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2013 On the spatial dynamics of the solution to the stochastic heat equation
Sigurd Assing, James Bichard
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Electron. J. Probab. 18: 1-32 (2013). DOI: 10.1214/EJP.v18-2797


We consider the solution of $\partial_t u=\partial_x^2u+\partial_x\partial_t B,\,(x,t)\in\mathbb{R}\times(0,\infty)$, subject to $u(x,0)=0,\,x\in\mathbb{R}$, where $B$ is a Brownian sheet. We show that $u$ also satisfies $\partial_x^2 u +[\,( \partial_t^2)^{1/2}+\sqrt{2}\partial_x( \partial_t^2)^{1/4}\,]\,u^a=\partial_x\partial_t{\tilde B}$ in $\mathbb{R}\times(0,\infty)$ where $u^a$ stands for the extension of $u(x,t)$ to $(x,t)\in\mathbb{R}^2$ which is antisymmetric in $t$ and $\tilde{B}$ is another Brownian sheet. The new SPDE allows us to prove the strong Markov property of the pair $(u,\partial_x u)$ when seen as a process indexed by $x\ge x_0$, $x_0$ fixed, taking values in a state space of functions in $t$. The method of proof is based on enlargement of filtration and we discuss how our method could be applied to other quasi-linear SPDEs.


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Sigurd Assing. James Bichard. "On the spatial dynamics of the solution to the stochastic heat equation." Electron. J. Probab. 18 1 - 32, 2013.


Accepted: 28 July 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 06247239
MathSciNet: MR3091716
Digital Object Identifier: 10.1214/EJP.v18-2797

Primary: 60H15
Secondary: 60H30

Keywords: Brownian sheet , Enlargement of filtration , Gaussian analysis , Stochastic partial differential equation

Vol.18 • 2013
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