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

DOI10.1214/EJP.V18-2797zbMATH Open1416.60063arXiv1305.3325MaRDI QIDQ388954

Sigurd Assing, James Bichard

Publication date: 17 January 2014

Published in: Electronic Journal of Probability (Search for Journal in Brave)

Abstract: We consider the solution of $partia{l}_{t}u=partia{l}_x^{2}u+partia{l}_{x}partia{l}_{t}B,,(x,t)inRimes(0,infty)$, subject to $u(x,0)=0,,xinR$, where $B$ is a Brownian sheet. We show that $u$ also satisfies $partia{l}_x^{2}u+[,(-partia{l}_t^2{)}^{1/2}+sqrt2partia{l}_x(-partia{l}_t^2{)}^{1/4},],u^a=partia{l}_{x}partia{l}_{t}ildeB$ in $Rimes(0,infty)$ where $u^a$ stands for the extension of $u(x,t)$ to $(x,t)in{R}^2$ which is antisymmetric in $t$ and $ildeB$ is another Brownian sheet. The new SPDE allows us to prove the strong Markov property of the pair $(u,partia{l}_{x}u)$ when seen as a process indexed by $xge{x}_0$, $x_0$ 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.

Full work available at URL: https://arxiv.org/abs/1305.3325

zbMATH Keywords

Gaussian analysisBrownian sheetstochastic partial differential equationenlargement of filtration

Mathematics Subject Classification ID

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

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