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 $\prod_{ j=1}^d|x_j|^{ -\beta_j}$.
Categories: math.PR