The compact support property for solutions to the heat equation with noise

Abstract

We consider all solutions of a martingale problem associated with the stochastic pde $$u_t = \tfrac{1}{2}u_{xx} + u^\gamma \dot W$$ and show thatu(t,·) has compact support for allt≧0 ifu(0,·) does and if γ<1. This extends a result of T. Shiga who derived this compact support property for γ≦1/2 and complements a result of C. Mueller who proved this property fails if γ≧1.