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 $$\partial_t Y(t,x) = \Delta Y(t,x) + Y(t,x)^\gamma \dot{L}(t,x),$$ 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 $\gamma \in [1/\alpha,1)$ 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.