Comparison principle for stochastic heat equations driven by α-stable white noises

Abstract

For a class of non-linear stochastic heat equations driven by α-stable white noises for $\mathit{\alpha }\in (1,2)$ with Lipschitz coefficients, we prove the existence and pathwise uniqueness of $L^p$-valued càdlàg solution to such an equation for $p\in (\mathit{\alpha },2]$ by considering a sequence of approximating stochastic heat equations driven by truncated α-stable white noises obtained by removing the big jumps from the original α-stable white noise. If the α-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we further prove a comparison theorem on the $L^2$-valued càdlàg solutions to such an 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.