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On the spatial dynamics of the solution to the stochastic heat equation. (English) Zbl 1416.60063

Summary: 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 \(x\geq x_0, x_0\) 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.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
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