooming in’ to points near the boundary using an affine transformation in order to achieve a limiting point process, with a density that is Gaussian in $(d-1)$ dimensions and exponential in the remaining dimension. This limiting density is a natural entropy maximizer with an interesting self-similarity property.

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Speaker: Subhro Ghosh (Princeton University)

Title: Rigidity phenomena in random point sets and applications

Date: 2/3/2015

Abstract: In several naturally occurring (infinite) point processes, we show that the number (and other statistical properties) of the points inside a finite domain are determined, almost surely, by the point configuration outside the domain. This curious phenomenon we refer to as “rigidity”. We will discuss rigidity phenomena in point processes and their applications. Depending on time, we will talk about applications to stochastic geometry and to random instances of some classical questions in Fourier analysis.

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Speaker: Diogo Pinheiro (Brooklyn College)

Title: Stochastic optimal control problems with model uncertainty

Date: 2/10/2015

Abstract: I will discuss stochastic optimal control problems with model uncertainty either in the form of a discrete sequence of random time horizons or in the form of a parametric dependence on a certain switching process. Such problems are interesting not only for their mathematical novelty, but also for their potential application to subjects such as Finance, Actuarial Science, Economics, Population Dynamics and Engineering. I will focus mainly on the derivation of generalized dynamic programming principles, as well as on the corresponding Hamilton-Jacobi-Bellman equations.

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Speaker: Milan Bradonjic (Bell Labs, Alcatel-Lucent)

Title: Asymptotic laws for coloring sparse random geometric graphs with constant number of colors

Date: 2/24/2015

Abstract: We study coloring of sparse random geometric graphs, in an arbitrary but constant dimension, with a constant number of colors. We show the law of large numbers, as well as the central limit theorem type results for the maximum number of nodes that can be properly colored. This object is neither scale-invariant nor smooth, so we design the tools that with the main method of subadditivity allow us to show the law of large numbers. Additionally, by proving the Lindeberg conditions, we show the normal limiting distribution. For the constants that appear in these results, we provide the exact value in dimension one, and upper and lower bounds in higher dimensions. This work is motived by wireless networks, and the results show that by excluding an arbitrarily small fraction of users from service, the max-min throughput capacity of a large wireless network can be improved by orders of magnitude. \ Joint work with Sem Borst.

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Speaker: Heng Yang

Title: Detection With Post-Change Distribution Uncertainty

Date: 3/14/2015

Abstract: We consider the problem of quickest detection of an abrupt change when there is uncertainty about the post-change distribution. In particular we examine this problem in the continuous-time Wiener model where the drift of observations changes from zero to a drift randomly chosen from a collection. We set up the problem as a stochastic optimization in which the objective is to minimize a measure of detection delay subject to a frequency of false alarm constraint. We consider a composite rule involving the CUSUM reaction period, the time between the last reset of the CUSUM statistic process and the CUSUM alarm, and show that by choosing parameters appropriately, such a composite rule can be asymptotically optimal of third order in detecting the change point as the average time to the first false alarm increases without bound. Because of the uncertainty, we would like to not only detect the change point but to also identify the post-change distribution. We define an identification function associated with the composite rule and show that it has arbitrarily small probability of identification error for the post-change drift as the average first false alarm increases without bound.

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Speaker: Jay Rosen (CUNY)

Title: Conditions for permanental processes to be unbounded

Date: 3/21/2015

Abstract: Permanental processes are generalizations of Gaussian processes. They replace Gaussian processes in Isomorphism Theorems for non-symmetric Markov Processes. To apply these Isomorphism Theorems it is important to know sample path properties of permanental processes. We present a Sudakov type inequality which gives lower bounds on permanental processe

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Speaker: Mykhaylo Shkolnikov (Princeton University)

Title: On multilevel Dyson Brownian motions.

Date: 3/28/2015

Abstract: I will discuss how Dyson Brownian motions describing the evolution of eigenvalues of random matrices can be extended to multilevel Dyson Brownian motions describing the evolution of eigenvalues of minors of random matrices. The construction is based on intertwining relations satisfied by the generators of Dyson Brownian motions of different dimensions. Such results allow to connect general beta random matrix theory to particle systems with local interactions, and to obtain novel results even in the case of classical GOE, GUE and GSE random matrix models. Based on joint work with Vadim Gorin.

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Speaker: Hugo Duminil-Copin (Universite de Geneve)

Title: A new proof of exponential decay of correlations in subcritical percolation and Ising models

Date: 4/5/2015

Abstract: We provide a new proof of exponential decay of correlations for subcritical Bernoulli percolation on $Z^d$. The proof is based on an alternative definition of the critical point. The proof extends to the Ising model and to infinite-range models on infinite locally-finite transitive graphs. It also provides a mean-field lower bound for the explosion of the infinite-cluster density in the supercritical regime. \newline Joint work with Vincent Tassion.

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Speaker: Elena Kosygina (Baruch College and the CUNY Graduate Center)

Title: Excited random walks in Markovian cookie environments on Z.

Date: 4/12/2015

Abstract: We consider a nearest-neighbor random walk on $Z$ whose probability $p(x,n)$ to jump to the right from site x depends not only on $x$ but also on the number of prior visits n to x. The collection $(p(x,n))_{x\in Z, n\ge 0}$ is sometimes called the “cookie environment” due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the transition probability according to the “flavour” of the cookie eaten. Assume that the cookie stacks are i.i.d. and that the cookie “flavours” at each stack $(p(x,n))_{n\ge 0}$ follow a finite state Markov chain in n. Thus, the environment at each site is dynamic, but it evolves according to the local time of the walk at each site rather than the random walk time. We discuss recurrence/transience, ballisticity, and limit theorems for such walks. $$ $$ This is a joint work with Jonathon Peterson, Purdue University.