Averaging Principle on Infinite Intervals for Stochastic Ordinary Differential Equations

In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.


Introduction
Highly oscillating systems may be "averaged" under some suitable conditions, and the evolution of the averaged system can reflect in some sense the dynamics of the original system. This idea of averaging dates back to the perturbation theory developed by Clairaut, Laplace and Lagrange in the 18th century, and is made rigorous by Krylov,Bogolyubov,Mitropolsky [17,1,2] for nonlinear oscillations. There are vast amount of works on averaging for deterministic systems which we will not mention here. Meantime, there are also many works on averaging principle for stochastic differential equations so far, see e.g. [3,4,11,13,15,26,27,28,29] among others. But to our best knowledge all the existing works on stochastic averaging are concerned with the so-called first Bogolyubov theorem, i.e. the convergence of the solution of the original equation to that of the averaged equation on finite intervals.
In the present paper, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for stochastic differential equations: if there exists a stationary solution for the averaged equation, then there exists in a small neighborhood (in the super-norm topology) a solution of the original equation which is defined on the whole axis and has the same recurrence property (in distribution sense) as the coefficients of the original equation. Furthermore, this recurrent solution is more general than the classical second Bogolyubov theorem, which only treats the almost periodic case. Note that the work [14] studies the averaging principle for stochastic differential equations with almost periodic coefficients, but they only show the convergence on the finite interval, not the super-norm topology on the whole axis.
To be more precise, we investigate the semilinear stochastic ordinary differential equation with Poisson stable (in particular, periodic, quasi-periodic, Bohr almost periodic, almost automorphic, Birkhoff recurrent, Levitan almost periodic, almost recurrent, pseudo periodic, pseudo recurrent) in time coefficients. Under some suitable conditions, this equation has a unique L 2bounded solution which has the same recurrent properties as the coefficients, see [8,19] for details. In this paper, we show that this recurrent solution converges to the unique stationary solution of the averaged solution uniformly on the whole real axis when the time scale goes to zero.
The paper is organized as follows. In the second section we collect some known notions and facts. Namely we present the construction of shift dynamical systems, definitions and basic properties of Poisson stable functions, Shcherbakov's comparability method, and the existence of compatible solutions for stochastic differential equations. In the third and fourth sections, we investigate the averaging principle on infinite intervals for linear and semilinear stochastic differential equations respectively.
Let us now introduce two examples of shift dynamical systems which we will use later in this paper.
The hull of ϕ, denoted by H(ϕ), is the set of all the limits of ϕ τn in C(R, X ), i.e.
Note that the set H(ϕ) is a closed and translation invariant subset of C(R, X ) and consequently it naturally defines on H(ϕ) a shift dynamical system -(H(ϕ), R, σ).
Example 2.3. Like in [8], we denote by BU C(R × X , X ) the space of all continuous functions f : R × X → X which are bounded on every bounded subset from R × X and continuous in t ∈ R uniformly with respect to x on each bounded subset Q of X . We equip this space with the topology of uniform convergence on bounded subsets of R × X , which can be generated by the following metric For given f ∈ BU C(R × X , X ) and τ ∈ R, we denote by f τ the τ -translation of f , i.e. f τ (t, x) := f (t + τ, x) for (t, x) ∈ R × X . Note that the space BU C(R × X , X ) endowed with the distance (2.1) is a complete metric space and invariant with respect to translations. Now we define a mapping σ : It is immediate to see (e.g. [6,ChI]) that the mapping σ is continuous and consequently the triplet (BU C(R × X , X ), R, σ) is a dynamical system. Similar to Example 2.2, for given f ∈ BU C(R × X , X ), the hull H(f ) is a closed and translation invariant subset of BU C(R × X , X ) and consequently it naturally defines on H(f ) a shift dynamical system -(H(f ), R, σ).
Denote by BC(X , X ) the space of all continuous functions f : X → X which are bounded on every bounded subset of X and equipped with the distance where Q k are the same as above. Note that (BC(X , X ), d) is a complete metric space. For given F ∈ BU C(R × X , X ), define F : R → BC(X , X ), t → F(t) by letting F(t) := F (t, ·) : X → X . Clearly, F ∈ C(R, BC(X , X )).

2.2.
Poisson stable functions. Let us recall the types of Poisson stable functions to be used in this paper; we refer the reader to [20,22,24,25] for further details and the relations among these types of functions.
(ii) A function ϕ ∈ C(R, X ) is said to be Bohr almost periodic if the set of ε-almost periods of ϕ is relatively dense for each ε > 0, i.e. for each ε > 0 there exists a constant l = l(ε) > 0 such that T (ϕ, ε) ∩ [a, a + l] = ∅ for all a ∈ R.
(iii) A function ϕ ∈ C(R, X ) is said to be pseudo-periodic in the positive (respectively, negative) direction if for each ε > 0 and l > 0 there exists a ε-almost period τ > l (respectively, τ < −l) of the function ϕ. The function ϕ is called pseudo-periodic if it is pseudo-periodic in both directions.
Remark 2.7. A function ϕ ∈ C(R, X ) is pseudo-periodic in the positive (respectively, negative) direction if and only if there is a sequence t n → +∞ (respectively, t n → −∞) such that ϕ tn converges to ϕ uniformly with respect to t ∈ R as n → ∞.
(iii) A function ϕ ∈ C(R, X ) is called Birkhoff recurrent if it is almost recurrent and Lagrange stable.
Definition 2.9. A function ϕ ∈ C(R, X ) is called Poisson stable in the positive (respectively, negative) direction if for every ε > 0 and l > 0 there exists τ > l (respectively, τ < −l) such that d(ϕ τ , ϕ) < ε. The function ϕ is called Poisson stable if it is Poisson stable in both directions.
In what follows, we denote as well Y a complete metric space.
(ii) A function ϕ ∈ C(R, X ) is said to be almost automorphic if it is Levitan almost periodic and Lagrange stable.
Remark 2.11. Note that: (i) Every Bohr almost periodic function is Levitan almost periodic.
Let ϕ ∈ C(R, X ). Denote by N ϕ (respectively, M ϕ ) the family of all sequences {t n } ⊂ R such that ϕ tn → ϕ (respectively, {ϕ tn } converges) in C(R, X ) as n → ∞. We denote by N u ϕ (respectively, M u ϕ ) the family of sequences {t n } ∈ N ϕ (respectively, {t n } ∈ M ϕ ) such that ϕ tn converges to ϕ (respectively, ϕ tn converges) uniformly with respect to t ∈ R as n → ∞.

Remark 2.13.
(i) The function ϕ ∈ C(R, X ) is pseudo-periodic in the positive (respectively, negative) direction if and only if there is a sequence {t n } ∈ N u ϕ such that t n → +∞ (respectively, t n → −∞) as n → ∞.
(ii) Let ϕ ∈ C(R, X ), ψ ∈ C(R, Y) and N u ψ ⊆ N u ϕ . If the function ψ is pseudo-periodic in the positive (respectively, negative) direction, then so is ϕ.
Finally, we remark that a Lagrange stable function is not Poisson stable in general, but all other types of functions introduced above are Poisson stable.
uniformly with respect to x on every bounded subset Q of X if the motion σ(·, F ) through F with respect to the Bebutov dynamical system (BU C(R × X , X ), R, σ) possesses the property A. Here the property A may be stationary, periodic, Bohr/Levitan almost periodic etc.
Remark 2.17. Note that a function ϕ ∈ C(R, X ) possesses the property A if and only if the motion σ(·, ϕ) : R → C(R, X ) through ϕ with respect to the Bebutov dynamical system (C(R, X ), R, σ) possesses this property.
The following statements hold: and ψ is pseudo periodic, then so is ϕ. 2.4. Compatible solutions of semilinear stochastic differential equations. Let B be a real separable Banach space with the norm | · |, and L(B) be the Banach space of all bounded linear operators acting on the space B equipped with operator norm · . Consider the linear homogeneous equation on the space B, where A ∈ C(R, L(B)). Denote by U (t, A) the Cauchy operator (see, e.g. [9]) of equation (2.2). Definition 2.21. Equation (2.2) is said to be uniformly asymptotically stable if there are positive constants N and ν such that If A ∈ C(R, L(B)), then by H(A) we denote the closure in the space the Banach space of all continuous and bounded mappings ϕ : R → B equipped with the norm Let (Ω, F, P) be a probability space, and L 2 (P, B) be the space of B-valued random variables X such that Then L 2 (P, B) is a Banach space equipped with the norm X 2 := Ω |X| 2 dP 1/2 .
Let P(B) be the space of all Borel probability measures on B endowed with the β metric: where f are bounded Lipschitz continuous real-valued functions on B with the norm Recall that a sequence {µ n } ⊂ P(B) is said to weakly converges to µ if f dµ n → f dµ for is the space of all bounded continuous real-valued functions on B.
Definition 2.23. A sequence of random variables {X n } is said to converge in distribution to the random variable X if the corresponding laws {µ n } of {X n } weakly converge to the law µ of X, i.e. β(µ n , µ) → 0.
In the following, we assume that H is a real separable Hilbert space. We still denote the norm in H by | · | and the operator norm in L(H) by · which will not cause confusion. Let us consider the stochastic differential equation where A ∈ C(R, L(H)) and F, G ∈ C(R×H, H). Here W is a two-sided standard one-dimensional Brownian motion defined on the probability space (Ω, F, P). We set F t := σ{W (u) : u ≤ t}.
24. An F t -adapted process {X(t)} t∈R is said to be a mild solution of equation (2.4) on R if it satisfies the following stochastic integral equation for all t ≥ s and each s ∈ R.
Definition 2.25. We say that functions F and G satisfy the condition Proof. The proof is analogous to Theorem 4.6 in [8].
Proof. These statements follow from Theorems 2.19 and 2.28 (see also Remark 2.4).

Averaging for linear equations
Let ε 0 be some fixed positive number. Consider the equation where A ∈ C(R, L(H)), f, g ∈ C(R, L 2 (P, H)), 0 < ε ≤ ε 0 and W is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (Ω, F, P, F t ), where F t := σ{W (u) : u ≤ t}.
Following [16] we say that f (t; ε) integrally converges to 0 if for any L > 0 we have If additionally there exists a constant m > 0 such that for any t ∈ R and 0 < ε ≤ ε 0 , then we say that f (t; ε) correctly converges to 0 as ε → 0. uniformly with respect to t ∈ R, then f (t; ε) := f ( t ε ) integrally converges to 0 as ε → 0. If additionally the function f is bounded on R, then f (t; ε) correctly converges to 0 as ε → 0.
uniformly with respect to t ∈ R and the operatorĀ is Hurwitz, i.e. Re λ < 0 for any λ ∈ σ(Ā). Then the following statements hold: (i) there exists a positive constant α ≤ ε 0 such that the equation where A ε (τ ) := A( τ ε ) for any τ ∈ R, is uniformly asymptotically stable for any 0 < ε ≤ α. Moreover there are constants N > 0 and ν > 0 such that for any τ ≥ τ 0 and 0 < ε ≤ α; (ii) there exists γ 0 > 0 such that Remark 3.6. (i) Note that Theorem 3.5 was proved for finite-dimensional almost periodic equations (this means that the matrix-function A(·) is almost periodic). For the proof for infinite-dimensional almost periodic systems see [18,ChXI].
(ii) It is not difficult to show that Theorem 3.5 remains true in general case (see above) and can be proved with slight modifications of the reasoning from [16,ChIV].
(iii) Under the conditions of Theorem 3.5 there are positive constants α, N and ν so that for any 0 < ε ≤ α and t ≥ τ . Then for any ν > 0 we have Proof. To estimate the integral we make the change τ − t = s, then  Since l is arbitrary, it follows that lim ε→0 sup t∈R I(t, ε) = 0.
Then W ε is also a Brownian motion with the same distribution as W .
Proof. The first and second statements follow directly from Theorem 2.28. We now verify the third statement, i.e. the uniform convergence of the unique bounded solution φ ε to the unique stationary solutionφ of the averaged equation. By Theorem 2.28 equation (3.18) has a unique bounded and stationary solutionφ, which is given by the formula where GĀ(t, τ ) = exp Ā (t − τ ) for t, τ ∈ R. From (3.16) and (3.19) we get By equality (3.5) there exists a function N : (0, α) → R + such that N (ε) → 0 as ε → 0 and for any t ≥ τ (t, τ ∈ R).
Note that To estimate the integral making the change of variable s := τ − t we obtain Consequently, integrating by parts from (3.20) for any t ∈ R and s < 0. Let now l be an arbitrary positive number, then we have Letting ε → 0 in above inequality we get Since l is arbitrary, we get by letting l → ∞ lim ε→0 sup t∈R I 11 (t, ε) = 0.
Note by Theorem 3.5-(ii) that Similarly we can show that In fact, using Itô's isometry property, the Cauchy-Schwartz inequality and reasoning as above we get By Lemma 3.7 the integral goes to 0 as ε → 0 uniformly with respect to t ∈ R.
The proof is complete.

Averaging principle for semilinear stochastic differential equations
Consider the following stochastic differential equation where A ∈ C(R, L(H)), F, G ∈ C(R × H, H), 0 < ε ≤ ε 0 and W is a two-sided standard onedimensional Brownian motion defined on the filtered probability space (Ω, F, P, F t ), where ε 0 is a small positive constant and F t := σ{W (u) : u ≤ t}. Below we will use the following conditions: (G1) there exists a positive constant M such that for any t ∈ R; (G2) there exists a positive constant L such that for any x 1 , x 2 ∈ H and t ∈ R; (G3) there exist functions ω 1 ∈ Ψ andF ∈ C(H, H) such that for any T > 0, x ∈ H and t ∈ R; (G4) there exist functions ω 2 ∈ Ψ andḠ ∈ C(H, H) such that for any T > 0, x ∈ H and t ∈ R; (G5) A ∈ C(R, L(H)) and there existsĀ ∈ L(H) such that  We consider as well the following equations x ∈ H and ε ∈ (0, ε 0 ], and ε 0 is some fixed small positive constant. Here as before W ε (t) := √ εW ( t ε ) for t ∈ R. Along with equations (4.2)-(4.3) we also consider the following averaged equation (4.4) dX(t) = (ĀX(t) +F (X(t)))dt +Ḡ(X(t))dW (t).
then there exists a constant C > 0, depending only on M, L, ϕ ∞ , such that for any t ∈ R and h > 0.
Proof. This statement follows from Theorems 2.19 and 4.3 (see also Remark 2.4).
Remark 4.5. In the present paper, we only consider the second Bogolyubov theorem for semilinear stochastic ordinary differential equations, i.e. the linear part A(·) is bounded operator valued. We will consider the case when A(·) is an unbounded operator in future work, which can be applied to related stochastic partial differential equations.