Almost Automorphic Problems In Stochastic Differential Equations

The theory of almost periodicity, by Danish mathematician H. Bohr, since founded in 1920's, has given a strong promotion to the development of harmonic analsis on groups and to the development of both topological and smooth dynamical systems. Owing to S. Bochner established the theory of almost periodic functions with values in Banach spaces. Favard, S. Bochner and Yoshizawa([36,37,14,103]) stududied a vast amount of resarch on almost periodic in ODE. Resently, stronly stimulated by applications and important extensions, Fink, Levitan, Pankov, G. Da Prato, C. Tudor, T. Morozan and P. H. Bezandry have been given to the studiy of almost periodic in PDE([38,39,60,61,83]) and SDE([96,97,24,25,26,12,13,71,72]).The concept of almost automorphy is a generalization of almost perodicity. It has been introduced in the literature by S. Bochner in relation to some aspects of differential geometry in 1955([15,16,17]). Foudamental propertis of almost automorphic functions on groups and abstract almost automorphic minimal flows were studied by S. Bochner, W.A.Veech, R. Terras and R. B. Basit ([10,14,15,16,18,101,99,100,98,95]). In the course of it, indicating somewhat complextiy and chaos were founded in symbolic dnamics.The study of almost automorphic dynamics in differential equations has been ig-nored for a long time, perhaps because it is not clear about the improtant effect of almost automorphy. Until the early 80's, several examples of almost periodic scalar ODE's were constructed by R. A. Johnson, in which the associated skew-product flows admit non-almost periodc almost automotphic, ergodic or non-erogic minimal sets([55,56]). A significant development of almost automorphy has been indicated by Johnson on almost periodic Floquet theory of two dimensional linear systems of ODE's, in which, he via an almost automorphic strong Perron transfomation to transform the original system into a canonical form ([54]). Recently, a series of research, by Shen and Yi, on almost periodic scalar parabolic equations in one dimension space were system-ically investigated. It was shown that all minimal sets in the associated skew-product semiflows are almost automorphic, and their issues such as properties ofω—limit sets, asymptotic behavior fo bounded solutions, hyperbolicity and stability, and ergodicity of a minimal set were all studied in ([88,89,90,91,92]).Since an epoch-making book "On Stochasic Differential Equations", edited by Ito, was appeared in 1951, stochastic analysis methods were applied to research the random factor which worked on the stochastic model. Such some authors transit their fields from stability of deteminated differential equations into SDE's.In this paper we introduce the concept of stochastic almost automorphic functions, and under some conditions of coefficients, we establish the existence and uniqueness of the stochastic almost automorphic solutions for some stochastic differential equaitons. Now let us introduc the main results of the paper.Almost atomorphic in distribtion senseDefinition 1 A continuous mappingμ:R→Pr(X), t→μt is called to be almost automorphic, if for every sequence of real numbers{sn'}, there exists a subsequence {sn} (?) {sn'} such that for some continuous mappingμ:R→Pr(X) hold for every t∈R.Definition 2 A stochastic continuous stochastic process x:R→C(R, X) is said to be almost automorphic in distribution sense, if the mapping is almost automorphic, whereμ(x(t)) is the distribution of x(t). That is ,if for every sequence of real numbers{sn'}(?)R,there exists a subsequence {sn)(?){sn'} such that for some process y and hold for every t∈R.The collection of all almost automorphic in distribution stochastic process x:R→Pr(Cb(R,X))is denoted by AA(R;Pr(Cb(R,X)).Theorem 1 If x,x1 and x2 are all almost automorphic in distribution stochastic process then (ⅰ)x1+x2 is almost automorphic in distribution sense; (ⅱ)λx is almost automorphic in distribution sense for every scalarλ.Definition 3 A jointly continuous funtions f:R×X→X,(t,x)→f(t,x)is said to be almost automorphic in distribution sense in t∈R for each x∈X,if for every sequence of real numbers{sn'},there exists a subsequence{sn},such that for some function g:R×X→X and for each t∈R and each x∈X.Theorem 2 Let,:R×X→X,(t,x)→f(t,x)be almost automorphic in distribution sense in t∈R for each x∈X,and f satisfies the Lipschitz conditions in the following sense: for all x, y∈X and each t∈R, where L> 0 is independent of t. Then for any almost automorphic in distribution sense stochastic process x:R→X, the stochastic process F:R→X given by F(t):= f(t,x(t)) is almost automorphic in distribution sense.The almost automorphic in distribution solutions of SDEFirstly, we consider the following autonomous SDE and where A is an infinitesimal generator which generates a C0-semigroup T(t), f,g are bounded and measurable stochastic processes, W(t)是is a two-sided standard one-dimensional Brown motion defined on the filtered probability space (Ω, F, P,Ft), where Ft=σ{W(u)-W(v);u,v≤t}.Theorem 1 If f,g∈AA(R:H) and the foudamental solutionΦof (0.0.8) is L2-exponential stable. Let x is the unique bounded solution of (0.0.8), and if{Po[x(t)]-1}t∈R is rela-tively compact in Pr(H). Then the mapping is almost automorphic.Definition 1 The unique almost automorphic in distribution solution xaa(t) of (0.0.8) is said to be stable in the distribution sense if for arbitaryε> 0, there existsδ> 0 such that whenever dBL{Poc-1,Po [xaa(0)]-1)<δ, where xc(t) stands for the solution of (0.0.8) with initial condition xc(0)=c. The solution xaa(t) is said to be asymptotically stable in the distribution sense if it is stble in the distribution sense and Theorem 2 Assume that the assumptions of Theoreml hold, then the unique almost automorphic in the distribution soltuion xaa(t) of (0.0.8) is asymptotically stable in the distribution sense.Theorem 3 Supose the foundamental solutionΦof (0.0.9) is L2-stable, f and g are all almost automorphic in the distribution stochastic process, which satisfy Lipschitz con-ditions in x. If x(t) is the unique L2-bounded solution of(0.0.9), and{Po [x(t)]-1}t∈R is relatively compact, then the mapping is almost automorphic.Theorem 4 Assume that the assumptions of Theorem 3 hold, then the unique almost automorphic in the distribution solution xaa(t) of (0.0.9) is asymptotically stable in the distribution sense. Now, we consider the nonautonomous SDE where A(t):D(A(t)) (?)H→H is a bounded linear operator in HI, and satisfies the dis-sipative conditions,f:R×H→H,(t, x)→f(t, x) and g:R×HI→L20, (t, x)→g(t, x) are all almost automorphic in the distribution functions, which satisty the boundedness and Lipschitz conditions in x.Let An(t)=nA(t)(nI-A(t))-1 is the Yosida approximation of A(t), we consider the following SDE and assume limn→∞fn(t, x)=f(t,x),limn→∞gn(t.x)= g(t,x).Theorem 5 Suppose A, An,f,fn, g, gn satisfy above conditions, x(t), xn(t) is the soltion of (0.0.10), (0.0.11), respectivley. and{Po [xn(t)]n∈N-1}is relatively compact. Then the mapping is almost automorphic.Almost atomorphic in square-mean senseDefinition 1 A stochastic continuous stochastic process x:R→L2(P,H) is said to be square-mean almost automophic, if every sequence of real numbers{s'n} has a subsequence{sn} such that for some stochastic process y:R→L2(P, H) hold for each t∈R. The collection of all square-mean almost automophic stochastic process x:R→L2(P,H) is denoted by AA(R; L2(P,H)). Lemma 1 If x, x1 and x2 are all square-mean almost automophic stochastic process, then(i) x1+x2 is square-mean almost automophic;(ii) Ax is square-mean almost automophic for every scalarλ;(iii) There exists a constant M> 0 such that supt∈R||x(t)||2≤M. That is, x is bounded in L2(P,H).Theorem 1 AA(R;L2(P,H)) is a Banach space when it is equipped with the norm for every x E AA(R:L2(P,H)).Definition 2 A function f:R×L2(P,H)→L2(P,H), (t,x)→f(t,x) which is jointly continuous, is said to be square-mean almost automophic in t R R for each x∈L2(P,H), if for every sequence of real numbers{s'n}, there exists a subsequence {sn} such that for some function f for each t∈R and each x∈L2(P, H). Theorem 2 Let f:R×L2(P,H)→L2(P,H),(t,x)→f(t,x)be square-mean almost automophic in t∈R for each x∈L2(P,H),and assume that f satisfies the Lipschitz condition in the following sense: for all x,y∈L2(P,H)and for each t∈R,where L>0 is independent of t.Then for any square-mean almost automophic process x:R→L2(P,H),the stochastic process F(t):=f(t,x(t))is square-mean almost automophic.square-mean almost automophic solutions of SDEFirstly,we consider the following autonomous SDE and where A is an infinitesimal generator which generates a C0-semigroup(T(t)t≥0)such that with K>0,ω>0.In addition,f,g are all stochastic process in L2(P,H),W(t) is a two-sided standard one-dimensional Brown motion defined on the filtered probability space(Ω,F,P,Ft),where Ft=σ{W(u)-W(v);u,v≤t}.Theorem 1 If f,g∈AA(R;L2(P,H)),then(0.0.12)has a unique square-mean almost automophic mild solution.Theorem 2 Assume f and g are all square-mean almost automophic process in in t∈R for each x∈L2(P,H).Moreover f and g satisfy the Lipschitz condition in the following sense: for constant L,L'≥0.Then(0.0.13)has a unique square-mean almost automophic mild solution,provided (2K2L)/(ω2)+(K2L')/ω<1.Definition 1 The unique square-mean almost automophic mild solution xaa(t)of (0.0.13)is said to be stable in square-mean sense if for arbitaryε>0,there existsδ>0 such that whenever‖c—xaa(0)‖2<δ,where xc(t)stands for the solution of(0.0.13)with initial condition xc(0)=c. The solution xaa(t)is said to be asymptotically stable in the square-mean sense,if it is stable in the square-mean sense andTheorem 3 Assume that the assumptions of Theorem 2 hold,then the unique square-mean almost automophic solution xaa(t)of(0.0.13)is asymptotically stable in the square-mean sense. Now,we consider the nonautunomous SDE where A(t):D(A(t))(?)H→H is a family of closed linear dense operator and satisfied "Acquistapace-Terrni" conditions, An(t)=nA(t)(nI-A(t))-1 is the Yosida approximation of A(t),and R(ω,A(·))∈AA(R,L(H)),f,g are all square-mean almost automophic.Theorem 4 Assume that the above conditons hold:and f,g are all satisfying the Lip-schitz conditions.then(0.0.14)has a unique square-mean almost automophic solution, provided (2K2L)/(δ2)+(K2L')/δ<1.