The compact support property for solutions to stochastic heat equations with stable noise

Abstract

We consider weak non-negative solutions to the stochastic partial differential equation \[ \partial_t Y(t,x) = ΔY(t,x) + Y(t,x)^γ\dot{L}(t,x), \] for $(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^d$, where $\gamma >0$ and $\dot{L}$ is a one-sided stable noise of index $\alpha \in (1,2)$. We prove that solutions with compactly supported initial data have compact support for all times if $\gamma \in (2-\alpha ,1)$ for $d=1$, and if $\gamma \in [1/\alpha ,1)$ in dimensions $d\in [2,2/(\alpha -1))\cap \mathbb{N}$. This complements known results on solutions to the equation with Gaussian noise. We also establish a stochastic integral formula for the density of a solution and associated moment bounds which hold in all dimensions for which solutions are defined.

arXiv comment63 pages. Final version

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