The Existence Of Pathwise Random Periodic Solutions Of Stochastic Differential Equations

Periodic solutions have been a central concept in the theory of the deterministic dynam-ical system for over a century starting from Poincare’s sem-inal work [18]. They have been studied for many important problems arising in numerous physical problems, e.g. van der Pol equations [20], Lienard equations [13].Now after over a century,Periodic behavior arises naturally in many real world prob-lems e.g. in biological, environmental and economic systems. But these problems are often subject to random perturbations or under the influence of noises. Needless to say, for random dynamical systems, to study the pathwise random periodic solutions is of great importance.Zhao and Zheng [3] started to study the problem and gave a definition of the pathwise random periodic solutions for C1-cocycles. It is well known that in the deterministic case, the most powerful method to prove the existence of the periodic solution is to study the fixed point of the Poincaremap. However, for random dynamical systems, it is very difficult.Feng, Zhao and Zhou [2] study the stochastic periodic solutions of periodic stochastic differential equations in Rn: Denoteâ–³:={{t, s)∈R2, s≤t}. This equation generates a semi-flow Y:△×Rn×Ωâ†'Rn. when the solution exists uniquely. Here W is a two-sided Brownian motion on a probability space (Ω,(?),P). Define θ:(-∞,∞)×Ω,â†'Ω,θtω(s)=W(t+s)-W(t). Therefore (Ω,(?) P,(θt)t∈R) is a metric dynamical system. Letting A be a n×n matrix, and A is hyperbolic.Assume:There exists a constant Ï„>0such that for any t∈R, Y∈Rn F(t, Y)=F(t+Ï„,Y), B0(t)=B0(t+Ï„). Definition A random periodic solution of period Ï„ of a semi-flow Y:△×Rn×Ωâ†'Rn is an (?)-measurable map φ:(-∞,∞)×Ωâ†'Rn such thatFeng, Zhao and Zhou [2] proof the existence of stochastic periodic solutions of (E)’is equivalent to the existence of stochastic periodic solutions of a Coupled forward-backward infinite horizon stochastic integral equations and random periodic solutions Further more, apply generalized Schauder’s fixed point theorem proof the existence of stochastic periodic solutions of a Coupled forward-backward infinite horizon stochastic integral equations and random periodic solutions, finally proof the existence of stochastic periodic solutions of (E)’This paper mainly studies the following stochastic periodic solutions of periodic s-tochastic differential equations Here A(t) is a continuous matrix function in (-∞,∞), f(t,X) is a continuous map, W is a two-sided Brownian motion on a probability space (Ω,(?), P). B(t) is a n x m matrix function. satisfy:(H1) There exists a constant Ï„>0such that A(t+Ï„)=A(t), f(t+Ï„,X)=f(t,X), B(t+t)=B(t),(?) t∈R,X∈Rn.(H2) The real part of Characteristic exponent of A(t) is not zero.(H3) Let f(t, X) be a continuous map, f(t, X) andâ–½f(t, X) is globally bounded.(H4) For any t1, t2, there exists a constant M>0, such that||B(t1)-B(t2)||2<M|t1-t2|In the third chapter give the main conclusions, following: Theorem1Assume Φ(t) is the fundamental matrix of the following linear periodic stochastic differential equations dx=A(t)xdt (LHP) There exists a t periodic non-singular matrix functions P(t), a constant matrix R, such that Φ(0=P(t)etR.further more, under the transformation X=P(t) Y,(E) translate into the following stochastic differential equation: dY(t)=-AY(t)dt+F(t, Y(t))dt+B0(t)dW(t),(E)’ Here A=-R, R is a constant matrix. F(t, Y)=p-1(t)f(t, P(t)Y), B0(t)=P-1(t)B(t).Theorem2Assume that (H1),...,(H4), then we have (E) has a stochastic periodic solutions.