Some Problems About Almost Periodic Solutions For Stochastic Differential Equations

In this paper, we research about the existence of solutions to stochastic differen-tial equations almost with periodic distribution. Bohr in 1924-1926 [19-21] founded the theory of almost periodic functions. In the early stage of the theory, researchers concentrated on the Fourier series theory of almost periodicity, untill they found that almost periodicity can perfectly discribe some kinds of physical phenomenon. From then on many people studied about almost periodic solutions of ordinary or functional differential equations. Their results could be found in Levitan and Zhikov’s book [63], Fink’s [39], and Yoshizawa’s [102]. Nowadays, stochastic differential equations are widely applied in many fields such as physics, economics, chemistry and so on. More and more researchers become curious about the almost periodic solutions for stochas-tic differential systems. As far as we know, till now only the fixed point method has been applied to investigate the existence of almost periodic solutions for stochastic differential equations (see Halanay [43], Morozan and Tudor [71], Da Prato and Tu-dor [29], and so on). But for deterministic systems, Favard [33,34] separation method, Amerio [1,2] semi-separation method and the stability method have perfectly given sufficient conditions to the existence of almost periodic solutions for ordinary differen-tial equations (see Fink [38], Seifert [77-81], Coppel [26], Miller [70]). We are curious of the question that, whether the classical Favard separation method, semi-separation method and stability method above could hold when the system is perturbed by white noise. And we are also curious about whether the stability in deterministic systems are still inherited properties in stochastic ones. In the first chapter, we introduced some preliminary knowledge. In Chapter 2, we successfully developed the stochastic Favard separation method and Amerio semi-separation method. We hope that, in stochastic systems, there are also inherited stabilities similar to ones Fink, Coppel and Miller have discussed, but the fact is negative. We can’t assure that the limits of distributions of stochastic differential equations’ solutions would keep the stabilities. So in Chapter 3, we get the stabilities of hull equations’ solutions and the existence of solutions almost periodic in distribution by giVing some inherited requirement to Lyapunov functions, and developed the stability method for almost periodic problems of stochastic systems.Throughout this paper,we denote(M,ρ)as a complete metric space,assume that (Ω,F,P)is a probability space,and the space L2(P,Rd)stands for the space of all Rd-valued random variables Y such that E|Y|=∫Ω|Y|2dP<∞, whose norm is denoted by |Y|2:=(∫Ω|Y|2dP)1/2. A stochastic process X:R�'L2(P,Rd)is said to be L2-bounded if |X(t)|∞:=(?)sup |X(t)|2<∞. For function f:R�'M,we denote R(f):={f(x)；(?)x∈R}. Besides,for function f(t)and sequence α={αn},if the limit limn�'∞ f(t+α)exists, we denote the limit as Tαnf(t).Consider the Ito stochastic differential equation on Rd dX=f(t,X(t))dt+g(t,X(t))dW, (A) where f(t,x)is an Rd-valued uniformly almost periodic function,g(t,x)is a(d×m)-matrix-valued uniformly almost periodic function,and W is a standard m-dimensional Brownian motion.For given r>0,let |·|∞:=supt|·|2,we have introduce the following notations:Br:={X∈L2(P,Rd):|X|2≤r}, Dr:={μ∈P(Rd):∫Rd|x|2dμ(x)≤r2},Br(A)=Br(f,g):={X(·):(X,W)weakly solves equation(f,g)on R on some filtered probability space for some W and |X|∞≤r},Dr(A)=Dr(f,g):={μ:μ(·)=L(X(·))for some X∈Br(f,g)}.Assume that f,g are uniformly almost periodic functions,sequence α satisfying that,Tαf=,Tαg uniformly on R×S,for compact set S(?)R.We call(Tαf,Tαg)as a hull of SDE(f,g),some times we denote it as(f,g)∈H(f,g)for short.Otherwise,if (f,g)are nonhomogeneous,that is,then are called homogeneous hull equations of (B) or (f, g).The main results of this paper are as follows.1. Favard separation method and solutions almost periodic in distribu-tionDefinition 1.1) The triple (X’, W’), (Ω’, F’, P’),{Ft’:t ∈ R} is a weak solution of (A) if(Ω’, F’, P’) is a probability space and{Ft’:t ∈ R} is a filtration of sub-σ-algebras of F’, W’={W’(t):t ∈ R} is an Ft’-adapted m-dimensional Brownian motion and X’={X’(t):t ∈ R} is an Ft’-adapted d-dimensional process such that X’(t)=X’(s)+∫stf(r,X’(r))dr+∫st(r, X’(r))dW’(r) for all t≥s and each s∈R almost surely.2) The weak solution (X’, W’), (Ω’, F’, P’),{Ft’ t ∈ R} is a strong solution if for given t0 ∈ R, there exists a measurable function h such that X’(·)= h(X’(to), W’(·)) on R almost surely. For strong or weak solutions on the positive real line, see [45] or [53] for details.We often assume (B)’coefficients satistying following condition: (H)f is Re-valued, g is (d×m)-matrix-valued, they’re both uniformly almost periodic functions, and W is a standard m-dimensional Brownian motion. Furthermore, f, g satisfy the conditions of global Lipschitz and linear growth with common Lipschitz and linear growth constants; that is, there are constants L and K, independent of t∈R, such that for all x, y ∈ Rd |fn(t,x)-fn(t,y)|∨|gn(t,x)-gn(t,y)|≤L|x-y|, (a) |fn(t,x)|∨|gn(t,x)|≤K(1+|x|). (b)For the Cauchy problem of (A) on the positive real line, it is well-known that the pathwise uniqueness implies uniqueness in the sense of probability law on the path space which we simply call "weak uniqueness", see e.g. [45,.§V.1]. In the meantime, we note that the weak uniqueness implies the uniqueness of law on Rd, and we always suppose that the coefficients of (A) satisfy the condition (a), so for any random variable Y, and Brownian motion W, (A) always admits a strong solution with initial value Y and Brownian motion W, hence we don’t distinguish strong solutions from weak one in following discussion.Definition 2.(1) A continuous function f:R�' M is called (Bochner) almost periodic if for any sequence a’, there exists a subsequence a (?)α’ such that Tαf exists uniformly on R.(2)) We say a set T(ε, f) is relatively dense on R if there is a number l= l(V )> 0 such that (a, a+l)n T(ε,f)≠(?) for any a E R. The set T(ε,f) is called the set of ε-almost periods of f.(3) A continuous function f:R �' M is called (Bohr) almost periodic if for any given ε> 0, the set is relatively dense on R.(4) For a given function f:R x D �'Rd almost periodic in t uniformly for x E D, the hull of f is defined as follows: H(f):={g:there exists a sequence a such that Tαf= g uniformly on R x S for every compact set S(?)D}.Definition 3. A continuous function f:R+ �'M is asymptotically almost periodic if there exists an almost periodic function p:R �' M such thatThe function p is called the almost periodic part of f. The asymptotically almost pe-riodic function on R_ is defined similarly.Definition 4 For some r> 0, if Br(A)≠(?), then constant is called the minimal value of (A). And if X0 ∈Br(A) with |X0|∞=λ, then X0 is a minimal solution of (A).Definition 5. We say that the Favard (separation) condition in the sense of stochastic differential equations or L2-norm holds for (B) if for any homogeneous hull equation corresponding to (B) every nontrivial L2-bounded (weak) solution X of (B*) on R satisfies inft∈R|X(t)|2> 0.Theorem 1. Consider a family of Ito stochastic equations on Rd: dX=fn(t, X)dt+gn(t, X)dW, n=1,2,…,where fn are Rd-valued, gn are (d×m.)-matrix-valued, and W is a standard m-dimensional Brownian motion. Assume that fn,gn satisfy the conditions of global Lipschitz and lin-ear growth with common Lipschitz and linear growth constants. Assume that fn �'f, gn�'g pointwise on R × Rd as n �' ∞ and that Xn ∈ Br0(fn,gn) for some constant r0 independent of n. Then there is a subsequence of {Xn} which converges in distribution, uniformly on compact intervals, to some X ∈Br0(f,g) given by the limit equation: dX= f(t, X)dt+g(t, X)dW.Theorem 2. Consider (2.1.1) with almost periodic coefficients. Assume that (B) admits an L2-bounded solution and that the Favard condition holds for (B). Then (B) admits an almost periodic in distribution solution.2. Amerio separation method and solutions almost periodic in distri-butionTheorem 3. Consider the linear equation on Rd dX= (AX+f(t))dt+g(t)dW, (B1)where A is a constant matrix, f, g are almost periodic, and W is a given m-dimensional Brownian motion. If X is a strong L2-bounded solution of (B1) on R so that X(τ) -X0(τ) is independent of X0(τ) and W for some τ ∈R, where X0 is the strong minimal solution of (B’). Then X is almost periodic in distribution.Lemma 1. Consider (A) and assume (H).If(A) admits an L2-bounded solution X on R with distribution asymptotically almost periodic on R+, then (A) admits a solution Y on R which is almost periodic in distribution such that lim(?) ρ(L(X(t)),L(Y(t)))= 0 and |Y|∞≤|X|∞. In particular, L(Y) is the almost periodic part of L(X). The similar result holds when X is asymptotically almost periodic in distribution on R_Theorem 4. Consider (A). Assume (H) and that the Amerio positive (or negative) semi-separation condition holds for (A) in Dr. Then |Dr(A)| is finite. If Br(A) is non-empty, then it consists of almost periodic in distribution solutions of (A).Corollary 1. Consider (A). Assume (H) and that each hull equation (f,g) of (A) admits a unique distribution in Dr, i.e. Dr(f,g) 9 only has one single element. Then Br(A) consists of almost periodic in distribution solutions with the unique common distribution in Dr(1.4.1)3. Solutions almost periodic in strong distribution senseFor any solution X of (A), it determines a distribution on C(R, Rd). Denote the shift mapping A:R�'P{C(R, Rd)), t �' μ(t):=L(X(t+·)),where P(C(R, Rd)) stands for the space of probability measures on the path space and L(X(t+·)) means the law of the C(R, Rd)-valued random variable X(t+ ·) on space C(R,Rd). Note that P(C(R,Rd)) is a separable complete metric space (see, e.g. [74, Chapter II, Theorems 6.2 and 6.5]). The solution X is said to be almost periodic in strong distribution sense if μ is a P(C(R, Rd))-valued almost periodic mapping. It is clear that if X is almost periodic in strong distribution sense, then it is almost periodic in distribution. And we’ve proved that, the consequences of Theorem 1,2 and Corollary 1 are still valid if we exchange the distribution on Rd in them with strong distribution.Now we’ve answered our first question at the beginning of this chapter, that the classical methods given by Favard and Amerio for deterministic systems would hold stochastic differential equations under necessary adaption.4. Lyapunov stability and almost periodic solutionsIn this section, we get the answer to the second question we are curious about. We find that there’s no stability in stochastic differential equations inherited among hulls of distributions of L2-bounded solutions.In later discussion, we usually use function V satisfying following conditions:(I) V:R × Rd�'R+is C2 in t (it means twicely continuously differentiable in t, and similar to C3), C3 in x, V itself and all the differentials DiV of V for i=0,1,2 are bounded on R×S for every compact S (?)Rd, and so as the derivatives Vtxixj, Vxixjxk for i, j, k=1,2, …, d, and we have And we define ·{gjl(t, x)-jl(t, y))), for (?)x, y ∈ Rd.Definition 6. Assume that f,g in (A) are uniformly almost periodic. We say that the Amerio positive (resp. negative) semi-separation condition holds for (A) in Dr if any hull equation (f, g) of (A) only admits positive (resp. negative) semi-separated in distribution solutions in Br; that is, for any μ∈ Dr(f,g), there is a con-stant d(μ)> 0, called separation constant, such that in ft≥0 ρ(μ(t),v(t))≥d(μ) (resp. inft<o ρ(μ(t), v(t))> d(μ)) for any other v ∈Dr(f,g).Definition 7. A property P is called negative semi-separating in Dr(A) if for any distinct μ, v ∈Dr(A) which satisfy P, there exists a constant dμ,v> 0 such that inf (?)ρ(μ(t), v(t))≥dμ,v.And P is inherited in distribution in Dr if μ ∈ Dr(f,g) has property P with respect to the elements of Dr(f,g), (f,g) ∈ H(f,g) with Tα(f,g)= (f, g) and Tαμ=v uniformly on compact intervals for some sequence a, then v also has property P with respect to the elements of Dr(f,g).Theorem 5. Consider (A) and assume (H). Assume that the property P is negative semi-separating in Dr(A) and inherited in distribution in Dr, and that the number of elements of Dr(A) satisfying property P is finite. Then every element of Dr(A) with property P is almost periodic. In particular, (A) admits almost periodic in distribution solutions in Br.Definition 9. For (?)t0 ∈ R, we say element μ(t)∈Dr(A) is uniformly stable on [to,+∞) within Dr(A) if for (?)>0, (?)t1≥t0,(?)δ(ε)> 0 such that for any other element η(t) ∈Dr(A) satisfying that ρ(μ(t1),η(t1))< δ(ε), then Usually we call such μ(t) uniformly stable on [to,+∞) for short, and if μ(t) is uniformly stable on [t1,+∞) for every t1 ∈ R, we call it uniformly stable for short. Theorem 6. Suppose (A)’s coefficients satisfy the standard hypothesis and there is a function V(·, ·) satisfying condition (I). Assume that there exists some constant b> 0 such that for V(t, x)∈R×Rd, V(t,x)≤b|x|2, (h) (?)V(t,x-y)≤0.(ha) Then if Dr(A)≠(?) for some r> 0, all the elements of it are uniformly stable within Dr(A) and if the number of these elements is finite, all of these elements are almost periodic ones.Corollary 2. Suppose (A)’s coefficients satisfy the standard hypothesis. Assume that there is a function V(·,·) satisfying condition (I), and there are constants a, b> 0 such that a|x|2≤V(t, x)< b|x|2, for Vx ∈ R2, (?)t ∈ R. (h3) Assume that there is some positively definite function c(-):R+ �' R+which is convex, increasing on R+, and(?)V(t, x-y)≤-c(|x-y|2) for (?)t ∈ R, (?)x, y ∈ Rd. (h4) Then if D(A) 0, it has unique element which is almost periodic in t.Theorem 7. Suppose (A)’s coefficients satisfy the standard hypothesis, and there exists a function V(·,·) satisfying (I). Suppose there is some constant b> 0 such that (h) is valid on R+ × Rd and for (?)t ∈ R+, (gjl(t+s1,x)-gjl{t+s2,y))≤0, for (?)x,y ∈Rd, (?)s1,s2 ∈ R+. Then if (A) has L2-bounded solutions, the distributions of these solutions are a.a.p. in t on R+, hence (A) has solution with almost periodic distribution.In applications, we often meet the question that how to get an L2-bounded solu-tion for a given stochastic differential equation, the following results would help.Lemma 2. Assume that (A)’s coefficients satisfy the standard hypothesis, (A) admits a solution φ on [t0,+∞) for some t0 ∈ R, and supt≥t0 E|φ(t)|2≤M for M>0., then (A) has a solution φ on R with supt∈R E|φ(t)|2≤M.Lemma 3. Assume that (A)’s coefficients satisfy the standard hypothesis, and there is a function V(·,·) satisfying condition (I), such that a|x|2≤V(t, x)≤b(t)|x|2+c(t), when|x|≤R for R>0, where constant a> 0, b(·), c(·) are positive on R, and supposeThen if X(t) is a solution of (A) with initial condition E|X(to)|2<+∞, X(t) is L2-bounded on [to,+∞).