## 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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