Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise

TL;DR

This paper investigates the precise spatial asymptotic behavior of the parabolic Anderson model driven by generalized Gaussian noise, including white and fractional white noise, revealing exact growth rates and connections to the KPZ equation.

Contribution

It provides the first detailed asymptotic analysis of the parabolic Anderson model with generalized Gaussian noise, including explicit formulas for growth rates and links to KPZ equation solutions.

Findings

Established the almost sure asymptotic growth rate for the maximum of the solution.

Derived explicit formulas for the spatial asymptotics in terms of noise parameters.

Connected spatial asymptotics to the Cole-Hopf solution of the KPZ equation.

Abstract

Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225-2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483-533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation \[\cases{\displaystyle {\frac{\partial u}{\partial t}}(t,x)={\frac{1}{2}}\Delta u(t,x)+V(t,x)u(t,x),\cr u(0,x)=u_0(x),}\] 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 \[\lim_{R\to\infty}(\log R)^{-2/3}\log\max_{|x|\le R}u(t,x)={\frac{3}{4}}\root 3\of {\frac{2t}{3}}\qquad a.s.\] is obtained for the parabolic Anderson model $\partial_tu={\frac{1}{2}}\partial_{xx}^2u+\dot{W}u$ with the $(1+1)$-white noise $\dot{W}(t,x)$. In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.