Temporal asymptotics for fractional parabolic Anderson model

Abstract

In this paper, we consider fractional parabolic equation of the form $ \frac{\partial u}{\partial t}=-(-Δ)^{\fracα{2}}u+u\dot W(t,x)$, where $-(-Δ)^{\fracα{2}}$ with $α\in(0,2]$ is a fractional Laplacian and $\dot W$ is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by $α$-stable process. As a byproduct, we obtain the critical values for $θ$ and $η$ such that $\mathbb{E}\exp\left(θ\left(\int_0^1 \int_0^1 |r-s|^{-β_0}γ(X_r-X_s)drds\right)^η\right)$ is finite, where $X$ is $d$-dimensional symmetric $α$-stable process and $γ(x)$ is $|x|^{-β}$ or $\prod_{j=1}^d|x_j|^{-β_j}$.