Loughborough University
Leicestershire, UK
LE11 3TU
+44 (0)1509 263171
Loughborough University

Loughborough University Institutional Repository

Please use this identifier to cite or link to this item: https://dspace.lboro.ac.uk/2134/21236

Title: Anticipating random periodic solutions. 1, SDEs with multiplicative linear noise
Authors: Feng, Chunrong
Wu, Yue
Zhao, Huaizhong
Keywords: Random periodic solutions
Periodic measures
Relative compactness
Malliavin derivative
Issue Date: 2016
Publisher: © Elsevier
Citation: FENG, C., WU, Y. and ZHAO, H., 2016. Anticipating random periodic solutions. 1, SDEs with multiplicative linear noise. Journal of Functional Analysis, 271 (2), pp. 365–417.
Abstract: In 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.
Description: 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.
Version: Accepted for publication
DOI: 10.1016/j.jfa.2016.04.027
URI: https://dspace.lboro.ac.uk/2134/21236
Publisher Link: http://dx.doi.org/10.1016/j.jfa.2016.04.027
ISSN: 0022-1236
Appears in Collections:Published Articles (Maths)

Files associated with this item:

File Description SizeFormat
FengWuZhao-Short-revised.pdfAccepted version832.3 kBAdobe PDFView/Open


SFX Query

Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.