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Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients
journal contributionposted on 2014-07-21, 10:18 authored by Qi Zhang, Huaizhong Zhao
We prove a general theorem that the L (R ; R) ⊗ L (R ; R)-valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the L (R ; R) ⊗ L (R ; R)-valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.
Q.Z. would like to acknowledge the financial support of the National Basic Research Program of China (973 Program) with Grant No. 2007CB814904.
- Mathematical Sciences
Published inJournal of Differential Equations
Pages953 - 991
CitationZHANG, Q. and ZHAO, H.-Z., 2010. Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients. Journal of Differential Equations, 2010 (5), pp. 953-991.
Publisher© Elsevier Inc.
- AM (Accepted Manuscript)
NotesNOTICE: this is the author’s version of a work that was accepted for publication in Journal of Differential Equations. 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 at http://dx.doi.org/10.1016/j.jde.2009.12.013.