Mini-Workshop on Stochastic Analysis

(B-1238, 12/31/2019 (Tuesday))

8:30 – 9: 30   Yaozhong Hu (University of Alberta)

Title: Stochastic heat equation with general nonlinear spatial rough Gaussian noise

Abstract:   In this talk, we consider the following one dimensional (in space variable) nonlinear 

stochastic heat equation driven by the Gaussian noise which is white in time and fractional in space:

$$  \frac{\partial u(t,x)}{\partial t}=\frac{\partial^2 u(t,x)}{\partial x^2}+\sigma(u(t,x))\dot{W}(t,x), $$

where $W(t, x)$ s a centered Gaussian process with covariance given by

$$ {\bf E} [ W(t, x)W(s, y)] =\frac 12  \left( |x|^{2H}+|y|^{2H}-|x-y|^{2H} \right) (s\wedge t). $

Here the Hurst parameter H is between 1/4 and 1/2 and $\dot{W}(t,x)=\frac{\partial^2 W}{\partial t\partial x}$.  

We remove the technical condition $\sigma(0)=0$ previously assumed. The idea is to introduce a weight for the solution.

When $\sigma (t,u)$ is a constant the solution is a Gaussian random field and we obtain the bound of the solution

$\sup_{0\le t\le T, |x|\le L} |u(t,x)|$ when $T$ and $L$ goes to infinity.  This is a joint work with Xiong Wang.

9:30 – 10:30   Yimin Xiao (Michigan State University)

Title: Regularity Properties and Singularity Propagation of Stochastic Wave Equation

Abstract:  Consider the linear stochastic wave equation (SWE) driven by a Gaussian noise which

is white in time and colored in space. We show that the solution satisfies a certain form of strong local nondeterminism and we use this property to derive regularity properties such as the exact uniform modulus of continuity for the solution. 

On the other hand, by applying general Gaussian methods, we show the existence and the propagation of singularities of the solution of SWE.  The combined results show interesting fine structure of the sample function of the solution. 

This talk is based joint work with Cheuk-Yin Lee.

10:30 – 11:30  王凤雨（天津大学）

Title: On Stochastic Hamiltonian Systems

Abstract: Starting from Villani's hypocoercivity for kinetic Fokker-Planck equations, we introduce some recent progress on the study of Stochastic Hamiltonian Systems (SHS), including  weak Poincare inequality for weak hypocoercivity, hypercontractivity, gradient and Harnack inequalities, and SHS with singular coefficients.