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

TL;DR

This paper establishes the existence of weak solutions for a class of stochastic heat equations driven by truncated lpha-stable white noises with non-Lipschitz coefficients, demonstrating different solution types depending on the stability index.

Contribution

It proves the existence of weak solutions in different spaces for stochastic heat equations with non-Lipschitz coefficients driven by truncated lpha-stable noises, extending previous results.

Findings

Weak solutions exist in measure-valued and functional forms.

Finite uniform p-th moments for solutions with 0<p<5/3.

Solutions exhibit stochastic continuity and flow properties.

Abstract

We consider a class of stochastic heat equations driven by truncated $\alpha$-stable white noises for $1<\alpha<2$ with noise coefficients that are continuous but not necessarily Lipschitz and satisfy globally linear growth conditions. We prove the existence of weak solution, taking values in two different spaces, to such an equation using a weak convergence argument on solutions to the approximating stochastic heat equations. For $1<\alpha<2$ the weak solution is a measure-valued c\`{a}dl\`{a}g process. However, for $1<\alpha<5/3$ the weak solution is a c\`{a}dl\`{a}g process taking function values, and in this case we further show that for $0<p<5/3$ the uniform $p$-th moment for $L^p$-norm of the weak solution is finite, and that the weak solution is uniformly stochastic continuous in $L^p$ sense and satisfies a flow property.