THE COMPACT SUPPORT PROPERTY FOR SOLUTIONS TO THE STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH COLORED NOISE

We study the compact support property for solutions of the following stochastic partial differential equations: partial derivative tu=aijuxixj(t,x)+biuxi(t,x)+cu+h(t,x,u(t,x))F-center dot(t,x),(t,x)is an element of(0,infinity)xRd,where F-center dot 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 = 0 has a compact support, then, under the reinforced Dalang's condition on F-center dot (which guarantees the existence and the H & ouml;lder continuity of a weak solution), all nonnegative weak solutions u(t,& sdot;) 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,infinity)xR