Electronic Journal of Probability

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

Richard Bass and Edwin Perkins

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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.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 36, 54 pp.

Accepted: 26 May 2012
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

stochastic partial differential equations stochastic differential equ ations uniqueness perturbation Jaffard's theorem

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Bass, Richard; Perkins, Edwin. On uniqueness in law for parabolic SPDEs and infinite-dimensional SDEs. Electron. J. Probab. 17 (2012), paper no. 36, 54 pp. doi:10.1214/EJP.v17-2049. https://projecteuclid.org/euclid.ejp/1465062358

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