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