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: $$\partial_t u = a^{ij}u_{x^ix^j}(t,x)+b^{i}u_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x),\quad (t,x)\in (0,\infty)\times{\bf{R}}^d,$$ where $\dot{F}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $h(t, x, u)$ satisfies $K^{-1}|u|^{\lambda}\leq h(t, x, u)\leq K(1+|u|)$ for $\lambda\in(0,1)$ and $K\geq 1$. We show that if the initial data $u_0\geq 0$ has a compact support, then, under the reinforced Dalang's condition on $\dot{F}$ (which guarantees the existence and the H\"older 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 $(0, \infty)\times \bf{R}$.