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

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 càdlàg 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 càdlàg solutions of such a equation. As a consequence, the non-negativity of the $L^2$-valued càdlàg solution is established for the above stochastic heat equation with non-negative initial function.

Cited by in corpus (1)

• Existence of weak solutions to stochastic heat equations driven by truncated $\alpha$-stable white noises with non-Lipschitz coefficients