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) = ∆Y (t, x) + Y (t, x)^γ ˙_{L(t, x),} for (t, x) ∈ R₊ × R^d, 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.