Comparison principle for stochastic heat equations driven by $\alpha$-stable white noises

TL;DR

This paper establishes existence, uniqueness, and comparison principles for non-linear stochastic heat equations driven by $oldsymbol{ extit{ extalpha}}$-stable white noises, including non-negativity results under certain conditions.

Contribution

It introduces a novel approach using truncated $ extit{ extalpha}$-stable noises to prove existence and uniqueness, and establishes a comparison theorem for solutions.

Findings

Existence and pathwise uniqueness of solutions in $L^p$ spaces.

Comparison theorem for spectrally one-sided $ extit{ extalpha}$-stable noises.

Non-negativity of solutions with non-negative initial data.

Abstract

For a class of non-linear stochastic heat equations driven by $\alpha$-stable white noises for $\alpha\in(1,2)$ with Lipschitz coefficients, we first show the existence and pathwise uniqueness of $L^p$-valued c\`{a}dl\`{a}g solutions to such a equation for $p\in(\alpha,2]$ by considering a sequence of approximating stochastic heat equations driven by truncated $\alpha$-stable white noises obtained by removing the big jumps from the original $\alpha$-stable white noises. If the $\alpha$-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we prove a comparison theorem on the $L^2$-valued c\`{a}dl\`{a}g solutions of such a equation. As a consequence, the non-negativity of the $L^2$-valued c\`{a}dl\`{a}g solution is established for the above stochastic heat equation with non-negative initial function.