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

TL;DR

This paper investigates the behavior of solutions to a class of non-linear stochastic heat equations driven by space-time white noise and harmonic functions, establishing conditions for existence, uniqueness, and exponential growth bounds.

Contribution

It provides new conditions for existence and uniqueness of solutions and quantifies their weak growth rates over time for specific Lévy process generators.

Findings

Solutions grow at most exponentially in time under certain conditions.

Explicit growth bounds are derived for the case of alpha-stable process generators.

The study extends understanding of stochastic heat equations with harmonic function coefficients.

Abstract

A study of a non-linear parabolic SPDEs of the form $\partial_{t}u=\mathcal{L}\,u + \sigma(u)f(B_t^x,t)\dot{w}$ with $\dot{w}$ as the space-time white noise and $f(B_t^x,t)$ a space-time harmonic function was done. The function $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and $\mathcal{L}$ the $L^2$-generator of a L\'{e}vy 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 $t$) at most a precise exponential rate for the $\mathcal{L}$; and grows in time at most a precise exponential rate for the case of $\mathcal{L}=-(-\Delta)^{\alpha/2},\,\,\alpha\in(1,2]$ generator of an alpha-stable process.