PSEUDO ALMOST PERIODICITY FOR STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

. In this article, we introduce the concept of p -mean θ -pseudo almost periodic stochastic processes, which is slightly weaker than p -mean pseudo al- most periodic stochastic processes. Using the operator semigroup theory and stochastic analysis theory, we obtain the existence and uniqueness of square- mean θ -pseudo almost periodic mild solutions for a semilinear stochastic differential equation in inﬁnite dimensions. Moreover, we prove that the obtained solution is also pseudo almost periodic in path distribution. It is noteworthy that the ergodic part of the obtained solution is not only ergodic in square- mean but also ergodic in path distribution. Our main results are even new for the corresponding stochastic diﬀerential equations (SDEs) in ﬁnite dimensions.


Introduction
The theory of almost periodic functions was introduced by Bohr [5,6,7] in [1924][1925][1926]. Since then, many interesting generalizations of almost periodic functions appeared. The concept of pseudo almost periodic functions is among these, which was first introduced by Zhang [20,21]. In the last two decades, pseudo almost periodic functions have been extensively investigated and have many applications in the theory of differential equations (see [1,2,4,13,16,17,18,22] for example).
Recently, pseudo almost periodicity of stochastic differential equations (SDEs) has attracted more and more attention. In the random case, there are several different ways to define pseudo almost periodicity for stochastic processes, such as square-mean pseudo almost periodicity, pseudo almost periodicity in distribution (in various senses), and so on. It is difficult to study square-mean pseudo almost periodicity for SDEs since some SDEs never have square-mean pseudo almost periodic solutions (cf. [2, Example 3.1]), and thus it is reasonable to consider pseudo almost periodicity in distribution for SDEs. However, there are seldom results on pseudo almost periodicity in distribution of SDEs (cf. [16,17,18]); except for [16] almost all earlier works were concerned with pseudo almost periodic in distribution solutions whose ergodic part are only ergodic in p-mean rather than in distribution.
In this article, we aim to study square-mean pseudo almost periodicity and pseudo almost periodicity in distribution for the following semilinear stochastic differential equations in a separable Hilbert space H: dX(t) = AX(t)dt + F (t, X(t))dt + G(t, X(t))dW (t), t ∈ R Motivated by a recent work [15], where Raynaud de Fitte proposed a new method to study almost periodicity of equation (1.1), we introduce the concept of p-mean θpseudo almost periodicity and obtain the existence and uniqueness of square-mean θ-pseudo almost periodic solution to equation (1.1). Moreover, using the maximal inequality of stochastic convolution for semigroups, we show that the solution is also pseudo almost periodic in path distribution. Note that the ergodic part of pseudo almost periodic solution we obtain is not only ergodic in square-mean but also ergodic in path distribution (see Definition 2.4).
The article is organized as follows. In section 2 we introduce some notions and properties of pseudo almost periodic processes, including p-mean θ-pseudo almost periodic processes and pseudo almost periodic in distribution processes. In section 3 we first establish a convolution theorem of square-mean θ-pseudo almost periodic stochastic processes, and with its help, we obtain the existence of square-mean θ-pseudo almost periodic and pseudo almost periodic in distribution solutions.

Preliminaries
Let (X, d X ) be a complete metric space. A set J ⊂ R is said to be relatively dense in R if there exists a constant l > 0 such that for any a ∈ R, we have [a, a+l]∩J = ∅.
Definition 2.1. [12] A continuous function f : R → X is called almost periodic if for every > 0, the set is relatively dense in R. Denote by AP (R, X) the set of all such functions.
Let (B, · ) be a Banach space. Denote by BC(R, B) be the Banach space of all continuous and bounded functions f : R → B equipped with the norm f ∞ := sup s∈R f (s) . Define is called pseudo almost periodic if it can be expressed as f = g + φ , where g ∈ AP (R, B) and φ ∈ P AP 0 (B). Denote by P AP (B) the set of all such functions.
The functions g and φ are called the almost periodic component and ergodic perturbation of the function f respectively.
Let (E, d) be a Polish space and P(E) be the set of all probability measures onto σ-Borel field of E. Denote by BC(E, R) the space of bounded continuous functions f : E → R with the norm f ∞ = sup x∈E |f (x)|. Let f ∈ BC(E, R) be Lipschitz continuous, we define We denote by C(R, E) the space of all continuous functions f : R → E equipped with the distance .
) is a complete metric space.
Let (Ω, F, P ) be a probability space and X : R × Ω → E be a stochastic process.
(a) We call that X is almost periodic in one-dimensional distribution if the mapping t → law(X(t)) from R to P(E) is almost periodic. (b) We call that X is almost periodic in finite-dimensional distribution, if for every finite sequence (t 1 , . . . , t n ), the mapping R → P(E n ) given by is almost periodic. (c) Assume that X has continuous trajectories. We call that X is almost periodic in path distribution if the mapping t → law(X(t + ·)) from R to P(C(R, E)) is almost periodic, where C(R, E) is endowed with the distance d C(R,E) and P(C(R, E)) is endowed with the distance d BL .
Definition 2.4. Let (Ω, F, P ) be a probability space and X : R × Ω → E be a stochastic process.
(a) We call that X is pseudo almost periodic in one-dimensional distribution if the mapping t → law(X(t)) is continuous and there exists a stochastic process Y : R × Ω → E which is almost periodic in one-dimensional distribution such that lim T →∞
(c) Assume that X has continuous trajectories. We call that X is pseudo almost periodic in path distribution if the mapping t → law(X(t + ·)) is continuous and there exists a stochastic process Y : R × Ω → E which is almost periodic in path distribution such that In this case, the function Z : R → R, Z(t) := d BL (law(X(t + ·)), law(Y (t + ·)), is called the ergodic part of X and we say that Z is ergodic in the distribution sense.
For p ≥ 1, we denote by L p (Ω, B) the space of all B-valued random variables X such that E X p = Ω X p dP < ∞. For X ∈ L p (Ω, B), let ). Let (Ω, F, P ) be a probability space. A family of measurable mappings on the sample space, θ t : Ω → Ω, t ∈ R, is called a measurable dynamical system if the following conditions are satisfied: (i) identity property: θ 0 = Id Ω , (ii) flow property: θ t θ s = θ t+s , for t, s ∈ R, (iii) measurability: (ω, t) → θ t ω is measurable. It is called a measure-preserving dynamical system if, furthermore, (iv) measure-preserving property: P (θ t A) = P (A), for every A ∈ F and t ∈ R.
In the sequel, we always assume that θ = (θ t ) t∈R is a measure-preserving dynamical system.
We say that X is p-mean θ-almost periodic (or simply θ p -almost periodic) if conditions (i) and (ii) below are satisfied: We denote by AP θ (R, L p (Ω, B)) the set of all such processes.
If p = 2, then we say that X is square-mean θ-almost periodic.
Proposition 2.7 (Equicontinuity and uniform continuity [15]). Let X : R×Ω → B be a θ p -almost periodic random process. Then (a) the mapping t → X (t + s, θ −s ·) is continuous from R to L p (Ω, B), uniformly with respect to s ∈ R.
uniformly with respect to t ∈ R.
Proposition 2.8 (Compactness [15]). Let X : R × Ω → B be a θ p -almost periodic random process, and let J be a compact interval of R.
Definition 2.9. A stochastic process X ∈ BC(R, L p (Ω, B)) is said to be p-mean θ-pseudo almost periodic if it can be expressed as where Y ∈ AP θ (R, L p (Ω, B)) and Z ∈ P AP 0 (L p (Ω, B)). The processes Y and Z are called the almost periodic part and ergodic part of X respectively. Denote by P AP θ (R, L p (Ω, B)) of all such processes. If p = 2, then we say that X is squaremean θ-pseudo almost periodic.
Proof. Without loss of generality, assume p = 1. Let (t 1 , . . . , t n ) be a finite sequence in R. Let us endow B n with the norm Similarly, one can show that the mapping t → law(X(t + t 1 ), . . . , X(t + t n )) is continuous.
Proposition 2.11. Let X n ∈ P AP θ (R, L p (Ω, B)), n = 1, 2, . . . . Assume further that there exists a stochastic process X such that By Proposition 2.7 and Definition 2.6, we deduce that there are numbers l > 0 and δ > 0 such that any interval in R of length l contains a subinterval of length δ whose numbers belong to P θ ( λ 2 , Y ). Thus, for every a ∈ R, there exists some number b such that Then Hence This implies that lim sup But Z ∈ P AP 0 (L p (Ω, B)), and we have a contradiction. Now, assume that X n = Y n +Z n , n = 1, 2, . . . , where Y n ∈ AP θ (R, L p (Ω, B)) and Z n ∈ P AP 0 (L p (Ω, B)). By (2.1), we obtain that (X n ) ∞ n=1 is a Cauchy sequence in BC(R, L p (Ω, B)). Thus we deduce that (Y n ) ∞ n=1 and (Z n ) ∞ n=1 are Cauchy sequences in BC(R, L p (Ω, B)) since and Z n = X n − Y n for n, m ∈ N + . Denote by Y and Z the limits of (Y n ) ∞ n=1 and (Z n ) ∞ n=1 , respectively. Then Y ∈ AP θ (R, L p (Ω, B)) and Z ∈ P AP 0 (L p (Ω, B)) because AP θ (R, L p (Ω, B)) and P AP 0 (L p (Ω, B)) is closed in BC(R, L p (Ω, B)). It is easy to see that X = Y + Z, hence X ∈ P AP θ (R, L p (Ω, B)).

Main results
In this section, H is a separable Hilbert space, Ω = C(R, H) is endowed with the compact-open topology, F is the Borel σ-algebra of Ω, and P is the Wiener measure on Ω with trace class covariance operator Q, and the process W with values in H defined by W (t, ω) = ω(t), ω ∈ Ω, t ∈ R is a Brownian motion with covariance operator Q. Let (F t ) t∈R be the augmented natural filtration of W . We refer to [8,14] for more information about stochastic integration and stochastic equations in Hilbert spaces. Let L(H) be the Banach space of continuous linear operators from H to itself with the operator norm · L(H) . Define θ = (θ t ) t∈R by for all τ, t ∈ R and ω ∈ Ω. Then by Definition 2.5, θ = (θ t ) t∈R is a measurepreserving dynamical system. To study equation (1.1), we first list our assumptions: (H1) A is the infinitesimal generator of a C 0 -semigroup (T (t)) t≥0 and there exists δ > 0 such that (H2) There exists a constant K > 0 such that F : R × H → H and G : R × H → L(H) satisfy, for every t ∈ R and x, y ∈ H, (H3) The functions F, G is pseudo almost periodic in t ∈ R for each x ∈ H, that is, the mappings F (·, x) : R → H and G(·, x) : R → L(H) are pseudo almost periodic for every x ∈ H.
Definition 3.1. A H-valued F t -progressively measurable stochastic process X(t), t ∈ R is called the mild solution of equation (1.1) if it satisfies for t, s ∈ R with t ≥ s.
Next, we give some technical lemmas for later use. is also in P AP 0 (R).
Lemma 3.3. Let g ∈ P AP 0 (R). Then for every T > 0, the function is also in P AP 0 (R).
Proof. Since P AP 0 (R) is translation invariant, g(· + t) ∈ P AP 0 (R) for every t ∈ R. Then, by Lebesgue's dominated convergence theorem, we obtain 1 2r Define the operator ψ : BC(R, L p (Ω, H)) → BC(R, L p (Ω, H)) by It is easy to see that ψ is well defined. Proof. Let F = F 1 + F 2 , where F 1 (·, x) and F 2 (·, x) are the almost periodic component and ergodic perturbation of the function F (·, x) for every x ∈ H. Let G = G 1 + G 2 , where G 1 (·, x) and G 2 (·, x) are the almost periodic component and ergodic perturbation of the function G(·, x) for every x ∈ H. By Lemma 5.2 in [19, Page 57], the functions F 1 , G 1 satisfy assumption (H2). Consequently, the functions F 2 , G 2 satisfy assumption (H2) with constant 2K.
By [19,Lemma 5.10] again, I 4 ∈ P AP 0 (L 2 (Ω, H)) if and only if the mapping is in P AP 0 (R). Using Itô's isometry, we obtain For the same reason as for I 2 , we have I 4 ∈ P AP 0 (L 2 (Ω, H)).
Proof. The proof of the existence and uniqueness of a mild solution to (1.1) in the functions space BC(R, L 2 (Ω, H)) is based on Banach fixed point theorem, which is the same as that of Theorem 3.1 in [11]. Using the factorization method (see section 5.3 in [8] or section 3.2 in [10]), we deduce that X has a continuous version. Moreover, we have lim where X 0 = 0 and X n = ψX n−1 , n = 1, 2, . . . . By Proposition 2.11 and Theorem 3.4, we deduce that X ∈ P AP θ (R, L 2 (Ω, H)). Now, let us prove that X is also pseudo almost periodic in path distribution. Let F = F 1 + F 2 , where F 1 (·, x) and F 2 (·, x) are the almost periodic component and ergodic perturbation of the function F (·, x) for every x ∈ H. Let G = G 1 + G 2 , where G 1 (·, x) and G 2 (·, x) are the almost periodic component and ergodic perturbation of the function G(·, x) for every x ∈ H.
By [19,Lemma 5.2], the functions F 1 , G 1 satisfy assumption (H2). Consequently, the functions F 2 , G 2 satisfy assumption (H2) with constant 2K. By [15,Theorem 5.1], there exists a stochastic process Y which is the mild solution of In other words, Y satisfies for every t, s ∈ R with t ≥ s, Moreover, Y ∈ AP θ (R, L 2 (Ω, H)) and Y is also almost periodic in path distribution.
Step 1. For every positive integer N , the mapping By Theorem 3.4, we have Z = X − Y ∈ P AP 0 (L 2 (Ω, H)). Then by [19,Lemma 5.10], we have E Z(·) 2 ∈ P AP 0 (R). Since P AP 0 (R) is translation invariant, we deduce that Σ 1 ∈ P AP 0 (R). Using condition (H2) and Hölder's inequality, we obtain For every > 0, let ϑ, K, x 1 , x 2 , . . . , x J be the same as in Theorem 3.4. Then by a similar argument of I 2 in Theorem 3.4, we have where C is a constant. By Lemma 3.3, we have that the mappings are in P AP 0 (R). Since is arbitrary, we deduce that Σ 2 ∈ P AP 0 (R). By [8, Theorem 6.10], for every L > 0, there exists a constant C L such that, for every a ∈ R and every predictable stochastic process Φ with E a+L a Then we obtain For the same reason as for Σ 2 , we have Σ 3 ∈ P AP 0 (R).
Gathering the estimations for Σ 1 -Σ 3 , we conclude that the mapping is in P AP 0 (R).