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

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

\[\cases{{\frac{\partial u}{\partial t}}(t,x)={\frac{1}{2}}\Delta u(t,x)+V(t,x)u(t,x),\cr u(0,x)=u_{0}(x),}\] 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

\[\lim_{R\to\infty}(\log R)^{-2/3}\log\max_{|x|\le R}u(t,x)={\frac{3}{4}}\root 3\of{\frac{2t}{3}}\qquad\mbox{a.s.}\] is obtained for the parabolic Anderson model $\partial_{t}u={\frac{1}{2}}\partial_{xx}^{2}u+\dot{W}u$ 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.