Temporal asymptotics for fractional parabolic Anderson model

In this paper, we consider fractional parabolic equation of the form $\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=-(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}u+u\dot{W}(t,x)$, where $-(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}$ 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 $\mathrm{\mid }x{\mathrm{\mid }}^{-\beta }$ or ${\prod }_{j=1}^d\mathrm{\mid }{x}_j{\mathrm{\mid }}^{-{\beta }_j}$.