On uniqueness in law for parabolic SPDEs and infinite-dimensional SDEs

Richard F. Bass (University of Connecticut)
Edwin A. Perkins (University of British Columbia)


We give a sufficient conditions for uniqueness inlaw for the stochastic partial differential equation$$\frac{\partial u}{\partial t}(x,t)=\tfrac12 \frac{\partial^2 u}{\partial x^2}(x,t)+A(u(\cdot,t)) \dot W_{x,t},$$where $A$ is an operator mapping $C[0,1]$ into itself and $\dot W$ isa space-time white noise. The approach is to first prove uniquenessfor the martingale problem for the operator$$\mathcal{L} f(x)=\sum_{i,j=1}^\infty a_{ij}(x) \frac{\partial^2 f}{\partial x^2}(x)-\sum_{i=1}^\infty \lambda_i x_i \frac{\partial f}{\partial x_i}(x),$$where $\lambda_i=ci^2$ and the $a_{ij}$ is a positive definite boundedoperator in Toeplitz form.

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Pages: 1-54

Publication Date: May 26, 2012

DOI: 10.1214/EJP.v17-2049


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