Asymptotic Behavior for High Moments of the Fractional Heat Equation with Fractional Noise

Abstract

In this paper, we investigate the large time behavior of the solution to the fractional heat equation

$$\begin{aligned} \frac{\partial u}{\partial t}(t,x)=-(-\Delta )^{\beta /2}u(t,x)+u(t,x)\frac{\partial ^{d+1} W}{\partial t\partial x_1\cdots \partial x_d},\quad t>0,\quad x\in \mathbb {R}^d, \end{aligned}$$

where $\beta \in (0,2)$ and the noise W(t, x) is a fractional Brownian sheet with indexes $H_0, H_1,\ldots ,H_d\in (\frac{1}{2},1)$. By using large deviation techniques and variational method, we find a constant $M_1$ such that for any integer $p\ge 1$ and $\alpha _0\beta +\alpha <\beta ,$

$$\begin{aligned} \lim _{t\rightarrow \infty }t^{-\frac{2\beta -\beta \alpha _0-\alpha }{\beta -\alpha }} \log Eu(t,x)^p=p^{\frac{2\beta -\alpha }{\beta -\alpha }}\left( \frac{\alpha _H}{2} \right) ^{\frac{\beta }{\beta -\alpha }}M_1, \end{aligned}$$

where $\alpha _0=2-2H_0$, $\alpha =\sum \nolimits _{j=1}^d(2-2H_j)$ and $\alpha _H=\prod \nolimits _{i=0}^dH_i(2H_i-1)$.