Weak Moment of a Class of Stochastic Heat Equation with Martingale-valued Harmonic Function

Abstract

A study of a non-linear parabolic SPDEs of the form with  as the space-time white noise and a space-time harmonic function was done. The function  is Lipschitz continuous and  the  -generator of a Lévy process . Some precise condition for existence and uniqueness of the solution were given and we show that the solution grows weakly(in law/distribution) in time (for large ) at most a precise exponential rate for the  ; and grows in time at most a precise exponential rate for the case of  generator of an alpha-stable process .