Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise

Abstract

The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter $H \in \left ( \frac 14, \frac 12 \right )$ in the space variable.