Temporal asymptotics for fractional parabolic Anderson model

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 ${\prod }_{j=1}^d|x_j{|}^{-{\beta }_j}$.

References in corpus (1)

• The Continuum Directed Random Polymer