Stochastic heat equations with non-Lipschitz coefficients driven by symmetric $\alpha$-stable L\'evy white noise

Abstract

We study the stochastic heat equation on a bounded interval with zero Dirichlet boundary conditions, driven by symmetric $\alpha$-stable space--time L\'evy white noise, where $1<\alpha<2$. Both the drift and noise coefficients are continuous and have at most linear growth. For bounded deterministic initial data and every $r>1/2$, we construct a weak solution with c\`adl\`ag paths in the negative Sobolev space $\mathbb{H}_{-r}$. We also construct a predictable mild solution on the same stochastic basis and with the same noise. At each fixed deterministic time, this mild solution is a density of the distribution-valued weak solution.