We present a functional version of the central limit theorem for functionals of a stationary Gaussian sequence proved by Breuer and Major in 1987. The proof uses techniques of Malliavin calculus and is based on the continuity of the divergence operator proved by Meyer.

In this talk, I will talk about comparison principle of a Hamiltonian-Jacobi equation which plays a critical step in proving the LDP of stochastic vortex dynamics.

We present several interesting phenomena about almost sure convergence on homogeneous chaoses that include Gaussian Wiener chaos, homogeneous sums in independent random variables. Our main strategy is to use extra randomness and apply simple conditioning argument, whose ideas are close to the spirit of Stein's method of exchangeable pairs. Some natural questions are left open in this short note. This is a joint work with Guillaume Poly from IRMAR, University of Rennes 1, France.

The KPZ equation is a fundamental stochastic PDE used for modeling random growth processes, Burgers turbulence, interacting particle system, random polymers etc. It is related to another important SPDE, namely, the stochastic heat equation (SHE). In this talk, we focus on the tail probabilities of the Cole-Hopf solution (i.e., the logarithm of the SHE) of the KPZ equation. For instance, we consider the probability that the solution is much smaller or bigger than the expected value. This also gives the tail probabilities of the SHE. Our analysis is based on a exact identity between the KPZ equation and the Airy Point process (which appears at the edge of the spectrum of the random Hermittian matrix) and the Brownian Gibbs property of the KPZ line ensemble.

This talk will be based a joint work with my advisor Prof. Ivan Corwin.

In this talk, we introduce a new type of BSDEs, we call it mean-field anticipated backward stochastic differential equations (MF-BSDEs, for short) driven by a fractional Brownian motion with Hurst parameter H>1/2. We will show that it's possible to prove the existence and uniqueness of this new type of BSDEs using two different approaches. Then, we will present a comparison theorem for such BSDEs. Finally, as an application of this type of equations, a related stochastic optimal control problem is studied. This is a joint work with Yufeng Shi and Jiaqiang Wen : Institute for Financial Studies and School of Mathematics, Shandong University, China.

Random sup-measures are natural objects when investigating extremes of stochastic processes. In this talk, we review a few random sup-measures recently introduced in the investigation of stationary sequences of random variables with long-range dependence. In particular, these random sup-measures characterize certain long-range clustering of extremes. If time permits, we shall explain how they arise from certain discrete models.

Joint works with Olivier Durieu and Gennady Samorodnitsky.

Preliminary exam.