TY - JOUR

T1 - Anticipating random periodic solutions-I. SDEs with multiplicative linear noise

AU - Feng, Chunrong

AU - Wu, Yue

AU - Zhao, Huaizhong

N1 - Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/7/15

Y1 - 2016/7/15

N2 - 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,Lloc2(omega)) 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.

AB - 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,Lloc2(omega)) 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.

KW - Malliavin derivative

KW - Periodic measures

KW - Random periodic solutions

KW - Relative compactness

UR - http://www.scopus.com/inward/record.url?scp=84965104344&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2016.04.027

DO - 10.1016/j.jfa.2016.04.027

M3 - Article

AN - SCOPUS:84965104344

SN - 0022-1236

VL - 271

SP - 365

EP - 417

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -