Almost Periodic Solutions and Stable Solutions for Stochastic Differential Equations

In this paper, we discuss the relationships between stability and almost periodicity for solutions of stochastic differential equations. Our essential idea is to get stability of solutions or systems by some inherited properties of Lyapunov functions. Under suitable conditions besides Lyapunov functions, we obtain the existence of almost periodic solutions in distribution.


Introduction
In 1924-1926, Bohr founded the theory of almost periodic functions [5,6,7]. Roughly speaking, an almost periodic function means that it is periodic up to any desired level of accuracy. Since many differential equations arising from physics and other fields admit almost periodic solutions, almost periodicity becomes an important property of dynamical systems and is extensively studied in the area of differential equations and dynamical systems. We refer the reader to the books, e.g. Amerio and Prouse [1], Fink [12], Levitan and Zhikov [15], Yoshizawa [28] etc, for an exposition. For deterministic differential equations, the existence of almost periodic solutions was studied under various stability assumptions. Markov [17] defined a kind of stability which implies almost periodicity. Deysach and Sell [11] assumed that there exists one bounded uniformly stable solution. Miller [18] assumed the existence of one bounded totally stable solution. Seifert [22] proposed the so-called Σ-stability, while Sell [23,24] proposed the stability under disturbance from the hull; actually, these two concepts of stability are equivalent. Coppel [9] sharpened Miller's result without the uniqueness of solutions by using the properties of asymptotically almost periodic functions; Yoshizawa [27] developed the idea of Coppel and improved all the results mentioned above. On the other hand, the Lyapunov's second method was employed to investigate the existence of almost periodic solutions: Hale [14] and Yoshizawa [26] assumed the existence of Lyapunov functions for pairs of solutions to conclude the uniform asymptotic stability in the large of the bounded solution.
However, the various stability assumptions mentioned above are not easily verified directly in practice. It is known that some stabilities, such as uniform stability and uniform asymptotic stability, can be characterized by Lyapunov functions. So it seems that it is a good idea to give some explicit conditions on Lyapunov functions to study the existence of almost periodic solutions, as Hale and Yoshizawa did in [14,26]. This is exactly what we are to do in the present paper for stochastic differential equations (SDE).
For the stochastically perturbed semilinear equations, almost periodic solutions were studied by assuming that the linear part of these equations satisfies the property of exponential dichotomy; see Halanay [13], Morozan and Tudor [19], Da Prato and Tudor [10], and Arnold and Tudor [2], among others. For general SDEs, Vârsana [25] studied asymptotical almost periodic (weaker than almost periodic) solutions by assuming that the stochastic system is total stable. Very recently, Liu and Wang [16] investigated the almost periodic solutions to SDEs by the separation method.
This paper is organized as follows. Section 2 is a preliminary section. Section 3 contains main results of this paper, in which we study almost periodic solutions for SDEs by mainly the Lyapunov function method. In Section 4, we illustrate our results by some applications.

Preliminaries
Assume that (M, d) is a complete metric space. Here is the definition of M -valued almost periodic and uniform almost periodic functions in the sense of Bohr: We say set A ⊂ R is relatively dense in R if for any given ǫ > 0, there exists l = l(ǫ) > 0, such that for every a ∈ R, (a, a + l) ∩ A = ∅. If there is a set T (ǫ, ϕ) relatively dense such that for any τ ∈ T (ǫ, ϕ) we have then we say that the function ϕ is almost periodic.
(ii) A continuous function f (·, ·) : R×R d → R d is almost periodic in t uniformly on compact sets if for every compact set S ⊂ R d there exists a relatively dense set T (ǫ, f, S) such that for every τ ∈ T (ǫ, f, S) we have: We also say such f (t, x) is uniformly almost periodic for short.
Bochner [3,4] gave an equivalent condition to Bohr's almost periodicity. The above definition of uniform almost periodicity can be found in Yoshizawa's book [28]; Seifert and Fink made another definition of uniform almost periodicity (see Definition 2.1 in [12]).
For R d -valued uniformly almost periodic function f (t, x), we denote as the hull of f . The hull has the following properties: x) be uniformly almost periodic. Then: (i) any g ∈ H(f ) is also uniformly almost periodic and H(g) = H(f ); (ii) for any g ∈ H(f ), there exists a sequence α with α n → +∞ (or α n → −∞) such that T α f = g uniformly on R × S for any compact S ⊂ R d ; (iii) for any sequence α ′ , there exists a subsequence α ⊂ α ′ such that T α f exists uniformly on R × S for any compact S ⊂ R d .
We respectively denote [0, +∞) and (−∞, 0] as R + and R − , and recall the definition of asymptotically almost periodic function valued in M as follows. Then we say f (t) is aymptotically almost periodic (a.a.p. in short) on R + . The η(t) in (2.3) is called the almost periodic part of f . The function f being a.a.p. on R − can be defined similarly.
Remark 2.4 (See [15], Chapter 1). If f is a.a.p. on R + or R − , then its almost periodic part is unique.
Lemma 2.5. The following statements are equivalent to f being asymptotic almost periodic on R + : (i) For any sequence α ′ = {α ′ n } such that α ′ n → +∞, there exists suitable subsequence α ⊂ α ′ such that T α f (t) uniformly exists on R + .
In this paper, we study the SDE: where f (t, x) is an R d -valued continuous function, g(t, x) is a (d×m)-matrix-valued continuous function, and W is a standard m-dimensional Brownian motion. And we usually assume the coefficients are uniformly almost periodic. Note that almost periodicity is defined on the whole R, but the Brownian motions in SDEs usually defined on R + . So we need to introduce two-sided Brownian motion: for two independent Brownian motions W 1 (t), W 2 (t) on the probability space (Ω, F, P ), let Then W is a two-sided Brownian motion defined on the filtered probability space (Ω, F, P, Furthermore, we always assume (2.4)'s coefficients satisfy the following condition: (H) The functions f , g are uniformly almost periodic. And there exists a constant K > 0 such that, for every t ∈ R and x, y ∈ R d , where a ∨ b = max{a, b} for a, b ∈ R.
For SDE (2.4) satisfying condition (H), if there exists a sequence α such that T α f =f and T α g =g, we denote the SDE with coefficients (T α f, T α g) as (f ,g) ∈ H(f, g) or T α (f, g) = (f ,g) for short. Besides, by the definition of uniform almost periodic function, if coefficients of (2.4) satisfy the condition (H), they must satisfy the global linear growth condition, that is, there is some constantK > 0, such that For R d -valued random variable X on the probability space (Ω, F, P), we denote L(X) as the distribution (or law) of X on R d . We denote by P(R d ) the space of all Borel probability measures on R d . For an R d -valued random variable X or stochastic process Y (t), we define the following norms: In what follows, we denote: on some filtered probability space for some W and X ∞ ≤ r}, We focus on the almost periodicity of distributions of SDEs' solutions instead of solutions themselves. It's well known that P(R d ) can be metrized with some distance (which we denote as ρ(·, ·)), such that the convergence under distance ρ(·, ·) is equivalent to the convergence under the weak-* topology of P(R d ), and P(R d ) is a complete metric space under ρ(·, ·) (see [20, Theorem 2.6.2] for details).
For a P(R d )-valued continuous function f , one of the necessary conditions of the almost periodicity of f is that, the set {f (t); t ∈ R} is contained in some compact set. Naturally we need to consider distributions of solutions for SDEs within some compact set. We get compactness on the space P(R d ) by L 2 -boundedness (see [21] for details).
We define the uniform stability of distributions of solutions for SDEs as follows: if for every ǫ > 0, there exists δ = δ(ǫ) > 0 such that for any t 1 ≥ t 0 and any other element If µ(t) is uniformly stable on [t 1 , +∞) for every t 1 ∈ R, we call it uniformly stable for short.
In what follows, we get the stability of solutions' distributions mainly by Lyapunov functions, which satisfy the following condition: x) > 0 for each x = 0, and V (t, 0) = 0 for all t ∈ R.

Main Results
In this paper, we need following results from [16] for further discussion: , Theorem 3.1). Consider the following family of Itô stochastic equa- where f n are R d -valued, g n are (d × m)-matrix-valued, and W is a standard m-dimensional Brownian motion. Assume that f n , g n satisfy condition (H). Assume further that f n → f , g n → g point-wise on R × R d as n → ∞, and that X n (t) ∈ B (fn,gn) r for some constant r > 0, independent of n. Then there is a subsequence of {X n } which converges in distribution, uniformly on compact intervals, to some 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 − .

Consider (2.4) and let
Now we give a sufficient condition to the uniform stability in distribution we defined in Definition 2.6: Theorem 3.3. Suppose that (2.4)'s coefficients satisfy condition (H) and there is a function V (·, ·) satisfying condition (L). Assume that there exists some constant b > 0 such that for all (t, x) ∈ R × R d , Then if D ; if the number of these elements is finite, all of these elements are almost periodic.
We want to prove that ρ(µ n (t n ), µ(t n )) → 0 and hence get contradiction to (3.5). It suffices to prove that X n (t) uniformly converge to X(t) in probability on [t 0 , +∞), that is, for every ǫ > 0, when n is large enough, Firstly, we prove that V (t, X n (t) − X(t)) is a supermartingale on [t 0 , +∞) for each n. For t ≥ t 0 , we have g(s, X n (s)) − g(s, X(s))dW (s).

By (3.4)
, − − → +∞ as k → +∞ for every n, by Fatou's lemma we have: Now we want to prove that E V (t 0 ,X n −X) is sufficiently small when n is large enough. By Jensen's inequality and (3.9) we have ≤ V (s, X n (s) − X(s)), a.s.. (3.10) That is, V (t, X n (t) − X(t)) is a supermartingale. So by the martingale inequality we have P{ sup which implies that, if n is large enough such that we will have (3.6). Thus sup which is contradictory to (3.5). Thus each element of D Step 2. Inherited property and a.a.p. Now we want to prove that the consequence of step 1 is also valid for all the hull equations. Let the sequence α ′ be such that (T α ′ f, T α ′ g) unifromly exists on R × S for any compact set S ⊂ R d . Since V , V t , V x i are bounded on R × S, V (t + α ′ n , x) are uniformly bounded and equi-continuous on I × S for any compact interval I ⊂ R. By Arzela-Ascoli's theorem, there is suitable subsequence α ⊂ α ′ such that T α V (t, x) exists uniformly on I × S. By the diagonalization argument, the α could be chosen such that T α V exists uniformly on any compact subset of R × R d .
Similarly we can extract further subsequence from α, which we still denote by α itself, such that T α V t , T α V x i , T α V x i x j exist uniformly on compact subsets of R × R d . More precisely, we have · (T α g jl (t, x) − T α g jl (t, y))) ≤ 0 (3.14) for all (t, x, y) ∈ R × R d × R d . Repeating Step 1, we obtain that all the elements of D , there is a separating constant d(f, g), depending only on (f, g) but independent of µ ∈ D (2.4) r , such that for any two different elements η(t), µ(t) ∈ D (2.4) r we have By Proposition 2.2-(ii), we may assume with loss of generality that the above sequence α satisfies lim n→∞ α n = −∞, so it follows from (3.15) that On the other hand, it follows from Proposition 3.1 that T α µ(t) ∈ D are a.a.p.. For the above sequence α with α n → −∞ and given sequence δ = {δ n } with δ n < 0, by Proposition 2.2-(iii) there exist suitable subsequences which we denote as themselves such that (T α+δ f, T α+δ g) exists uniformly on R × S for any compact set S ⊂ R d . By Arzela-Ascoli's theorem there are subsequences β, γ ⊂ α such that T β+δ µ(t), T γ+δ µ(t) exist uniformly on compact intervals (see the proof of [16, Theorem 3.1] for details). By Proposition 3.1, T β+δ µ(t), T γ+δ µ(t) ∈ D (T α+δ f,T α+δ g) r , then by the separating property obtained above we have Then it follows from Lemma 2.5 that all the elements of D Assume that there is some positively definite function c(·) : R + → R + which is convex, increasing on R + , and Then if D (2.4) = ∅, it has a unique element which is almost periodic.
Proof. We prove the uniqueness by contradiction. If there are two elements µ(t), η(t) ∈ D (2.4) , then there's some r > 0 such that µ(t), η(t) ∈ D Assume that X(t), Y (t) are two strong L 2 -bounded solutions of (2.4) for given Brownian motion W (t) such that L(X(t)) = µ(t), L(Y (t)) = η(t). For given ǫ > 0, let T (ǫ) = 2br 2 /c(ǫ) + 1. Firstly, we prove that for every t ∈ R there is If this is not true, then there ist ∈ R and ǫ 0 > 0 such that Similar to the proof of Theorem 3.3, for given s ∈ R, we define Then it follows from Ito's formula that for t ≥ s, Because c(r) is convex, increasing on R + , by Jensen's inequality we have: Noting that τ k a.s.
Theorem 3.5. Assume that (2.4)'s coefficients satisfy condition (H), and there exists a function V (·, ·) satisfying condition (L). Suppose that there is some constant b > 0 such that (3.3) is valid on R + × R d and for all t ∈ R + , s 1 , s 2 ∈ R + and x, y ∈ R d , Then if (2.4) has L 2 -bounded solutions, the distributions of these solutions are a.a.p. on R + and (2.4) admits at least one solution with almost periodic distribution.
Proof. For sequence α = {α n } such that α n → +∞, assume that (T α f, T α g) exist uniformly on R × S for any compact set S ⊂ R d , and T α µ(t) exists uniformly on compact intervals (see, again, the proof of [16, Theorem 3.1] for details). For r > 0 and every µ(t) ∈ D , we what to prove that T α µ(t) uniformly exists on R + .
We now show that V (t, X n+p (t) − X n (t)) is a supermartingale on R + for given n and p. For every 0 ≤ s < t < +∞, we define stopping times τ n,p k := inf{t ≥ s; |X n (t)| ∨ |X n+p (t)| > k}, for every k, n, p ∈ N.
By Itô's formula we have [g ji (u + α n+p , X n+p (u)) − g ji (u + α n , X n (u))] [g ji (u + α n+p , X n+p (u)) − g ji (u + α n , X n (u))] it follows from (3.20) that Noting that τ n,p k a.s. − − → +∞ as k → +∞ for every n, p, we have by Fatou's lemma (similar to (3.9)): That is, V (t, X n+p (t) − X n (t)) is a supermartingale for given p and n. Similar to (3.10), by Jensen's inequality, is also a supermartingale. For any ǫ > 0, we define V ǫ > 0 as the one in (3.7). Then by the martingale inequality, we have When n is large enough, by (3.3) we have This together with (3.21) implies By Theorem 4.1.3 in [8], there exists a suitable stochastic process X(t) such that X n (t) P − → X(t) uniformly on R + . Thus µ(t + α n ) uniformly converges to some T α µ(t) on R + . By Lemma 2.5, we can see that each µ(t) ∈ D (2.4) r is a.a.p. on R + . So the distribution of any L 2 -bounded solution is a.a.p. on R + . By Proposition 3.2, there exists some L 2 -bounded solution of (2.4) with almost periodic distribution. The proof is complete.
We now conclude this section by giving a sufficient condition for the existence of L 2bounded solutions via Lyapunov functions: Theorem 3.7. Assume that (2.4)'s coefficients satisfy condition (H), and there is a function V satisfying condition (L) such that for some constant R > 0 where constant a > 0, b(·), c(·) are positive functions on R. Assume further that Then if X(t) is a solution of (2.4) with initial condition E|X(t 0 )| 2 < +∞, X(t) is L 2 -bounded on [t 0 , +∞).
Then D (4.3) r has a unique element which is almost periodic.
If A i (t) are almost periodic for i = 1, 2, then (4.5) has L 2 -bounded solutions, and all the L 2 -bounded solutions of (4.5) have the same distribution which is almost periodic.