The compact support property for solutions to the stochastic partial differential equations with colored noise

Abstract

We study the compact support property for solutions of the following stochastic partial differential equations: $${\partial }_{t}u=a^{ij}{u}_{x^i{x}^j}(t,x)+b^i{u}_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x),(t,x)\in (0,\infty )\times {\mathbf{R}}^d,$$ where $\dot{F}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $h(t,x,u)$ satisfies $K^{-1}|u{|}^{\lambda }\le h(t,x,u)\le K(1+|u|)$ for $\lambda \in (0,1)$ and $K\ge 1$. We show that if the initial data $u_0\ge 0$ has a compact support, then, under the reinforced Dalang's condition on $\dot{F}$ (which guarantees the existence and the Hölder continuity of a weak solution), all nonnegative weak solutions $u(t,\cdot )$ have the compact support for all $t>0$ with probability 1. Our results extend the works by Mueller-Perkins [Probab. Theory Relat. Fields, 93(3):325--358, 1992] and Krylov [Probab. Theory Relat. Fields, 108(4):543--557, 1997], in which they show the compact support property only for the one-dimensional SPDEs driven by space-time white noise on $(0,\infty )\times \mathbf{R}$.

39pages. In v3 we have corrected some errors from v1. We mainly fixed Theorem 3.1 and Lemma 3.1 with a modification of the solution space (the equation (2.10) in v3). We added additional comments from v2 where the manuscript is the same as in v3