|Wednesday, March 8|
|1:30||-||3:00||Tutorial Lecture I - Elchanan Mossel|
|3:30||-||5:00||Tutorial Lecture II - Elchanan Mossel|
|5:00||-||6:00||Short talks session|
|6:30||Women in Probability Dinner|
|Thursday, March 9|
|11:00||-||12:00||Short talks session|
|2:00||-||3:30||Open problems session|
|4:00||-||6:00||Short talks session|
|Friday, March 10|
|11:00||-||12:00||Kai Lai Chung Lecture - David Nualart|
|Saturday, March 11|
- Mixing in product spaces
Mixing properties of dynamical systems are a central topic of study in mathematics. In the lectures we will take a modern, probabilistic, short horizon quantitative view of this problem and see that it involves a number of areas in modern mathematics including isoperimetric theory, noise stability and additive combinatorics. We will explain where are we in a quest for a unified theory as well as some of the applications of this theory.
Nike Sun - Supercritical minimum mean-weight cycles
We consider the minimum mean-weight cycle (MMWC) in the stochastic mean-field distance model, that is, in the complete graph on n vertices with edges weighted by independent exponential random variables (of unit rate). Mathieu and Wilson [MW] (2012) showed that the MMWC exhibits very different characteristics according to whether its mean weight is smaller or larger than 1/(ne), where both cases occur with asymptotically positive probability. While the behavior in the subcritical (below 1/(ne)) regime is characterized in detail by [MW], much less was understood in the supercritical regime (above 1/(ne)). I will describe some of the obstacles, and present our results determining the length and weight asymptotics for the supercritical MMWC. Joint work with Jian Ding and David B. Wilson.
Massimiliano Gubinelli - Weak universality of fluctuations and singular stochastic PDEs
Mesoscopic fluctuations of microscopic (discrete or continuous) dynamics can be described in terms of non-Gaussian random fields. These random fields are fully described by certain nonlinear stochastic partial differential equations which are universal: they depend on very few details of the microscopic model. However, due to the extreme irregular nature of the random field sample paths, these equations turn out to not be well-posed in any classical analytic sense. In this talk I will review recent progress in the mathematical understanding of such singular equations and of their (weak) universality. If time permits I will discuss the case of the one dimensional Kardar-Parisi-Zhang equation and of the three dimensional Stochasic Allen-Cahn equation.
David Nualart (Kai Lai Chung Lecture) - Malliavin calculus and central limit theorems
The aim of this talk is to present some recent applications of the stochastic calculus of variations to derive central and noncentral limit theorems in probability for random variables that can be represented as divergence integrals. In the first part, we will discuss Gaussian and mixed Gaussian approximations using Stein’s and interpolation methods for Wiener functionals. In the second part we will present a new method for proving tightness in the functional central limit theorem for the self-intresection local time of the fractional Brownian motion, using techniques of Malliavin calculus.
Jason Schweinsberg - Yaglom-type limit theorems for branching Brownian motion with absorption.
We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, improving upon a result of Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, which also improves upon results of Kesten (1978). An important tool in the proofs of these results is the convergence of branching Brownian motion with absorption to a continuous state branching process.
Paul Dupuis - A variational representation for functionals of a Poisson random measure and applications
When combined with weak convergence methods, the variational representation for positive functionals of infinite dimensional Brownian motion is very convenient for the large deviation analysis of many complicated process models, such as stochastic partial differential equations. More recently, a variational representation for functionals of a Poisson random measure has been developed, and it has likewise proven useful for problems with difficult features. After revisiting the formulation of the representation for the Brownian case and reviewing an elementary application, we focus on recent uses of the Poisson representation. By formulating process models as the solution to a stochastic differential equation driven by Poisson noise and establishing weak regularity properties of the mapping from noise space to state space, one can characterize the large deviation properties of processes that are hard to treat otherwise. Examples presented include the exploration process for a random graph model and a multiscale stochastic model.