Precise Moment Asymptotics for the Stochastic Heat Equation of a Time-Derivative Gaussian Noise

Abstract

This article establishes the precise asymptotics

$$\mathbb{E}u^m(t,x)\;\;\;\;(t\rightarrow\infty\;{\rm{or}}\;m\rightarrow\infty)$$

for the stochastic heat equation

$$\frac{\partial{u}}{\partial{t}}(t,x)=\frac{1}{2}\Delta{u}(t,x)+u(t,x)\frac{\partial{W}}{\partial{t}}(t,x)$$

with the time-derivative Gaussian noise ${{\partial W} \over {\partial t}}(t,x)$ that is fractional in time and homogeneous in space.