Statistics and Data Science Seminar
Xia Chen
University of Tennessee
Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise
Abstract: This work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation
$$\frac{\partial u}{\partial t}(t,x)=\frac{1}{2}\triangle u(t,x)+V(t,x)u(t,x),\quad\quad\mathrm{with}\ u(0,x)=u_0(x),$$
where the homogeneous generalized Gaussian noise $V(t,x)$ is, among other forms, white or fractional white in time
and space. Associated with the Cole-Hopf solution to the KPZ equation, in particular, the precise asymptotic form
$$\lim_{R\to+\infty}(\log R)^{-2/3}\log\max_{|x|\leq R}\, u(t,x)=\frac{3}{4}\sqrt[3]{\frac{2t}{3}}\quad \mathrm{a.s.}$$
is obtained for the parabolic Anderson model $\partial_t u=\frac{1}{2}\partial^2_{xx}u+\dot{W}u$ with the $(1+1)$-white noise $\dot{W}(t,x)$.
Wednesday December 2, 2015 at 4:00 PM in SEO 636