Asymptotics of the density of parabolic Anderson random fields

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Yaozhong Hu(胡耀忠), 加拿大阿尔伯塔大学Centennial教授,1981年获江西大学计算数学学士学位,1984年获中科院武汉数学物理研究所硕士学位,1992年获法国路易斯巴斯德大学概率博士学位,师从国际著名概率学家P. A. Meyer教授。胡教授的研究兴趣广泛,主要研究领域是随机分析、数理金融、随机控制、随机微分方程数值分析等。在 Ann. Probability、Probab. Theory Related Fields、Ann. Applied Probability、Bernoulli、Stochatis Process. Appl.、Mem. Amer. Math. Soc.、Comm. PDEs、J. Funct. Anal、Trans. Amer. Math. Soc等概率论和数学综合类top期刊上发表论文100多篇,出版专著2部。2015年,当选为Fellow of Institute of Mathematical Statistics。


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.