一、题目

Asymptotics of The Density of Parabolic Anderson Random Fields

二、主讲人

胡耀忠

三、摘要

Parabolic Anderson model is a very simple stochastic heat equation with multiplicative Gaussian noise. The solution $u(t,x)$ of this equation can be represented by the Wiener-It\^o chaos expansion. It is related to the Anderson localization and is also related to the so-called KPZ equation describing the physical growth phenomena. We investigate the shape of the density ρ(t,x; y) of the solution u(t,x) to the stochastic partial differential equation $\frac{\partial}{\partial t} u(t,x) = (1/2) \Delta u(t,x)+u \diamond \dot W (t,x)$, where $\dot W (t,x)$ is a general Gaussian noise and $\diamond$ denotes the Wick product. We mainly concern with the asymptotic behavior of $\rho(t,x; y)$, the density of the random variable $u(t,x)$, when $y\to\infty$ or when $t\to 0+$. Both upper and lower bounds are obtained and these two bounds match each other modulo some multiplicative constants. If the initial condition is positive, then $\rho(t,x; y)$ is supported on the positive half-line $y\in [0, \infty)$ and in this case we show that $\rho(t,x; 0+)=0$ and obtain an upper bound for $\rho(t,x; y)$ when $y\to 0+$. Our tool is Malliavin calculus and I will also present a very brief and heuristic introduction.

This is joint work with Khoa Le.

四、主讲人简介

胡耀忠，加拿大University of Alberta at Edmonton特聘教授。1992年获得法国University of Strasbourg I博士学位，师从国际著名概率学家P. A. Meyer教授，1997年至2017年在美国University of Kansas工作，历任助理教授、副教授、教授。曾学术访问过英国、法国、德国、西班牙、瑞士、加拿大、挪威、墨西哥、以色列、南非等国家和地区，是概率论与随机分析、随机控制以及数理金融领域国际知名专家，在相关研究领域国际权威杂志发表论文170余篇、专著2部，在2015年当选为IMS Fellow。

五、邀请人

数学学院教授史敬涛 

六、时间

8月9日（周二）10:00--11:00

七、地点

腾讯会议

八、联系人

史敬涛，shijingtao@sdu.edu.cn

九、主办单位

山东大学数学学院