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On the spatial dynamics of the solution to the stochastic heat equation


 
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1. Title Title of document On the spatial dynamics of the solution to the stochastic heat equation
 
2. Creator Author's name, affiliation, country Sigurd Assing; University of Warwick; United Kingdom
 
2. Creator Author's name, affiliation, country James Bichard; University of Warwick; United Kingdom
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) stochastic partial differential equation, enlargement of filtration, Brownian sheet, Gaussian analysis
 
3. Subject Subject classification 60H15; 60H30
 
4. Description Abstract 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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7. Date (YYYY-MM-DD) 2013-07-28
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2797
 
10. Identifier Digital Object Identifier 10.1214/EJP.v18-2797
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 18
 
12. Language English=en en
 
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