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Anticipating random periodic solutions—I. SDEs with multiplicative linear noise
journal contribution
posted on 2016-05-16, 11:05 authored by Chunrong Feng, Yue Wu, Huaizhong ZhaoIn this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify them as solutions of coupled forward–backward infinite horizon stochastic integral equations (IHSIEs), using the “substitution theorem” of stochastic differential equations with anticipating initial conditions. In general, random periodic solutions and the solutions of IHSIEs, are anticipating. For the linear noise case, with the help of the exponential dichotomy given in the multiplicative ergodic theorem, we can identify them as the solutions of infinite horizon random integral equations (IHSIEs). We then solve a localised forward–backward IHRIE in C(R, L²loc (Ω)) using an argument of truncations, the Malliavin calculus, the relative compactness of Wiener–Sobolev spaces in C([0,T],L²(Ω)) and Schauder's fixed point theorem. We finally measurably glue the local solutions together to obtain a global solution in C(R,L²(Ω)). Thus we obtain the existence of a random periodic solution and a periodic measure.
History
School
- Science
Department
- Mathematical Sciences
Published in
Journal of Functional AnalysisVolume
271Issue
2Pages
365 - 417Citation
FENG, C., WU, Y. and ZHAO, H., 2016. Anticipating random periodic solutions—I. SDEs with multiplicative linear noise. Journal of Functional Analysis, 271 (2), pp. 365–417.Publisher
© ElsevierVersion
- AM (Accepted Manuscript)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Acceptance date
2016-04-27Publication date
2016-05-09Notes
This paper was accepted for publication in the journal Journal of Functional Analysis and the definitive published version is available at http://dx.doi.org/10.1016/j.jfa.2016.04.027.ISSN
0022-1236Publisher version
Language
- en