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

TL;DR

This paper investigates the support properties of solutions to a stochastic heat equation driven by stable noise, establishing conditions under which solutions maintain compact support over time and providing formulas for their density and moments.

Contribution

It extends known results by characterizing the compact support property for solutions with stable noise, including a stochastic integral formula and moment bounds across dimensions.

Findings

Solutions with certain initial conditions retain compact support over time.

Conditions on b3 and  determine support preservation in different dimensions.

Derived a stochastic integral formula for the solution's density and established moment bounds.

Abstract

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.