# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1706.02402