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 \[\cases{\displaystyle {\frac{\partial u}{\partial t}}(t,x)={\frac{1}{2}}Δu(t,x)+V(t,x)u(t,x),\cr u(0,x)=u_0(x),}\] 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 ${\partial }_{t}u=\frac{1}{2}{\partial }_{xx}^{2}u+\dot{W}u$ 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.

Published at http://dx.doi.org/10.1214/15-AOP1006 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Quenched asymptotics for Brownian motion in generalized Gaussian potential
• Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks

Cited by in corpus (3)

• Intermittency for the wave and heat equations with fractional noise in time
• A macroscopic multifractal analysis of parabolic stochastic PDEs
• Stochastic heat equations driven by space-time $G$-white noise under sublinear expectation