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On the spatial dynamics of the solution to the stochastic heat equation
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Assing, Sigurd and Bichard, James (2013) On the spatial dynamics of the solution to the stochastic heat equation. Electronic Journal of Probability, Volume 18 . pp. 1-32. Article number 70. doi:10.1214/EJP.v18-2797 ISSN 1083-6489.
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Official URL: http://dx.doi.org/10.1214/EJP.v18-2797
Abstract
We consider the solution of partial derivative(t)u = partial derivative(2)(x) u + partial derivative(x)partial derivative(t) B, (x, t) is an element of R x (0; infinity), subject to the initial condition u (x, 0) = 0; x is an element of R, where B is a Brownian sheet. We show that u also satisfies partial derivative(2)(x) u + [(-partial derivative(2)(t))(1/2) + root 2 partial derivative(x) (-partial derivative(2)(t))(1/4)]u(a) = partial derivative(x)partial derivative(t) (B) over tilde in R x (0; 1) where u(a) stands for the extension of u (x; t) to (x; t) is an element of R-2 which is antisymmetric in t and (B) over tilde is another Brownian sheet. The new SPDE allows us to prove the strong Markov property of the pair (u; partial derivative(x)u) when seen as a process indexed by x >= 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.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||||
Library of Congress Subject Headings (LCSH): | Stochastic partial differential equations | ||||||||
Journal or Publication Title: | Electronic Journal of Probability | ||||||||
Publisher: | University of Washington. Dept. of Mathematics | ||||||||
ISSN: | 1083-6489 | ||||||||
Official Date: | 28 July 2013 | ||||||||
Dates: |
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Volume: | Volume 18 | ||||||||
Page Range: | pp. 1-32 | ||||||||
Article Number: | Article number 70 | ||||||||
DOI: | 10.1214/EJP.v18-2797 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 26 December 2015 | ||||||||
Date of first compliant Open Access: | 26 December 2015 |
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