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

We consider non-negative weak solutions to the stochastic partial differential equation ∂tY(t,x)=ΔY(t,x)+Y(t,x)γL˙(t,x), for (t,x)∈R+×Rd, where γ>0 and L˙ is a one-sided white stable noise of index α∈(1,2). We prove that solutions with compactly supported initial data have compact support for all times if γ∈(2−α,1) for d=1, and if γ∈[1∕α,1) in dimensions d∈[2,2∕(α−1))∩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.