Summary
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.
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The author's research was supported by an NSF grant and an NSERC operating grant
The author's research was supported by an NSERC operating grant
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Mueller, C., Perkins, E.A. The compact support property for solutions to the heat equation with noise. Probab. Th. Rel. Fields 93, 325–358 (1992). https://doi.org/10.1007/BF01193055
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DOI: https://doi.org/10.1007/BF01193055