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 ${\hat{I}}^3>0$ and $\dot{L}$ is a one-sided stable noise of index $\hat{I}\pm \in (1,2)$. We prove that solutions with compactly supported initial data have compact support for all times if ${\hat{I}}^3\in (2-\hat{I}\pm ,1)$ for $d=1$, and if ${\hat{I}}^3\in [1/\hat{I}\pm ,1)$ in dimensions $d\in [2,2/(\hat{I}\pm -1))\cap \mathbb{N}$. 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.

63 pages. Final version