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Title: Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding
Authors: Feng, Chunrong
Zhao, Huaizhong
Keywords: Wiener-Sobolev compactness
Malliavin derivative
Differential equations
Integral equations
Random periodic solution
Issue Date: 2012
Publisher: © Elsevier
Citation: FENG, C.-R. and ZHAO, H.-Z., 2012. Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding. Journal of Functional Analysis, 262 (10), pp. 4377-4422.
Abstract: In this paper, we study the existence of random periodic solutions for semilinear SPDEs on a bounded domain with a smooth boundary. We identify them as the solutions of coupled forward-backward infinite horizon stochastic integral equations on L (D) in general cases. For this we use Mercer's Theorem and eigenvalues and eigenfunctions of the second order differential operators in the infinite horizon integral equations. We then use the argument of the relative compactness of Wiener-Sobolev spaces in C ([0, T], L (Ω×D)) and generalized Schauder's fixed point theorem to prove the existence of a solution of the integral equations. This is the first paper in literature to study random periodic solutions of SPDEs. Our result is also new in finding semi-stable stationary solution for non-dissipative SPDEs, while in literature the classical method is to use the pull-back technique so researchers were only able to find stable stationary solutions for dissipative systems.
Description: NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Functional Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Functional Analysis, 262 (10), pp. 4377-4422, URL: http://dx.doi.org/10.1016/j.jfa.2012.02.024.
Version: Accepted for publication
DOI: 10.1016/j.jfa.2012.02.024
URI: https://dspace.lboro.ac.uk/2134/15517
Publisher Link: http://dx.doi.org/10.1016/j.jfa.2012.02.024
ISSN: 0022-1236
Appears in Collections:Published Articles (Maths)

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