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

We consider weak non-negative solutions to the stochastic partial differential equation $${\mathrm{\partial }}_{t}Y(t,x)=\mathrm{\Delta }Y(t,x)+Y(t,x{)}^{\gamma }\dot{L}(t,x),$$ 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 $\gamma \in [1\mathrm{/}\alpha ,1)$ in dimensions $d\in [2,2\mathrm{/}(\alpha -1))\cap \mathbb{N}$. 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.