Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 113, pp. 1–9.

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

CONTINUOUS DEPENDENCE OF RECURRENT SOLUTIONS

FOR STOCHASTIC DIFFERENTIAL EQUATIONS

HAIJING QIU, YAN WANG

Abstract. Existence, uniqueness and asymptotic stability of recurrent solu-

tions have been investigated extensively for semi-linear stochastic differential

equations. In this article, we show that the unique recurrent solution depends
continuously on the coefficients of the equation in the compact-open topology

or uniform topology, which depends on how the coefficients vary with respect

to the parameter.

1. Introduction

Recurrent motions are important in dynamical systems since almost all interest-
ing and complicated dynamics happens on the recurrent set. On the other hand,
recurrence is also extensively studied for Markov processes since it is closely related
to invariant measures (or stationary distributions) of Markov processes. Because
of these facts, there are many studies on recurrent solutions for stochastic differen-
tial equations (SDEs) so far. Among others, let us mention some works which are
closely related to our work: see Khasminskii [13], Morozan [18], Da Prato and Tu-
dor [8], Chen et al [6] and Ji et al [12] for periodic solutions for SDEs, see Halanay
[10], Da Prato and Tudor [8], Arnold and Tudor [1], Bezandry and Diagana [2],
Wang and Liu [20], Liu and Wang [17], Li et al [14] for almost periodic solutions
for SDEs, see Fu and Liu [9], Chen and Lin [7], Wang and Gao [19], Liu and Sun
[16], Chang and Tang [4] for almost automorphic solutions for SDEs, and see the
very recent work Cheban and Liu [5] (see also Liu and Liu [15] for Lévy noise case)
for general recurrent solutions for SDEs.

When the system (or the SDE) depends on a parameter, a natural question
is: when the parameter varies, does the SDE still possess recurrent solutions and
whether the recurrent solutions vary continuously with respect to the parameter?
Answers to this question will enable us to understand deeply the robustness or
global bifurcation phenomenon of the system in consideration, which further shed
light on our insight into the real problem the system describes. Therefore, to partly
investigate this question we consider in present paper continuous dependence of
recurrent solutions (including stationary solutions as special case) on the coeffi-
cients of stochastic differential equations, in contrast to and based on existence and
uniqueness of recurrent solutions that are studied in above mentioned works.

2010 Mathematics Subject Classification. 35B30, 37L15, 37B20, 60H10.
Key words and phrases. Stochastic differential equation; continuous dependence;

recurrent solution.
c©2020 Texas State University.

Submitted August 2, 2020. Published November 17, 2020.

1



2 H. QIU, Y. WANG EJDE-2020/113

2. Preliminaries

Let H be a real separable Hilbert space with the norm | · |, (Ω,F ,P) be a
probability space, and Lp(P, H) be the space of H-valued random variables with
finite p-order moments (p ≥ 1), i.e. ‖x‖p := (E|x|p)1/p = (

∫
Ω
|x|pdP)1/p < ∞ for

x ∈ Lp(P, H). We denote by Cb(R, Lp(P, H)) the space of bounded continuous
mappings from R to Lp(P, H) with the norm ‖f‖∞ := supt∈R ‖f(t)‖p for f ∈
Cb(R, Lp(P, H)).

Let us consider the stochastic differential equation

dx(t) = (Ax(t) + F (t, x(t)))dt+G(t, x(t))dW (t), (2.1)

where F,G ∈ C(R ×H,H), W is a two-sided standard one-dimensional Brownian
motion defined on (Ω,F ,P), and A generates a C0-semigroup {U(t)}t≥0 on H which
is exponentially stable, i.e. there are positive constants N , ν such that

‖U(t)‖ ≤ N e−νt for t ≥ 0. (2.2)

Definition 2.1. We say that functions F and G satisfy the condition

(C1) if there exists a number A0 ≥ 0 such that |F (t, 0)|, |G(t, 0)| ≤ A0 for any
t ∈ R;

(C2) if there exists a number L ≥ 0 such that Lip(F ),Lip(G) ≤ L, where

Lip(F ) := sup
{ |F (t, x1)− F (t, x2)|

|x1 − x2|
: x1 6= x2, t ∈ R

}
;

(C3) if F and G are continuous in t uniformly w.r.t. x on each bounded subset
Q ⊂ H.

Proposition 2.2 ([5, Proposition 4.4]). Consider the equation (2.1). Suppose that
the functions F and G satisfy conditions (C1) and (C2). For p > 2, denote

cp :=
[p(p− 1)

2

( p

p− 1

)p−2
]p/2

.

If

θp := 2p−1N pLp
[(2(p− 1)

νp

)p−1

+ cp

(p− 2

νp

)p/2−1] 2

νp
< 1,

then (2.1) admits a unique bounded solution in Cb(R, Lp(P, H)).

Remark 2.3. Note that the contraction constant θp is continuous in p when p > 2.
Furthermore, cp = 1 when p = 2 in Proposition 2.2, so we have

lim
p→2+

θp =
2N 2L2

ν2
+

2N 2L2

ν
.

Proposition 2.4 ([5, Theorem 4.6, Corollary 4.7]). Consider the equation (2.1).
Suppose that the functions F and G satisfy the conditions (C1) and (C2). Then the
following statements hold:

(1) If L < ν
N
√

2+ν
, then (2.1) has a unique solution ξ ∈ C(R, B[0, r]), where

r =
NA0

√
2 + ν

ν −NL
√

2 + ν
, (2.3)

B[0, r] := {x ∈ L2(P, H) : ‖x‖2 ≤ r}. (2.4)



EJDE-2020/113 CONTINUOUS DEPENDENCE OF RECURRENT SOLUTIONS FOR SDES 3

(2) If L < ν
2N
√

1+ν
and additionally F,G satisfy (C3), then the solution ξ has

the same recurrence property in distribution as the coefficients (F,G) with
respect to t; that is, if (F,G) are jointly periodic, quasi-periodic, almost
periodic, almost automorphic, Poisson stable etc in t, then the distribution
t 7→ µ(t) of ξ(·) is periodic, quasi-periodic, almost periodic, almost auto-
morphic, Poisson stable etc.

By simple change of notation in the proof, we have the following result.

Proposition 2.5 ([11, Theorem 3.2 on page 7]). Let A be a closed subset of a
complete metric space (X , d) and Y a metric space. If for each fixed x ∈ A the
mapping y 7→ Tyx is continuous and Ty : A → A is a family of uniform contraction
on A, i.e. there exists a constant θ ∈ [0, 1) such that

d(Tyx, Tyx̄) ≤ θd(x, x̄) for all y ∈ Y and x, x̄ ∈ A.

Then the unique fixed point g(y) of Ty is continuous in y ∈ Y.

3. Main results

Theorem 3.1. Consider the equation (2.1) and the family of stochastic differential
equations

dx(t) = (Ax(t) + Fλ(t, x(t)))dt+Gλ(t, x(t))dW (t), (3.1)

where λ ∈ Λ, with Λ being a metric space. Assume that the conditions of Proposition
2.4 hold with (F,G) replaced by (Fλ, Gλ), and that the constants A0,L remain the
same for all λ ∈ Λ. Then the following statements hold for the unique L2-bounded
(also recurrent) solution ξλ of (3.1):

(1) Fix λ0 ∈ Λ. If for any R > 0,

sup
t∈R,|x|≤R

|Fλ(t, x)− F (t, x)| → 0 and sup
t∈R,|x|≤R

|Gλ(t, x)−G(t, x)| → 0

as λ → λ0, then ξλ converges to the unique bounded solution ξ of (2.1) in
the norm ‖ · ‖∞ as λ→ λ0.

(2) If the convergence of coefficients happens in the BUC space, i.e. for any
R > 0

sup
|t|≤R,|x|≤R

|Fλ(t, x)− F (t, x)| → 0, sup
|t|≤R,|x|≤R

|Gλ(t, x)−G(t, x)| → 0

as λ → λ0, then ξλ converges to ξ in L2-norm uniformly on any compact
interval as λ→ λ0.

Proof. Since Λ is a metric space, we only need to prove the results for sequences.
Let λn → λ0 as n→∞. Consider the sequence of equations

dx(t) = (Ax(t) + Fn(t, x(t)))dt+Gn(t, x(t))dW (t), (3.2)

with Fn := Fλn → F and Gn := Gλn → G as n→∞.
(i) Let the operator Φ : Cb(R, L2(P, H))→ Cb(R, L2(P, H)) given by

(Φφ)(t) :=

∫ t

−∞
U(t− τ)F (τ, φ(τ))dτ +

∫ t

−∞
U(t− τ)G(τ, φ(τ))dW (τ)



4 H. QIU, Y. WANG EJDE-2020/113

be the contraction mapping which produces the bounded (recurrent) solution for
(2.1); see the proof of [5, Theorem 4.6] for details. On the other hand, note that
limp→2+ θp < 1 (see Remark 2.3) if and only if

L < ν

N
√

2(1 + ν)
,

which is satisfied by our assumption on L. So it follows from Proposition 2.2 that
(2.1) admits a unique Lp-bounded solution for some p > 2. This Lp-bounded
solution is exactly the unique L2-bounded solution ξ of (2.1).

Similarly we denote the contraction mapping for the equation (3.2) by Φn. Then
for given φ ∈ Cb(R, Lp(P, H)) ⊂ Cb(R, L2(P, H)) and t ∈ R we have

E|Φn(φ)(t)− Φ(φ)(t)|2

= E
∣∣∣ ∫ t

−∞
U(t− τ)Fn(τ, φ(τ))dτ +

∫ t

−∞
U(t− τ)Gn(τ, φ(τ))dW (τ)

−
∫ t

−∞
U(t− τ)F (τ, φ(τ))dτ −

∫ t

−∞
U(t− τ)G(τ, φ(τ))dW (τ)

∣∣∣2
≤ 2E

∣∣∣ ∫ t

−∞
U(t− τ)[Fn(τ, φ(τ))− F (τ, φ(τ))]dτ

∣∣∣2
+ 2
∣∣∣ ∫ t

−∞
U(t− τ)[Gn(τ, φ(τ))−G(τ, φ(τ))]dW (τ)

∣∣∣2
≤ 2N 2

∫ t

−∞
e−ν(t−τ)dτ

∫ t

−∞
e−ν(t−τ)E|Fn(τ, φ(τ))− F (τ, φ(τ))|2dτ

+ 2N 2

∫ t

−∞
e−2ν(t−τ)E|Gn(τ, φ(τ))−G(τ, φ(τ))|2dτ

≤ 2N 2

ν2
sup
τ∈R

E|Fn(τ, φ(τ))− F (τ, φ(τ))|2

+
N 2

ν
sup
τ∈R

E|Gn(τ, φ(τ))−G(τ, φ(τ))|2.

(3.3)

That is,

‖Φn(φ)− Φ(φ)‖2∞ ≤
2N 2

ν2
sup
τ∈R

E|Fn(τ, φ(τ))− F (τ, φ(τ))|2

+
N 2

ν
sup
τ∈R

E|Gn(τ, φ(τ))−G(τ, φ(τ))|2.
(3.4)

For the above fixed φ ∈ Cb(R, Lp(P, H)), the family {|φ(τ)|2 : τ ∈ R} is uniformly
integrable (see e.g. [3, page 31] for details), and hence by conditions (C1) and (C2)
the family

{|Fn(τ, φ(τ))− F (τ, φ(τ))|2 : n ∈ N, τ ∈ R}

is uniformly integrable. So it follows that for any ε > 0, there exists R > 0 such
that

sup
τ∈R,n∈N

E[|Fn(τ, φ(τ))− F (τ, φ(τ))|2I|Fn(τ,φ(τ))−F (τ,φ(τ))|2≥R] < ε,



EJDE-2020/113 CONTINUOUS DEPENDENCE OF RECURRENT SOLUTIONS FOR SDES 5

where IA means the indicator function of the set A. On the other hand, by as-
sumption we have

lim
n→∞

sup
τ∈R,|x|≤R

|Fn(τ, x)− F (τ, x)| = 0.

So we have

lim
n→∞

sup
τ∈R

E|Fn(τ, φ(τ))− F (τ, φ(τ))|2 = 0.

Similarly,

lim
n→∞

sup
τ∈R

E|Gn(τ, φ(τ))−G(τ, φ(τ))|2 = 0.

Therefore, it follows from (3.4) that Φn(φ)→ Φ(φ) in the norm ‖ · ‖∞ as n→∞.
Applying Proposition 2.5 by setting X = Cb(R, L2(P, H)), A = Cb(R, Lp(P, H))

and Y = Λ, we obtain the desired result.
(ii) Let now k > 0, l > L > 0, t ∈ [−L,L], and f be a bounded nonnegative

function. Then we have∫ t

−∞
e−k(t−τ)f(τ)dτ =

∫ −l
−∞

e−k(t−τ)f(τ)dτ +

∫ t

−l
e−k(t−τ)f(τ)dτ

≤ sup
t∈R

f(t)
e−k(t+l)

k
+ sup
|t|≤l

f(t)
1− e−k(t+l)

k
.

Consequently,

max
|t|≤L

∫ t

−∞
e−k(t−τ)f(τ)dτ ≤ ekLe−kl

k
sup
t∈R

f(t) +
1− e−kLe−kl

k
sup
|t|≤l

f(t). (3.5)

For given φ ∈ C(R, B[0, r]) and t ∈ R (recalling that B[0, r] is defined in (2.4)),
by (3.3) we have

E|Φn(φ)(t)− Φ(φ)(t)|2

≤ 2N 2

ν

∫ t

−∞
e−ν(t−τ)E|Fn(τ, φ(τ))− F (τ, φ(τ))|2dτ

+ 2N 2

∫ t

−∞
e−2ν(t−τ)E|Gn(τ, φ(τ))−G(τ, φ(τ))|2dτ.

(3.6)

Applying (3.5) to (3.6) with k = ν and 2ν, f(τ) = E|Fn(τ, φ(τ))−F (τ, φ(τ))|2 and
f(τ) = E|Gn(τ, φ(τ))−G(τ, φ(τ))|2 respectively, we obtain

max
|t|≤L

E|Φn(φ)(t)− Φ(φ)(t)|2

≤ 2N 2

ν2
eν(L−l) sup

t∈R
EI2

n(t) +
N 2

ν
e2ν(L−l) sup

t∈R
EJ2

n(t)

+
2N 2

ν2
[1− e−ν(L+l)] sup

|t|≤l
EI2

n(t) +
N 2

ν
[1− e−2ν(L+l)] sup

|t|≤l
EJ2

n(t),

(3.7)

where In(t) := |Fn(t, φ(t))− F (t, φ(t))| and Jn(t) := |Gn(t, φ(t))−G(t, φ(t))|.
By conditions (C1) and (C2), we have

max
{

sup
t∈R

EI1
n(t), sup

t∈R
EI2

n(t)
}
≤ 4(A0 + Lr)2.



6 H. QIU, Y. WANG EJDE-2020/113

Let now {ln} be a sequence of positive numbers such that ln → +∞ as n → ∞.
Then by (3.7) we have

max
|t|≤L

E|Φn(φ)(t)− Φ(φ)(t)|2

≤
(2N 2

ν2
+
N 2

ν

)
eν(L−ln)4(A0 + Lr)2

+
(2N 2

ν2
+
N 2

ν

)
max

{
sup
|t|≤ln

EI2
n(t), sup

|t|≤ln
EJ2

n(t)
}
.

(3.8)

Since Fn → F and Gn → G in the BUC space, by [5, Remarks 2.2 and 2.30],
passing to limit in (3.8) as n→∞, for L > 0 we obtain

lim
n→∞

max
|t|≤L

E|Φn(φ)(t)− Φ(φ)(t)|2 = 0

by the uniform integrability of the families {I2
n(t) : n ∈ N, t ∈ R} and {J2

n(t) : n ∈
N, t ∈ R}.

Finally, applying Proposition 2.5 with X = C(R, L2(P, H)), A = C(R, Lp(P, H)),
both endowed with the Bebutov metric (see the metric d in [5, §2.1] for details),
and Y = Λ, we obtain the desired result. The proof is complete. �

Remark 3.2. To simplify the notation and highlight the idea, we consider only
the most simple noise: one-dimensional Brownian motion. Indeed, the main results
of this paper still hold for more general noise case, i.e. when the noise W in (2.1)
is replaced by a Q-Wiener process, which brings no essential but just notational
difference; see e.g. [15, 16, 20] for details.

As a simple but important case, we have the following result on the contin-
uous dependence of recurrent solutions for asymptotically autonomous stochastic
systems.

Corollary 3.3. Consider the equation (2.1) and the family of stochastic differential
equations (3.1). Assume that the coefficients F,G of (2.1) are independent of t
and that the conditions of Theorem 3.1 hold. Then the unique L2-bounded (also
recurrent) solution of (3.1) converges to the unique stationary solution of (2.1) as
λ→ λ0, and the mode of convergence is the same as their coefficients.

A much simpler case is the continuous dependence of stationary distributions.

Corollary 3.4. Consider the family of stochastic differential equations (3.1) and
its limit equation (2.1). Assume that the coefficients Fλ, Gλ of (3.1) and F,G of
(2.1) are independent of t and that the conditions of Theorem 3.1 hold. Then the
unique stationary solution (hence stationary distribution) of (3.1) converges to that
of (2.1) as λ→ λ0.

4. Applications

In this section we illustrate our results with two examples.

Example 4.1. Consider the stochastic differential equation

dy =
(
− 4y +

λ sin t

3 + cos
√

3t

y

y2 + 1

)
dt+

(1

2
y +

1 + λ

2 + cos t+ cos
√

2 t

)
dW

=: (Ay + fλ(t, y))dt+ gλ(t, y)dW,

(4.1)



EJDE-2020/113 CONTINUOUS DEPENDENCE OF RECURRENT SOLUTIONS FOR SDES 7

where λ ∈ [0, 1] is a parameter and W is a one-dimensional two-sided Brownian
motion. Clearly A generates an exponentially stable semigroup on R with N = 1
and ν = 4. Note that for each λ, fλ is quasi-periodic in t and gλ is Levitan
almost periodic in t, uniformly with respect to y on any bounded subset of R, so
fλ, gλ are jointly Levitan almost periodic. The Lipschitz constants of fλ, gλ satisfy
max{Lip(fλ),Lip(gλ)} ≤ 1/2 for all λ ∈ [0, 1], so the conditions of Proposition 2.4
are met. Hence (4.1) admits a unique L2-bounded mild solution ξλ and this unique
L2-bounded solution is Levitan almost periodic in distribution.

For fixed λ0 ∈ [0, 1], we note that for any R > 0,

lim
λ→λ0

sup
t∈R,|y|≤R

|fλ(t, y)− fλ0(t, y)| = 0,

lim
λ→λ0

sup
t∈R,|y|≤R

|gλ(t, y)− gλ0
(t, y)| =∞,

lim
λ→λ0

sup
|t|≤R,|y|≤R

|gλ(t, y)− gλ0
(t, y)| = 0.

Therefore, by Theorem 3.1 the unique Levitan almost periodic solution ξλ converges
to ξλ0

in the norm ‖·‖2 uniformly on any compact interval but cannot be uniformly
on R.

If fλ remains unchanged but gλ(t, y) = 1
2y + (1 + λ)(2 + cos t + cos

√
2 t), then

fλ and gλ are quasi-periodic. So it follows from Proposition 2.4 again that (4.1)
admits a unique L2-bounded mild solution ξλ which is quasi-periodic in distribution.
Furthermore, by Theorem 3.1 the unique quasi-periodic solution ξλ converges to
ξλ0

in the norm ‖ · ‖2 uniformly on R, because in this case we have

lim
λ→λ0

sup
t∈R,|y|≤R

|gλ(t, y)− gλ0
(t, y)| = 0.

Example 4.2. Consider the following stochastic heat equation on the interval [0,1]
with Dirichlet boundary condition,

∂u

∂t
=
∂2u

∂ξ2
+ u sin t+ (1− λ)(2− sin t− sin

√
3t) + λ

∂W

∂t

=:
∂2u

∂ξ2
+ fλ(t, u) + λ

∂W

∂t
,

u(t, 0) = u(t, 1) = 0, t > 0.

(4.2)

Here W is a one-dimensional two-sided Brownian motion and λ ∈ [0, 1] is a parame-
ter. Let A be the Laplace operator, then A : D(A) = H2(0, 1)∩H1

0 (0, 1)→ L2(0, 1).
Denote H := L2(0, 1) and the norm on H by | · |. Then the above equation can be
written as the evolution equation

dY (t) = (AY (t) + Fλ(t, Y (t)))dt+ λdW (t) (4.3)

on the Hilbert space H with

Y (t) := u(t, ·), Fλ(t, Y (t)) := fλ(t, u(t, ·)).

Note that, operator A has eigenvalues {−n2π2}∞n=1 and generates a C0-semigroup

{U(t)}t≥0 on H satisfying ‖U(t)‖ ≤ e−π
2t for t ≥ 0, i.e. N = 1 and ν = π2.

Note that Lip(F ) ≤ 1 for all λ ∈ [0, 1], so the conditions of Proposition 2.4 hold.
Thus (4.3), hence (4.2), admits a unique L2-bounded mild solution ξλ which is
quasi-periodic in distribution.



8 H. QIU, Y. WANG EJDE-2020/113

Note that as λ→ 0, for any R > 0 we have

lim
λ→0

sup
t∈R,|Y |≤R

|Fλ(t, Y )− F0(t, Y )| = 0,

so it follows from Theorem 3.1 that ξλ → ξ0 in the norm ‖ · ‖2 uniformly on R,
with ξ0 being the deterministic quasi-periodic solution for the equation (4.2) with
parameter λ = 0. Similarly, as λ → 1, we have ξλ → ξ1 uniformly on R, with ξ1
being the periodic solution of the equation (4.2) with parameter λ = 1.

If

fλ(t, u) = u sin t+
1− λ

2− sin t− sin
√

3t
in (4.2), then for any R > 0 we have

lim
λ→0

sup
t∈R,|Y |≤R

|Fλ(t, Y )− F0(t, Y )| =∞, lim
λ→0

sup
|t|≤R,|Y |≤R

|Fλ(t, Y )− F0(t, Y )| = 0.

So it follows from Theorem 3.1 that ξλ → ξ0 in the norm ‖ · ‖2 uniformly on any
compact interval but cannot be uniformly on R, with ξ0 being the deterministic
Levitan almost periodic solution for the equation (4.2) with parameter λ = 0.
Similarly, as λ → 1, we have ξλ → ξ1 uniformly on any compact interval but
cannot be uniformly on R, with ξ1 being the periodic solution of the equation (4.2)
with parameter λ = 1.

Acknowledgements. The authors are grateful to Professor Zhenxin Liu for help-
ful discussions. The authors sincerely thank the anonymous referees for their careful
reading and helpful suggestions. This work is partially supported by the NSFC,
Grant 11871132, and by the startup fund from Dalian University of Technology.

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Haijing Qiu

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024,

China
Email address: 18233150561@163.com

Yan Wang (corresponding author)
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024,

China

Email address: wangyan84@dlut.edu.cn


	1. Introduction
	2. Preliminaries
	3. Main results
	4. Applications
	Acknowledgements

	References