| Título: | On the spatial dynamics of the solution to the stochastic heat equation |
| Autores: |
Assing, Sigurd; University of Warwick Bichard, James; University of Warwick |
| Fecha: | 2013-01-04 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: | Peer-reviewed Article |
| Tema: |
stochastic partial differential equation, enlargement of filtration, Brownian sheet, Gaussian analysis 60H15; 60H30 |
| Descripción: | 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. |
| Idioma: | Inglés |
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