Temporal asymptotics for fractional parabolic Anderson model

TL;DR

This paper investigates the long-term behavior of solutions to a fractional parabolic Anderson model driven by Gaussian noise, deriving explicit Lyapunov exponents and critical parameters for moments of certain stochastic integrals.

Contribution

It provides the first precise calculation of moment Lyapunov exponents for solutions of fractional parabolic equations with space-time noise, using variational inequalities and large deviation principles.

Findings

Explicit formulas for Lyapunov exponents of solutions.

Identification of critical thresholds for moment finiteness.

Application of variational and large deviation techniques to fractional SPDEs.

Abstract

In this paper, we consider fractional parabolic equation of the form $ \frac{\partial u}{\partial t}=-(-\Delta)^{\frac{\alpha}{2}}u+u\dot W(t,x)$, where $-(-\Delta)^{\frac{\alpha}{2}}$ with $\alpha\in(0,2]$ 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 $\alpha$-stable process. As a byproduct, we obtain the critical values for $\theta$ and $\eta$ such that $\mathbb{E}\exp\left(\theta\left(\int_0^1 \int_0^1 |r-s|^{-\beta_0}\gamma(X_r-X_s)drds\right)^\eta\right)$ is finite, where $X$ is $d$-dimensional symmetric $\alpha$-stable process and $\gamma(x)$ is $|x|^{-\beta}$ or $\prod_{j=1}^d|x_j|^{-\beta_j}$.