Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 34, pp. 1–14.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

PSEUDO ALMOST PERIODICITY FOR STOCHASTIC

DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

YE-JUN CHEN, HUI-SHENG DING

Abstract. 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 dif-

ferential 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.

1. Introduction

The theory of almost periodic functions was introduced by Bohr [5, 6, 7] in 1924-
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 dif-
ferent 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 pe-
riodic 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

2020 Mathematics Subject Classification. 60H15, 34C27.
Key words and phrases. Pseudo almost periodic; solutions in distribution;

stochastic differential equations in infinite dimensions.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted October 19, 2022. Published April 10, 2023.

1



2 Y.-J. CHEN, H.-S. DING EJDE-2023/34

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 (1.1)

where A : D(A) ⊂ H → H is a linear operator, F : R×H → H, and G : R×H →
L(H) are continuous.

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 θ-
pseudo 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.

2. Preliminaries

Let (X, dX) 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 6= ∅.

Definition 2.1. [12] A continuous function f : R→ X is called almost periodic if
for every ε > 0, the set

P (ε, f) := {τ ∈ R : sup
t∈R

dX(f(t+ τ), f(t)) < ε}

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‖∞ :=
sups∈R ‖f(s)‖. Define

PAP0(B) :=
{
f ∈ BC(R,B) : lim

T→∞

1

2T

∫ T

−T
‖f(t)‖dt = 0

}
.

Definition 2.2 ([19]). A function f ∈ BC(R,B) is called pseudo almost periodic if
it can be expressed as f = g + φ , where g ∈ AP (R,B) and φ ∈ PAP0(B). Denote
by PAP (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‖∞ = supx∈E |f(x)|. Let f ∈ BC(E,R) be Lipschitz
continuous, we define

‖f‖L := sup
x 6=y

{ |f(x)− f(y)|
d(x, y)

}
, ‖f‖BL := ‖f‖∞ + ‖f‖L.



EJDE-2023/34 PSEUDO ALMOST PERIODIC SOLUTIONS FOR SDES 3

Then (P(E), dBL) is a complete metric space where

dBL(µ, ν) := sup
{∣∣ ∫

E

fdµ−
∫
E

fdν
∣∣ : ‖f‖BL ≤ 1

}
, µ, ν ∈ P(E).

We denote by C(R, E) the space of all continuous functions f : R → E equipped
with the distance

dC(R,E)(f, g) :=

∞∑
k=1

1

2k
supt∈[−k,k] d(f(t), g(t))

1 + supt∈[−k,k] d(f(t), g(t))
.

Then (C(R, E), dC(R,E)) is a complete metric space.

Definition 2.3. [3] 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 7→ 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 (t1, . . . , tn), the mapping R→ P(En) given by

t 7→ law(X(t1 + t), . . . , X(tn + t))

is almost periodic.
(c) Assume that X has continuous trajectories. We call that X is almost

periodic in path distribution if the mapping t 7→ law(X(t + ·)) from R to
P(C(R, E)) is almost periodic, where C(R, E) is endowed with the distance
dC(R,E) and P(C(R, E)) is endowed with the distance dBL.

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 7→ 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→∞

1

2T

∫ T

−T
dBL(law(X(t)), law(Y (t)))dt = 0.

(b) We say that X is pseudo almost periodic in finite-dimensional distribution,
if there exists a stochastic process Y : R × Ω → E which is almost periodic in
finite-dimensional distribution such that, for every finite sequence (t1, . . . , tn), the
mapping t 7→ law(X(t+ t1, . . . , X(t+ tn))) is continuous and

lim
T→∞

1

2T

∫ T

−T
dBL(law(X(t+t1), . . . , X(t+tn)), law(Y (t+t1), . . . , Y (t+tn)))dt = 0.

(c) Assume that X has continuous trajectories. We call that X is pseudo almost
periodic in path distribution if the mapping t 7→ law(X(t + ·)) is continuous and
there exists a stochastic process Y : R × Ω → E which is almost periodic in path
distribution such that

lim
T→∞

1

2T

∫ T

−T
dBL(law(X(t+ ·)), law(Y (t+ ·)))dt = 0.

In this case, the function Z : R → R, Z(t) := dBL(law(X(t + ·)), law(Y (t + ·)), is
called the ergodic part of X and we say that Z is ergodic in the distribution sense.



4 Y.-J. CHEN, H.-S. DING EJDE-2023/34

For p ≥ 1, we denote by Lp(Ω,B) the space of all B-valued random variables X
such that E‖X‖p =

∫
Ω
‖X‖pdP <∞. For X ∈ Lp(Ω,B), let

‖X‖Lp = (E‖X‖p)1/p
.

Then (Lp(Ω,B), ‖ · ‖Lp) is a Banach space.

Definition 2.5 ([9]). Let (Ω,F , P ) be a probability space. A family of measurable
mappings on the sample space, θt : Ω→ Ω, t ∈ R, is called a measurable dynamical
system if the following conditions are satisfied:

(i) identity property: θ0 = IdΩ,
(ii) flow property: θtθs = θt+s, for t, s ∈ R,

(iii) measurability: (ω, t) 7→ θtω is measurable.

It is called a measure-preserving dynamical system if, furthermore,

(iv) measure-preserving property: P (θtA) = P (A), for every A ∈ F and t ∈ R.

In the sequel, we always assume that θ = (θt)t∈R is a measure-preserving dy-
namical system.

Definition 2.6. [15] Let X : R × Ω → B be a stochastic process. Assume that
X(t) ∈ Lp(Ω,B) for every t ∈ R. We say that X is p-mean θ-almost periodic (or
simply θp-almost periodic) if conditions (i) and (ii) below are satisfied:

(i) the mapping R×R→ Lp(Ω,B) defined by (t, s) 7→ X(t+ s, θ−s·) is contin-
uous,

(ii) for every ε > 0, the set

Pθ(ε,X) :=
{
τ : sup

t∈R
(E‖X(t+ τ, θ−τ ·)−X(t, ·)‖p)1/p ≤ ε

}
is relatively dense in R.

We denote by APθ(R, Lp(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 θp-almost periodic random process. Then

(a) the mapping t 7→ X (t+ s, θ−s·) is continuous from R to Lp(Ω,B), uniformly
with respect to s ∈ R.

(b) the mapping s 7→ X (t+ s, θ−s·) is uniformly continuous from R to Lp(Ω,B),
uniformly with respect to t ∈ R.

Proposition 2.8 (Compactness [15]). Let X : R× Ω→ B be a θp-almost periodic
random process, and let J be a compact interval of R. Then

(i) the set LJ = {X (s+ t, θ−t·) : s ∈ J, t ∈ R} is relatively compact in Lp(Ω,B),
(ii) the set S = {law(X(t, ·)) : t ∈ R} is uniformly tight, that is, for each ε > 0,

there exists a compact subset K of B such that

sup
t∈R

P ({ω ∈ Ω : X(t, ω) /∈ K}) ≤ ε.

Definition 2.9. A stochastic process X ∈ BC(R, Lp(Ω,B)) is said to be p-mean
θ-pseudo almost periodic if it can be expressed as

X = Y + Z,

where Y ∈ APθ(R, Lp(Ω,B)) and Z ∈ PAP0(Lp(Ω,B)). The processes Y and Z
are called the almost periodic part and ergodic part of X respectively. Denote by



EJDE-2023/34 PSEUDO ALMOST PERIODIC SOLUTIONS FOR SDES 5

PAPθ(R, Lp(Ω,B)) of all such processes. If p = 2, then we say that X is square-
mean θ-pseudo almost periodic.

Proposition 2.10. If X ∈ PAPθ(R, Lp(Ω,B)), then X is pseudo almost periodic
in finite-dimensional distribution.

Proof. Without loss of generality, assume p = 1. Let (t1, . . . , tn) be a finite sequence
in R. Let us endow Bn with the norm

‖(x1, . . . , xn)‖n =

n∑
k=1

‖xk‖.

Let X = Y + Z where Y ∈ APθ(R, L1(Ω,B)) and Z ∈ PAP0(L1(Ω,B)). Fur-
thermore, it follows from [15, Theorem 4.1] that Y is almost periodic in finite-
dimensional distribution. Since Z = X−Y ∈ PAP0(L1(Ω,B)) and PAP0(L1(Ω,B))
is translation invariant, we have

lim
T→∞

1

2T

∫ T

−T
dBL(law(X(t+ t1), . . . , X(t+ tn)), law(Y (t+ t1), . . . , Y (t+ tn)))dt

≤ lim
T→∞

1

2T

∫ T

−T
E‖(X(t+ t1), . . . , X(t+ tn))− (Y (t+ t1), . . . , Y (t+ tn))‖ndt

≤ lim
T→∞

1

2T

∫ T

−T

n∑
k=1

E‖X(t+ tk)− Y (t+ tk)‖dt = 0.

Similarly, one can show that the mapping t 7→ law(X(t + t1), . . . , X(t + tn)) is
continuous. �

Proposition 2.11. Let Xn ∈ PAPθ(R, Lp(Ω,B)), n = 1, 2, . . . . Assume further
that there exists a stochastic process X such that

lim
n→∞

sup
t∈R
‖Xn(t)−X(t)‖Lp = 0. (2.1)

Then X ∈ PAPθ(R, Lp(Ω,B)).

Proof. Let Ψ ∈ PAPθ(R, Lp(Ω,B)) and Ψ = Y + Z, where Y ∈ APθ(R, Lp(Ω,B))
and Z ∈ PAP0(Lp(Ω,B)). Let us show that ‖Y ‖∞ ≤ ‖Ψ‖∞.

To obtain a contradiction, assume that ‖Y ‖∞ > ‖Ψ‖∞. Then there exists t0 ∈ R
such that ‖Y (t0)‖Lp > supt∈R ‖Ψ(t)‖Lp . Let

λ = ‖Y (t0)‖Lp − sup
t∈R
‖Ψ(t)‖Lp > 0. (2.2)

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 Pθ(

λ
2 , Y ). Thus, for every a ∈ R, there exists some number b

such that [b, b+ δ] ⊂ [a, a+ l] and

‖Y (t0 + τ, θ−τ ·)− Y (t0)‖Lp ≤ λ

2
, τ ∈ [b, b+ δ].

Then

‖Y (t0)‖Lp − λ

2
≤ ‖Y (t0 + τ, θ−τ ·)‖Lp ≤ ‖Y (t0)‖Lp +

λ

2
. (2.3)



6 Y.-J. CHEN, H.-S. DING EJDE-2023/34

Since θ is measure-preserving, we have ‖Y (t0 + τ)‖Lp = ‖Y (t0 + τ, θ−τ ·)‖Lp . Using
(2.2) and (2.3), we have

‖Y (t0 + τ)‖Lp − sup
t∈R
‖Ψ(t)‖Lp ≥ λ

2
.

Hence

‖Z(t0 + τ)‖Lp = ‖Ψ(t0 + τ)− Y (t0 + τ)‖Lp ≥ λ

2
.

This implies that

lim sup
n→∞

1

2nl

∫ nl

−nl
‖Z(t)‖Lpdt ≥ λδ

2l
.

But Z ∈ PAP0(Lp(Ω,B)), and we have a contradiction.
Now, assume that Xn = Yn+Zn, n = 1, 2, . . . , where Yn ∈ APθ(R, Lp(Ω,B)) and

Zn ∈ PAP0(Lp(Ω,B)). By (2.1), we obtain that (Xn)∞n=1 is a Cauchy sequence in
BC(R, Lp(Ω,B)). Thus we deduce that (Yn)∞n=1 and (Zn)∞n=1 are Cauchy sequences
in BC(R, Lp(Ω,B)) since

‖Yn − Ym‖∞ ≤ ‖Xn −Xm‖∞
and Zn = Xn − Yn for n,m ∈ N+. Denote by Ỹ and Z̃ the limits of (Yn)∞n=1

and (Zn)∞n=1, respectively. Then Ỹ ∈ APθ(R, Lp(Ω,B)) and Z̃ ∈ PAP0(Lp(Ω,B))
because APθ(R, Lp(Ω,B)) and PAP0(Lp(Ω,B)) is closed in BC(R, Lp(Ω,B)). It is

easy to see that X = Ỹ + Z̃, hence X ∈ PAPθ(R, Lp(Ω,B)). �

3. 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 (Ft)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 fromH to itself with the operator norm ‖·‖L(H).
Define θ = (θt)t∈R by

θτ (ω)(t) = ω(t+ τ)− ω(τ) = W (t+ τ, ω)−W (τ, ω)

for all τ, t ∈ R and ω ∈ Ω. Then by Definition 2.5, θ = (θt)t∈R is a measure-
preserving dynamical system.

To study equation (1.1), we first list our assumptions:

(H1) A is the infinitesimal generator of a C0-semigroup (T (t))t≥0 and there exists
δ > 0 such that

‖T (t)‖L(H) ≤ e−δt, t ≥ 0.

(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,

‖F (t, x)‖+ ‖G(t, x)‖L(H) ≤ K(1 + ‖x‖),
‖F (t, x)− F (t, y)‖+ ‖G(t, x)−G(t, y)‖L(H) ≤ K‖x− y‖.



EJDE-2023/34 PSEUDO ALMOST PERIODIC SOLUTIONS FOR SDES 7

(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 Ft-progressively measurable stochastic process X(t),
t ∈ R is called the mild solution of equation (1.1) if it satisfies

X(t) = T (t− s)X(s) +

∫ t

s

F (r,X(r)) dr +

∫ t

s

G(r,X(r)) dW (r)

for t, s ∈ R with t ≥ s.

Next, we give some technical lemmas for later use.

Lemma 3.2 ([2]). Let h ∈ PAP0(R). Then the function

t 7→
(∫ t

−∞
e−δ(t−s)h2(s)ds

)1/2

is also in PAP0(R).

Lemma 3.3. Let g ∈ PAP0(R). Then for every T > 0, the function

t 7→
∫ t+T

t−T
g(σ)dσ

is also in PAP0(R).

Proof. Since PAP0(R) is translation invariant, g(·+ t) ∈ PAP0(R) for every t ∈ R.
Then, by Lebesgue’s dominated convergence theorem, we obtain

1

2r

∫ r

−r

∫ t+T

t−T
g(σ)dσdt =

1

2r

∫ r

−r

∫ T

−T
g(σ + t)dσdt

=
1

2r

∫ T

−T

∫ r

−r
g(σ + t)dtdσ

=

∫ T

−T

( 1

2r

∫ r

−r
g(σ + t)dt

)
dσ → 0 as r →∞. �

Define the operator ψ : BC(R, Lp(Ω, H))→ BC(R, Lp(Ω, H)) by

(ψX)(t) =

∫ t

−∞
T (t− s)F (s,X(s)) ds+

∫ t

−∞
T (t− s)G(s,X(s))dW (s).

It is easy to see that ψ is well defined.

Theorem 3.4. Assume that conditions (H1)–(H3) hold. Then the operator ψ maps
PAPθ(R, L2(Ω, H)) into itself.

Proof. Let F = F1 + F2, where F1(·, x) and F2(·, x) are the almost periodic com-
ponent and ergodic perturbation of the function F (·, x) for every x ∈ H. Let
G = G1 + G2, where G1(·, x) and G2(·, 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 F1, G1 satisfy assumption (H2). Consequently, the functions
F2, G2 satisfy assumption (H2) with constant 2K.



8 Y.-J. CHEN, H.-S. DING EJDE-2023/34

Let X ∈ PAPθ(R, L2(Ω, H)) and X = Y +Z, where Y ∈ APθ(R, L2(Ω, H)) and
Z ∈ PAP0(L2(Ω, H)). By [15, Proposition 5.1], we obtain ψ1Y ∈ APθ(R, L2(Ω, H)),
where

(ψ1Y )(t) =

∫ t

−∞
T (t− s)F1(s, Y (s)) ds+

∫ t

−∞
T (t− s)G1(s, Y (s)) dW (s).

Now, we prove that ψX − ψ1Y ∈ PAP0(L2(Ω, H)). By definition of ψ and ψ1, we
have

ψX(t)− ψ1Y (t)

=

∫ t

−∞
T (t− s)[F (s,X(s))− F1(s, Y (s))] ds

+

∫ t

−∞
T (t− s)[G(s,X(s)−G1(s, Y (s))] dW (s)

=

∫ t

−∞
T (t− s)[F (s,X(s))− F (s, Y (s))]ds+

∫ t

−∞
T (t− s)F2(s, Y (s)) ds

+

∫ t

−∞
T (t− s)[G(s,X(s)−G(s, Y (s))] dW (s)

+

∫ t

−∞
T (t− s)G2(s, Y (s)) dW (s)

=: I1(t) + I2(t) + I3(t) + I4(t).

Since Z ∈ PAP0(L2(Ω, H)), we have that s 7→ (E‖Z(s)‖2)1/2 is in PAP0(R).

Moreover, using Lemma 3.2, we deduce that t 7→ (
∫ t
−∞ e−δ(t−s)E‖Z(s)‖2ds)1/2 is

also in PAP0(R). Then, using conditions (H1), (H2), and Hölder’s inequality, we
obtain

lim
r→∞

1

2r

∫ r

−r

(
E‖I1(t)‖2

)1/2
dt

≤ lim
r→∞

1

2r

∫ r

−r

(
E(

∫ t

−∞
e−δ(t−s)K‖X(s)− Y (s)‖ds)2

)1/2

dt

≤ K 1√
δ

lim
r→∞

1

2r

∫ r

−r

(
E

∫ t

−∞
e−δ(t−s)‖X(s)− Y (s)‖2ds

)1/2

dt

≤ K 1√
δ

lim
r→∞

1

2r

∫ r

−r

(∫ t

−∞
e−δ(t−s)E‖Z(s)‖2ds

)1/2

dt = 0.

This implies that I1 ∈ PAP0(L2(Ω, H)).
Furthermore, using Itô’s isometry, we have

lim
r→∞

1

2r

∫ r

−r

(
E‖I3(t)‖2

)1/2
dt

≤ (trQ)1/2 lim
r→∞

1

2r

∫ r

−r

(
E

∫ t

−∞
e−2δ(t−s)‖G(s,X(s))−G(s, Y (s))‖2L(H)ds

)1/2

dt

≤ K(trQ)1/2 lim
r→∞

1

2r

∫ r

−r

(∫ t

−∞
e−2δ(t−s)E‖X(s)− Y (s)‖2ds

)1/2

dt

≤ K(trQ)1/2 lim
r→∞

1

2r

∫ r

−r

(∫ t

−∞
e−2δ(t−s)E‖Z(s)‖2ds

)1/2

dt = 0.



EJDE-2023/34 PSEUDO ALMOST PERIODIC SOLUTIONS FOR SDES 9

Thus I3 ∈ PAP0(L2(Ω, H)).
Let us estimate the second term I2. By [19, Lemma 5.10], I2 ∈ PAP0(L2(Ω, H))

if and only if the mapping

t 7→ E‖
∫ t

−∞
T (t− s)F2(s, Y (s))ds‖2

is in PAP0(R). Using condition (H1) and Hölder’s inequality, we have

E‖
∫ t

−∞
T (t− s)F2(s, Y (s))ds‖2 ≤ E(

∫ t

−∞
e−δ(t−s)‖F2(s, Y (s))‖ds)2

≤ 1

δ

∫ t

−∞
e−δ(t−s)E‖F2(s, Y (s))‖2ds.

Using Proposition 2.8, we have that the family (Y (s, θ−s·))s∈R is uniformly square
integrable. Thus the family (Y (s, ·))s∈R is uniformly square integrable since θ is
measure-preserving. Then for every ε > 0, there exists a constant ϑ ∈ (0, ε) such
that for every A ∈ F with P (A) < ϑ, we have

sup
s∈R

E‖Y (s)‖21A < ε. (3.1)

Moreover, the set {law(Y (s, ·)) : s ∈ R} is uniformly tight. Then there exists a
compact subset K ⊂ B such that

sup
s∈R

P (Y (s) ∈ K) > 1− ϑ. (3.2)

Since K is compact, there are points x1, x2, . . . , xJ ∈ K such that K is covered by
the balls of radius ε with centers at the points xi, i = 1, 2, . . . , J . We denote the set
{Y (s, ·) ∈ K} by As and {Y (s, ·) /∈ K} by Acs. Then using (3.1) and (3.2), we have

∆1 :=
1

δ

∫ t

−∞
e−δ(t−s)E

(
‖F2(s, Y (s))‖21As

)
ds

≤ 1

δ

∫ t

−∞
e−δ(t−s)(2Kε+

J∑
i=1

‖F2(s, xi)‖)2ds

≤ 2

δ

∫ t

−∞
e−δ(t−s)(4K2ε2 + J

J∑
i=1

‖F2(s, xi)‖2)ds

≤ 8K2

δ2
ε2 +

2J

δ

J∑
i=1

∫ t

−∞
e−δ(t−s)‖F2(s, xi)‖2ds

and

∆2 :=
1

δ

∫ t

−∞
e−δ(t−s)E

(
‖F2(s, Y (s))‖21Ac

s

)
ds

≤ 4K2 1

δ

∫ t

−∞
e−δ(t−s)E

(
(1 + ‖Y (s)‖)21Ac

s

)
ds

≤ 8K2 1

δ

∫ t

−∞
e−δ(t−s)E

(
(1 + ‖Y (s)‖2)1Ac

s

)
ds

≤ 8K2 1

δ2
ε+ 8K2 1

δ2
ε = 16K2 1

δ2
ε.



10 Y.-J. CHEN, H.-S. DING EJDE-2023/34

For every xi, i = 1, 2, . . . , J , F2(·, xi) ∈ PAP0(H). By Lemma 3.2 and [19, Lemma
5.10], the function

t 7→
∫ t

−∞
e−δ(t−s)‖F2(s, xi)‖2ds

is in PAP0(R). Thus

lim sup
r→∞

1

2r

∫ r

−r
E‖
∫ t

−∞
T (t− s)F2(s, Y (s))ds‖2dt

≤ lim sup
r→∞

1

2r

∫ r

−r
(∆1 + ∆2)dt

≤ Cε+
2J

δ

J∑
i=1

lim
r→∞

1

2r

∫ r

−r

∫ t

−∞
e−δ(t−s)‖F2(s, xi)‖2dsdt

= Cε,

where C is a constant. Since ε is arbitrary, we have

lim
r→∞

1

2r

∫ r

−r
E‖
∫ t

−∞
T (t− s)F2(s, Y (s)) ds‖2dt = 0.

Hence I2 ∈ PAP0(L2(Ω, H)).
By [19, Lemma 5.10] again, I4 ∈ PAP0(L2(Ω, H)) if and only if the mapping

t 7→ E‖
∫ t

−∞
T (t− s)G2(s, Y (s))dW (s)‖2

is in PAP0(R). Using Itô’s isometry, we obtain

E‖
∫ t

−∞
T (t− s)G2(s, Y (s))dW (s)‖2 ≤ (trQ)

∫ t

−∞
e−2δ(t−s)E‖G2(s, Y (s))‖2L(H)ds.

For the same reason as for I2, we have I4 ∈ PAP0(L2(Ω, H)).
Gathering the estimates for I1-I4, we deduce that ψX−ψ1Y ∈ PAP0(L2(Ω, H)).

Thus ψX ∈ PAPθ(R, L2(Ω, H)). �

Theorem 3.5. Assume that conditions (H1)–(H3) hold. If η = 2K2

δ2 + K2

δ trQ < 1,

then (1.1) has a unique L2-bounded mild solution X which satisfies, for every t ∈ R,

X(t) =

∫ t

−∞
T (t− s)F (s,X(s))ds+

∫ t

−∞
T (t− s)G(s,X(s))dW (s).

Moreover, X ∈ PAPθ(R, L2(Ω, H)) and X is pseudo almost periodic in path distri-
bution.

Proof. The proof of the existence and uniqueness of a mild solution to (1.1) in the
functions space BC(R, L2(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
n→∞

sup
t∈R

E‖Xn(t)−X(t)‖2 = 0,

where X0 = 0 and Xn = ψXn−1, n = 1, 2, . . . . By Proposition 2.11 and Theorem
3.4, we deduce that X ∈ PAPθ(R, L2(Ω, H)).

Now, let us prove that X is also pseudo almost periodic in path distribution.
Let F = F1 + F2, where F1(·, x) and F2(·, x) are the almost periodic component



EJDE-2023/34 PSEUDO ALMOST PERIODIC SOLUTIONS FOR SDES 11

and ergodic perturbation of the function F (·, x) for every x ∈ H. Let G = G1 +
G2, where G1(·, x) and G2(·, 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 F1, G1 satisfy assumption (H2). Consequently,
the functions F2, G2 satisfy assumption (H2) with constant 2K. By [15, Theorem
5.1], there exists a stochastic process Y which is the mild solution of

dY (t) = AY (t)dt+ F1(t, Y (t))dt+G1(t, Y (t))dW (t).

In other words, Y satisfies for every t, s ∈ R with t ≥ s,

Y (t) = T (t− s)Y (s) +

∫ t

s

T (t−σ)F1(σ, Y (σ)) dσ+

∫ t

s

T (t−σ)G1(σ, Y (σ))dW (σ).

Moreover, Y ∈ APθ(R, L2(Ω, H)) and Y is also almost periodic in path distribution.

Step 1. For every positive integer N , the mapping

t 7→ E sup
s∈[t−N,t+N ]

‖X(s)− Y (s)‖2

is in PAP0(R). By the definition of mild solutions and condition (H1), we have

E sup
s∈[t−N,t+N ]

‖X(s)− Y (s)‖2

≤ 3E sup
s∈[t−N,t+N ]

‖T (s− (t−N))X(t−N)− T (s− (t−N))Y (t−N)‖2

+ 3E sup
s∈[t−N,t+N ]

‖
∫ s

t−N
T (s− σ)F (σ,X(σ))− T (s− σ)F1(σ, Y (σ))dσ‖2

+ 3E sup
s∈[t−N,t+N ]

‖
∫ s

t−N
T (s− σ)G(σ,X(σ))− T (s− σ)G1(σ, Y (σ))dW (σ)‖2

≤ 3E‖X(t−N)− Y (t−N)‖2

+ 3E
(∫ t+N

t−N
‖F (σ,X(σ))− F1(σ, Y (σ))‖dσ

)2

+ 3E sup
s∈[t−N,t+N ]

‖
∫ s

t−N
T (s− σ)G(σ,X(σ))− T (s− σ)G1(σ, Y (σ))dW (σ)‖2

=: Σ1(t) + Σ2(t) + Σ3(t).

By Theorem 3.4, we have Z = X − Y ∈ PAP0(L2(Ω, H)). Then by [19, Lemma
5.10], we have E‖Z(·)‖2 ∈ PAP0(R). Since PAP0(R) is translation invariant, we
deduce that Σ1 ∈ PAP0(R).

Using condition (H2) and Hölder’s inequality, we obtain

Σ2(t) = 3E
(∫ t+N

t−N
‖F (σ,X(σ))− F1(σ, Y (σ))‖ dσ

)2

≤ 12N2E

∫ t+N

t−N
‖F (σ,X(σ))− F1(σ, Y (σ))‖2dσ

≤ 24N2E

∫ t+N

t−N
‖F (σ,X(σ))− F (σ, Y (σ))‖2 + ‖F2(σ, Y (σ))‖2dσ



12 Y.-J. CHEN, H.-S. DING EJDE-2023/34

≤ 24K2N2E

∫ t+N

t−N
‖X(σ)− Y (σ)‖2dσ + 24N2E

∫ t+N

t−N
‖F2(σ, Y (σ))‖2dσ.

For every ε > 0, let ϑ,K, x1, x2, . . . , xJ be the same as in Theorem 3.4. Then by a
similar argument of I2 in Theorem 3.4, we have

24N2E

∫ t+N

t−N
‖F2(σ, Y (σ))‖2dσ ≤ C̃ε+ 2J

J∑
i=1

∫ t+N

t−N
‖F2(σ, xi)‖2dσ,

where C̃ is a constant. By Lemma 3.3, we have that the mappings

t 7→ E

∫ t+N

t−N
‖X(σ)− Y (σ)‖2dσ and t 7→

∫ t+N

t−N
‖F2(σ, xi)‖2dσ

are in PAP0(R). Since ε is arbitrary, we deduce that Σ2 ∈ PAP0(R).
By [8, Theorem 6.10], for every L > 0, there exists a constant CL such that, for

every a ∈ R and every predictable stochastic process Φ with E
∫ a+L

a
‖Φ‖2ds <∞,

we have

E sup
t∈[a,a+L]

‖
∫ t

a

T (t− s)Φ(s) dW (s)‖2 ≤ CLE
∫ a+L

a

‖Φ(s)‖2ds.

Then we obtain

Σ3(t) ≤ 3CnE

∫ t+N

t−N
‖G(σ,X(σ))−G1(σ, Y (σ))‖2dσ.

For the same reason as for Σ2, we have Σ3 ∈ PAP0(R).
Gathering the estimations for Σ1-Σ3, we conclude that the mapping

t 7→ E sup
s∈[t−N,t+N ]

‖X(s)− Y (s)‖2

is in PAP0(R).

Step 2. We claim that

lim
r→∞

1

2r

∫ r

−r
dBL(law(X(t+ ·)), law(Y (t+ ·)))dt = 0.

For every positive integer N , we have

dBL(law(X(t+ ·)), law(Y (t+ ·)))

≤
∫

Ω

∞∑
k=1

1

2k
sups∈[−k,k] ‖X(t+ s)− Y (t+ s)‖

1 + sups∈[−k,k] ‖X(t+ s)− Y (t+ s)‖
dP

≤
N∑
k=1

∫
Ω

1

2k
sup

s∈[−k,k]

‖X(t+ s)− Y (t+ s)‖dP +

∞∑
k=N+1

1

2k

≤ N
(
E sup
s∈[t−N,t+N ]

‖X(s)− Y (s)‖2
)1/2

+

∞∑
k=N+1

1

2k
.

Let ε > 0. Choose N large enough such that
∑∞
k=N+1

1
2k < ε. Then by Step 1

and [19, Lemma 5.10], we have

lim sup
r→∞

1

2r

∫ r

−r
dBL(law(X(t+ ·)), law(Y (t+ ·)))dt



EJDE-2023/34 PSEUDO ALMOST PERIODIC SOLUTIONS FOR SDES 13

≤ lim
r→∞

1

2r

∫ r

−r
N
(
E sup
s∈[t−N,t+N ]

‖X(s)− Y (s)‖2
)1/2

dt+ ε = ε.

Since ε is arbitrary, we obtain

lim
r→∞

1

2r

∫ r

−r
dBL(law(X(t+ ·)), law(Y (t+ ·)))dt = 0.

Step 3. Let us show that t 7→ law(X(t + ·)) is continuous. For every positive
integer N , every t0 ∈ R and every t ∈ R with |t− t0| < 1,

dBL(law(X(t+ ·)), law(X(t0 + ·)))

≤
N∑
k=1

∫
Ω

1

2k
sup

s∈[−k,k]

‖X(t+ s)−X(t0 + s)‖dP +

∞∑
k=N+1

1

2k

≤ N
∫

Ω

sup
s∈[−N,N ]

‖X(t+ s)−X(t0 + s)‖dP +

∞∑
k=N+1

1

2k
.

Let ε > 0. Choose N large enough such that
∑∞
k=N+1

1
2k < ε

2 . By a simple
calculation we obtain

E sup
s∈[t0−N−1,t0+N+1]

‖X(s)‖2 <∞.

Thus E sups∈[t0−N−1,t0+N+1] 2‖X(s)‖ < ∞. Since X has continuous trajectories
and

sup
s∈[−N,N ]

‖X(t+ s)−X(t0 + s)‖ ≤ sup
s∈[t0−N−1,t0+N+1]

2‖X(s)‖,

we have
∫

Ω
sups∈[−N,N ] ‖X(t + s) − X(t0 + s)‖dP → 0 as t → t0 by Dominated

convergence theorem. Then there exists a constant h > 0 such that |t− t0| < h,∫
Ω

sup
s∈[−N,N ]

‖X(t+ s)−X(t0 + s)‖dP <
ε

2
.

Hence dBL(law(X(t + ·)), law(X(t0 + ·))) < ε. This implies that t 7→ law(X(t +
·)) is continuous. Then by Definition 2.4, X is pseudo almost periodic in path
distribution. �

Acknowledgments. Y.-J. Chen acknowledges support from the Graduate Inno-
vation Fund of Jiangxi Province (YC2021-B078) and by the Graduate Innovation
Fund of Jiangxi normal university (YJS2022058). H.-S. Ding acknowledges sup-
port from the NSFC (11861037) and from the Two Thousand Talents Program of
Jiangxi Province (jxsq2019201001).

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Ye-Jun Chen
School of Mathematics and statistics, Jiangxi Normal University, Nanchang, Jiangxi
330022, China

Email address: chenyejun999@jxnu.edu.cn

Hui-Sheng Ding (corresponding author)

School of Mathematics and statistics, and Jiangxi Provincial Center for Applied Math-
ematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

Email address: dinghs@mail.ustc.edu.cn


	1. Introduction
	2. Preliminaries
	3. Main results
	Acknowledgments

	References