Particle representations for stochastic
partial differential equations
Tom Kurtz
Kai Lai Chung Lecture
Abstract: Stochastic partial differential equations arise naturally as limits of finite systems of weighted interacting particles. For a variety of purposes, it is useful to keep the particles in the limit obtaining an infinite exchangeable system of stochastic differential equations for the particle locations and weights. The corresponding de Finetti measure then gives the solution of the SPDE. These representations frequently simplify existence, uniqueness and convergence results. The talk will focus on situations where the particle locations are given by an iid family of diffusion processes, and the weights are chosen to obtain a nonlinear driving term and to match given boundary conditions for the SPDE. (Recent results are joint work with Dan Crisan.)
Statistical mechanics on sparse and random graphs
Andrea Montanari
Abstract: Given a finite graph, and variables associated to its vertices, factor models provide a flexible way to define joint probability distributions over those variables, that are Markovian with respect to the underlying graph. Well known examples include the Ising model, the Potts model, the hardcore model, and so on. Such models have been classically studied on grids or other amenable graphs, because of their applications within statistical physics. Recent developments in theoretical computer science and graph theory have motivated the study of graph sequences that converge locally to trees. Over the last twenty years, physicists have developed an surprisingly rich set of sophisticated conjectures on these models. I will review motivations, recent progress towards proving these conjectures, and a number of open problems.
Hitting probabilities for systems of stochastic waves
Marta Sanz-Sole
Abstract: We will report on several results about hitting probabilities of random fields that explain the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. As an application, we will consider a \(d\)-dimensional random field \(u=\{u(t,x)\}\) that solves a non-linear system of stochastic wave equations in spatial dimensions \(k=\{1,2,3\}\), driven by a spatially homogeneous Gaussian noise that is withe in time. The necessity of having sharp results on the rate of degeneracy of the eigenvalues of the Malliavin matrix corresponding to the vector \((u(t,x), u(s,y))\) will be highlighted.
Combinatorial aspects of random polymers
Neil O’Connell
I will describe an algebraic-combinatorial framework for the study of random polymers which is based on a geometric lifting of the Robinson-Schensted-Knuth correspondence introduced by A.N. Kirillov (2000) and its surprising connections to GL(n,R) Whittaker functions which we have been developing in recent work [O’C 2009, Corwin-O’C-Seppalainen-Zygouras 2011, O’C-Seppalainen-Zygouras 2012]. This talk will be mainly based on the recent paper arXiv:1210.5126 with Timo Seppalainen and Nikos Zygouras.
Lipschitz embeddings of random sequences
Allan Sly
Abstract: We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. As an example we show that there exists an increasing M-Lipschitz embedding from one i.i.d. Bernoulli sequences into an independent copy with positive probability provided that M is large enough. In a closely related problem we show that two independent Poisson processes on R are roughly isometric (or quasi-isometric). Joint work with Riddhipratim Basu
Malliavin calculus tools for the sharp estimation of
densities, tails, distances, and other functionals on Wiener space
Frederi Viens
Abstract: The stochastic calculus of variations of Paul Malliavin has seen many successes in its decades-long venerable history, starting with methods for proving existence and smoothness of densities of stochastic differential systems. We will discuss some of the newest applications of this Malliavin calculus, which return to questions about densities and distributions, but now in a fully quantitative way. Under the impetus of an unexpected 2005 result of Nualart and Peccati, known as the 4th moment theorem, Nourdin and Peccati started a systematic program of using estimates of Malliavin-based functionals of variables on Wiener space to derive sharp quantitative results of convergence in law, on Wiener chaos and beyond. We will describe basic elements of this program, including some based on the interplay between the Malliavin calculus and Stein’s lemma. We will also present related results without Stein’s lemma, based on the same functionals used by Nourdin and Peccati, resulting in non-asymptotic estimates of densities, tails, and other concrete functionals such as expected suprema. Time permitting, applications to various stochastic systems, such as SPDEs, polymers, and spin systems in random environments, will be given, and open problems will be summarized. This talk will cover joint works with H. Airault, R. Eden, P. Malliavin, I. Nourdin, E. Nualart, G. Peccati, C. Tudor, and X. Zhang.
