RIMS Kôkyûroku Bessatsu
B79 (2020), 199–213

A note on random strings with values in a bounded
convex domain
By

Bin Xie∗

Abstract
In this note, the ergodic properties of the Markov semigroup associated with the random string in a bounded convex domain are mainly studied. Under proper assumptions, theMarkov property, the existence and uniqueness of invariant measures are shown by using theapproximating system of stochastic partial diﬀerential equations.

§ 1.

Introduction

Let (B(t))t≥0 be a K-dimensional Brownian motion on a complete probability space(Ω, F, P) with the filtration (Ft )t≥0 satisfying usual conditions, and let the mappingsb : Rd → Rd and σ : Rd → Rd ⊗ RK be Borel measurable. Let us denote by O a smooth,bounded convex domain in Rd . Moreover, without loss of generality, we suppose theorigin point 0 is in the domain O. In this note, we mainly study some ergodic propertiesof the following system of the stochastic partial diﬀerential equation with reflection inO:
du(t, x) ={κ∆u(t, x) + b(u(t, x))}dt






+ σ(u(t, x))dB(t) + η(dtdx), t > 0, x ∈ (0, 1),


(1.1)
u(t, 0) =u(t, 1) = 0, t ≥ 0,u(0, x) =ξ(x), x ∈ [0, 1],u(t, x) ∈Ō, t > 0, x ∈ [0, 1],
Received February 21, 2019. Revised May 27, 2019.
2010 Mathematics Subject Classification(s): Primary 60H15; Secondary 60J35, 35R60.
Key Words: Random motion of strings, SPDEs with reflection, invariant measure, convex domain,
existence and uniquenessSupported by JSPS KAKENHI Grant Number 16K05197∗ Department of Mathematical Sciences, Faculty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan.
e-mail: bxie@shinshu-u.ac.jp; bxieuniv@outlook.com
c 2020 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
⃝

200

B. Xiewhere κ > 0 is a constant, Ō denotes the closure of O, u : [0, 1] → Rd denotes ad-dimensional function taking values in Ō, ∆u(x) = (u′′1 (x), · · · , u′′d (x))t denotes the
Laplace operator on [0, 1] satisfying homogeneous Dirichlet boundary conditions, theinitial value ξ is a continuous function defined on [0, 1] such that ξ(x) ∈ Ō for allx ∈ [0, 1], and satisfying the compatibility condition ξ(0) = ξ(1) = 0, and η denotes
a d-dimensional signed random measure on [0, ∞) × [0, 1], which prevents the solution
from leaving the closed set Ō.
The reflected stochastic partial diﬀerential equation (1.1) is initially studied in [17]and it can be used to describe the random motion of strings in a bounded convex
domain. Random motion of strings is originally introduced in [5], which is describedby systems of stochastic partial diﬀerential equations driven by d-dimensional spacetime white noises. In that paper, the string is allowed to move randomly in the whole
space Rd , so C([0, 1], Rd )-valued solutions are considered. In addition, many important
problems are also studied in that paper. After that, many works on random strings
have been published, for example, hitting properties of a random string [8].
But, the studies on the random motion of strings in a bounded domain of Rd arevery rare. To study such motions, it is natural to introduce the system of reflectedstochastic partial diﬀerential equations like (1.1). From now on, let us give a shortreview on the reflected stochastic partial diﬀerential equations. Stochastic partial differential equations with reflections can be regarded as infinite-dimensional Skorokhod
problems, so it is widely studied. The real-valued (i.e., d = 1) stochastic partial diﬀerential equation with reflection driven by a space-time white noise is first investigated in[9]. In that paper, the existence and uniqueness of solutions is established for the caseof additive noises. For the case of multiplicative noises, the minimum solution is constructed in [4] and the about 16 years later the uniqueness of it is proved in [12]. Many
important properties relative to real- valued reflected stochastic partial diﬀerential equations have been investigated until now. For instance, see [6, 13] for reversible measures,[14] for occupation densities, [2] for hitting properties, [12] for large deviation principlesof the Freidlin-Wentzell type, [10, 16] for dimension-free Harnack inequalities, and [11]for hypercontractivity of the corresponding transition semigroup. Reflected stochastic
partial diﬀerential equations are also closely connected with the study of random interface models. In [6], it is shown that the fluctuation of a ∇ϕ-interface model near a hardwall weakly converges to the stationary solution of a reflected stochastic partial diﬀerential equation driven by the additive space-time white noise. For more information on
the study of real-valued reflected stochastic partial diﬀerential equations, we refer thereaders to the monograph [15] and references therein.
However, as we mentioned above, there is little research on the systems of reflectedstochastic partial diﬀerential equations, because of some diﬃculties such as the lackA note on random strings

201

of comparison theorems for systems. To the best of our knowledge, only two research
papers on such topics under diﬀerent frameworks are published at present, see [1, 17].
In [1], a random string with reflection in a convex domain is described by a systemof stochastic partial diﬀerential equations driven by a d-dimensional additive spacetime white noise by using the Dirichlet form approach. On the other hand, in [17],instead of d-dimensional space-time white noises, the finite dimensional time white
noise is considered and thanks to that, the case of multiplicative noises can be studied.
Both of these papers mainly investigated the existence and uniqueness of solutions.
But few properties of solutions have been studied in the above two papers. In thisnote, we will devote to ergodic properties of the system (1.1). More precisely, we will
mainly investigate the Markov property of the solution, the existence and uniquenessof invariant measures.
The remainder of this note is organized as follows. In Section 2, the precise formulation of (1.1) and its existence and uniqueness of solutions are first stated as preparation,and then the Markov property of the solution of (1.1) is shown. In Section 3, the existence and uniqueness of invariant measures for the Markov semigroup associated withthe solution of (1.1) is investigated.
§ 2.

Preliminaries and Markov property

In this section, we first formulate the precise mathematical meaning of the system(1.1) and state the result on the existence and uniqueness of solutions. Then, we showthat the solution u(t; ·) can be uniquely extended to a map from L2O to L2O , and finally
we prove that the stochastic process (u(t; ·))t≥0 is a Markov process with its state space
L2O .
Let us here introduce some notions, which will be used throughout this note. Let
H = L2 ([0, 1], Rd ) be the usual L2 -space with norm | · |H and inner product ⟨·, ·⟩H . Let
L2O = {u ∈ H : u(x) ∈ Ō for all x ∈ [0, 1]}. It is known that L2O is a closed subset of
H. So, it is a Polish space. Let us denote by Bb (E) (respectively Cb (E)) the family of
all bounded and measurable (respectively continuous) functions from a subset E ⊂ H
to R. Let V be the usual Sobolev space H01 , i.e., the completion of C0∞ with respect to∑d ∫ 1
1
norm ∥u∥V = ( i=1 0 (∂x ui (x))2 dx) 2 . In addition, we use ⟨·, ·⟩ and | · | to denote theusual inner product and norm of Rd .
Let us now state the mathematical definition of the system (1.1) according to theoriginal research [17].
Definition 2.1.
We call a pair (u, η) a solution of the reflected system (1.1) if
the following (i)-(iv) are satisfied:(i) u is a continuous d-dimensional random field on [0, ∞) × [0, 1] such that u(t, x) ∈202

B. XieŌ a.s., and u(t, x) is Ft -measurable.
(ii) η is a Rd -valued signed random measure on [0, ∞) × [0, 1] such that(a) For any T > 0,[]E ∥η([0, T ] × [0, 1])∥TV < ∞,where ∥ · ∥TV denotes the total variation of η on [0, T ] × [0, 1].
(b) η is (Ft )-adapted. More precisely, for any Borel set B ⊂ [0, t] × [0, 1], η(B) is
Ft -measurable.
(iii) For any test function ϕ ∈ C02 ([0, 1], Rd ) with ϕ(0) = ϕ(1) = 0, the following stochastic integral equation is fulfilled:∫ t∫ t⟨u(t), ϕ⟩H = ⟨ξ, ϕ⟩H +⟨u(s), ∆ϕ⟩H ds +⟨b(u(s)), ϕ⟩H ds
+

K ∫ t
∑
k=1

0

0

⟨σk (u(s)), ϕ⟩H dBk (s) +0d ∫ t∫ 1∑i=1

0

ϕi (x)ηi (dsdx) a.s.,0where u(t) := u(t, ·) and σk denotes the column vector of the matrix σ, that is, σk =(σ1k , · · · , σdk )t , k = 1, 2, . . . , K.
(iv) The following relation is satisfied by (u, η):for any T > 0 and ϕ ∈ C([0, T ] × [0, 1], Ō),d ∫ T ∫ 1∑
(ϕi (t, x) − ui (t, x))ηi (dtdx) ≥ 0 a.s.
i=1

0

0

Throughout this note, according to [17], we suppose that the coeﬃcients b and σ
are Lipschitz continuous, that is, there exists a constant L > 0 such thatmax{|b(u) − b(v)|, ∥σ(u) − σ(v)∥} ≤ L|u − v|,√
where ∥σ∥ = Tr(σσ t ).
Then, the existence and uniqueness of solutions of (1.1) is proved in [17].
Theorem 2.2 (Theorem 2.1 [17]).
Suppose ξ ∈ V . Then the reflected stochastic
partial diﬀerential equation (1.1) has a unique solution (u, η) such that
[
]
[∫
]
T
E sup |u(t)|2H + E∥u(t)∥2V dt < ∞.
t∈[0,T ]0

Remark. By the proof of Theorem 2.1 [17], it is known that η(dtdx) is absolutely
continuous with respect to the Lebesgue measure dx.
From now on, we devote to the study on the Markov property of solution of (1.1).
Indeed, we can show that the solution (u(t))t≥0 of the reflected system (1.1) can be
regarded as an L2O -valued Markov process. We first show that the solution u(t; ·) is
continuous in its initial value. More precisely, the next theorem holds.

A note on random strings

203

Theorem 2.3.
For any two non-random initial data ξi ∈ V, i = 1, 2, there
exists a constant C > 0 independent of t such that
[
]
2
(2.1)
E |u(t; ξ1 ) − u(t; ξ2 )|H ≤ |ξ1 − ξ2 |2H exp(Ct),where and in the sequel u(t; ξ) denote the solution of (1.1) with the initial value ξ.
In particular, u(t; ·) can be uniquely extended to a map from L2O to itself, which
will be still denoted by the same symbol in the sequel.
As a preparation for the proof Theorem 2.3, let us state two lemmas. For anyx ∈ Rd , let us denote by Π(x) the projection of x onto the closed set Ō, that is, Π(x)is the unique element of Ō such that|x − Π(x)| = inf |x − y|.
y∈O

It is well-known that an element Π(x) ∈ Ō is the projection of x on Ō if and only if forany y ∈ Ō⟨x − Π(x), Π(x) − y⟩ ≥ 0.

(2.2)

In the next lemma, some fundamental properties of Π are summarized according to our
purposes.
Lemma 2.4.
Let Π be the projection map from Rd onto Ō. Then the following
properties hold:(i) Π is contractive on Rd , i.e.,|Π(x) − Π(y)| ≤ |x − y|, x, y ∈ Rd .

(2.3)

In particular, Π is Lipschitz continuous on Rd .
(ii) For any x ∈ Rd and y ∈ Ō,⟨x − y, x − Π(x)⟩ ≥ 0.
(iii) For any x, y ∈ Rd ,(2.4)

⟨x − y, x − Π(x) − (y − Π(y))⟩ ≥ 0.

Proof. The properties (i) and (ii) are taken from Lemma 2.1 [17], see Lemma 2.1[17] for their proofs. Therefore, it is enough to show (iii). From (2.2), it follows that forany x, y ∈ Rd ,⟨x − Π(x), Π(x) − Π(y)⟩ ≥ 0 and ⟨y − Π(y), Π(y) − Π(x)⟩ ≥ 0,

204

B. Xiewhich gives that⟨x − y, Π(x) − Π(y)⟩ ≥ |Π(x) − Π(y)|2 .
Then it is known that0 ≤ ⟨x − y, Π(x) − Π(y)⟩ ≤ |x − y||Π(x) − Π(y)|,which implies (2.4) by using the contractive property (2.3) of Π.
To give the proof of Theorem 2.3, let us introduce the approximating system ofstochastic partial diﬀerential equations according to the strategy of the proof of Theorem
2.2 used in [17].
For each n ∈ N, we consider the following penalized system:dun (t, x) ={κ∆un (t, x) + b(un (t, x))}dt + σ(un (t, x))dB(t)





− n{un (t, x) − Π(un (t, x))}dt, t > 0, x ∈ (0, 1),(2.5)
 un (t, 0) =un (t, 1) = 0, t ≥ 0,



 n

u (0, x) =ξ(x), x ∈ [0, 1].
Since the coeﬃcients b, σ and the projection map Π are Lipschitz continuous on Rd , weknow that for each n ∈ N, the penalized system (2.5) has a unique solution. In thefollowing, we sometimes denote its solution by un (t, x; ξ) or un (t; ξ) according to our
purposes. From the proof of Theorem 2.1 [17], we have the next relation between u(t)and un (t).
Lemma 2.5.
For any initial value ξ ∈ V , the sequence {un (t)}n∈N converges tothe unique solution u(t) of (1.1) in the following sense:
[
]
[∫
]
T

lim E

n→∞

sup |un (t) − u(t)|2H + lim En→∞t∈[0,T ]∥un (t) − u(t)∥2V dt = 0.
0

Let us now formulate the proof of Theorem 2.3.
Proof of Theorem 2.3.

For simplicity, let us set

fn (t, x) := un (t, x; ξ1 ) − un (t, x; ξ2 ).
ufn (t), we have that
Applying Itô’s formula to u
(2.6)

2

fn (t)uH∫ t=|ξ1 − ξ2 |2H + 2κ0⟨⟩
fn (s), κ∆ufn (s) ds
u
H

205

A note on random strings

∫ t
+2
0

⟨

∫ t

+ 2n
+2

⟨

⟩
fn (s), −{un (s; ξ1 ) − Π(un (s; ξ1 ))} + {un (s; ξ2 ) − Π(un (s; ξ2 ))} ds
u
H

0
K
∑∫ t
k=1

+2

⟩
fn (s), b(un (s; ξ1 )) − b(un (s; ξ2 )) ds
u
H

0

K ∫ t
∑

k=1

0

2

|σk (un (s; ξ1 )) − σk (un (s; ξ2 ))|H ds⟨⟩
fn (s), σk (un (s; ξ1 )) − σk (un (s; ξ2 )) dBk (s)uH:=|ξ1 − ξ2 |2H + I1n (t) + I2n (t) + I3n (t) + I4n (t) + I5n (t).
Noting that κ > 0 and⟨⟩
fn (s), ∆ufn (s)u2

fn (s)
=− u
a.s.,
H
V

we know that I1n (t) ≤ 0 a.s..
Thanks to Lemma 2.4 (iii), we can also easily know that I3n (t) ≤ 0 a.s.
By the Lipschitz continuity of the coeﬃcients b and σ, we have that∫ t
I2n (t) + I4n (t) ≤ C

fn (s)|2H ds a.s.
|u0holds for a constant C > 0 independent of n and t > 0.
Since
]
[
sup E

sup

n∈N

t∈[0,T ]2

fn (t)<∞uH

by Lemma 2.2 [17], we know that the last term I5n (t) in (2.6) is a real valued martingale.
Therefore, taking expectations of both sides of (2.6) and then using the above estimates,
we have that
∫ t
[
]
[
]
2
2
n
f (t)|H ≤ |ξ1 − ξ2 |H + C
fn (s)|2H ds.
E |u
E |u
0

Therefore, the Gronwall inequality yields that
(2.7)

[
]
fn (t)|2H ≤ |ξ1 − ξ2 |2H exp(Ct), t ≥ 0.
E |uConsequently, by Lemma 2.5, the desired result (2.1) can be obtained by letting n → ∞in the above inequality and noting that C is independent of n and t.
From now on, we are going to show that the stochastic process (u(t; ·))t≥0 introduced in Theorem 2.3 is Markovian. Here let us formulate a lemma based on [13].

206

B. XieLemma 2.6 (Lemma 1 [13]).
Let us suppose the sequence of probability measures {µn }n∈N weakly converges to a probability measure µ on a Polish space S. If thesequence of real valued continuous functions {ψn }n∈N is uniformly bounded and equicontinuous on S andlim ψn (x) = ψ(x), x ∈ supp(µ),
n→∞

then

∫
lim

n→∞

∫
ψn (s)µn (ds) =S

ψ(s)µ(ds),supp(µ)where supp(µ) denotes the support of µ.
Under our assumptions, it is known that the solution (un (t))t≥0 of (2.5) is a Markov
process. Let us denote by Ptn the corresponding Markov transition semigroup associatedwith the penalized system (2.5), that is,
Ptn ϕ(ξ) = E[ϕ(un (t; ξ))], ξ ∈ H, ϕ ∈ Bb (H).
In addition, let us define the operator Pt by(2.8)

Pt ϕ(ξ) = E[ϕ(u(t; ξ))], ξ ∈ L2O , ϕ ∈ Bb (L2O ), t ≥ 0.

Theorem 2.7.
(2.9)

For any uniformly continuous function ϕ ∈ Cb (H), we have thatlim Ptn ϕ(ξ) = Pt ϕ(ξ), t ≥ 0, ξ ∈ L2O

n→∞

and
(2.10)Pt Ps ϕ(ξ) = Pt+s ϕ(ξ), s, t ≥ 0, ξ ∈ L2O .

In particular, (Pt )t≥0 is a Markov transition semigroup.
Proof. Let µn and µ denote the distributions of un (t; ξ) and u(t; ξ) respectively.
Then by Lemma 2.5, it is known that µn converges weakly to µ. By Theorem 2.3, it iseasy to know (2.9) holds for any continuous ϕ ∈ Cb (H).
From now on, let us prove (2.10). Since ϕ ∈ Cb (H) is uniformly continuous, we
can show that for each n ∈ N, Ptn ϕ(·) is also uniformly continuous on H by the relation(2.7), and then the sequence {Ptn ϕ(·)}n∈N is uniformly bounded and equicontinuouson H by the Markovian property of (Ptn )t≥0 , Thus, we know that for any uniformly
continuous function ϕ ∈ Cb (H), the sequence {Psn ϕ(·)}n∈N and the sequence {µn }n∈Nsatisfy all of the conditions in Lemma 2.6. Consequently, we can complete the proof of(2.10) by Lemma 2.6.

207

A note on random strings

Remark. By Theorem 2.7, it is known that (Pt )t≥0 defined by (2.8) is a Markov
transition semigroup acting on Cb (L2O ), and also on Bb (L2O ). In particular, (u(t))t≥0
can be regarded as a Markov process with its state space L2O .
§ 3.

Existence and uniqueness of invariant measures

The main goal of this section is to investigate the existence and uniqueness ofinvariant measures of the system (1.1).
Let us first note that (Pt )t≥0 is a Feller Markovian semigroup. More precisely, we
have the next lemma.
Lemma 3.1.
(Pt )t≥0 is a Feller transition semigroup on Cb (L2O ), i.e., Pt ϕ ∈Cb (L2O ) for any ϕ ∈ Cb (L2O ) and t > 0.
Proof. It can be easily shown by Theorem 2.3.
From now on, we will first show the existence of invariant measures under thefollowing assumption:(A): There exist two constants γ ∈ R and γ ′ > 0 such that⟨x, b(x)⟩ ≤ γ ′ + γ|x|2 , x ∈ Rd .
Theorem 3.2.
If further κ > γ + 2L2 , then there exists an invariant probability
measure µ concentrated on L2O and()µ exp(δ| · |2H ) < ∞(3.1)
holds for some small enough constant δ > 0.
Proof. Since (Pt ) is a Feller transition semigroup, it is enough for us to show thatfor some probability measure ν supported on L2O , the family of probability measures(µT )T >0 defined by∫
1 T ∗P νdt, T > 0
µT :=
T 0 t
is tight based on the Krylov-Bogoliubov Theorem (for example, see Theorem 3.1.1 [3]),∫where Pt∗ ν(A) = L2 Pt 1A (ξ)µ(dξ).
OLet us still use un (t) to denote the unique solution of the approximating system(2.5). Applying Itô’s formula, we have that
∫ t
∫ t
n
2
2
n
n
|u (t)|H =|ξ|H + 2κ(3.2)
⟨u (s), ∆u (s)⟩H ds + 2⟨un (s), b(un (s))⟩H ds
0

0

208

B. Xie

∫ t
− 2n
+2

⟨un (s), un (s) − Π(un (s))⟩H ds

0
K
∑∫ t

|σk (un(s))|2H ds + 2

0

k=1

K ∫ t
∑
k=1

⟨un (s), σk (un (s))⟩H dBk (s).

0

For any ξ ∈ V , by the Dirichlet boundary conditions on the Laplace operator ∆, we see
that
∫ t
∫ t
n
n
(3.3)
⟨u (s), ∆u (s)⟩H ds = −∥un (s)∥2V ds ≤ 0 a.s.
00Noting that 0 ∈ O, by Lemma 2.4 (ii), we have that for any x ∈ Rd ,⟨x, x − Π(x)⟩ ≥ 0,
and then
∫ t
−2n

(3.4)

⟨un (s), un (s) − Π(un (s))⟩H ds ≤ 0 a.s.
0

Applying the estimates (3.3) and (3.4) to (3.2), we have
(3.5)

|un (t)|2H∫ t≤|ξ|2H − 2κ
+2

∥u

K ∫ t
∑
k=1

|σk (un(s))|2H ds + 2

K ∫ t
∑

k=1

∫ t

∫ t
∥un (s)∥2V ds + 20(γ ′ + γ|un (s)|2H )ds

0

0

K ∫ t
∑
k=1

(s)∥2V ds + 2

0≤|ξ|2H − 2κ
+2

∫ t
n

⟨un (s), σk (un (s))⟩H dBk (s)

0

{ ′
}
γ + 2∥σ(0)∥2 + (γ + 2L2 )|un (s)|2H ds

0⟨un (s), σk (un (s))⟩H dBk (s)0≤|ξ|2H + (γ ′ + 2∥σ(0)∥2 )t − 2(κ − γ − 2L2 )
+2

K ∫ t
∑
k=1

∫ t
∥un (s)∥2V ds
0⟨un (s), σk (un (s))⟩H dBk (s),0where the assumption (A) has been applied for the first inequality, and the Lipschitz
continuity of σ and the relation |u|H ≤ ∥u∥V have used for the second and the lastinequality respectively.
Since supn∈N E[supt∈[0,T ] ∥un (t)∥2H ] < ∞, it is known that the last term in (3.5) isa real-valued martingale. Therefore, taking now the expectations of both sides of (3.5),

209

A note on random strings

we obtain that
∫ t [
[
]
]
2
2
n
2
′
2
2
E |u (t)|H ≤|ξ|H + (γ + 2∥σ(0)∥ )t − 2(κ − γ − 2L )E ∥un (s)∥V ds, t ≥ 0.
0

Note that the constants γ ′ , γ and L are independent of n and t. Therefore, by lettingn → ∞ in the above inequality and using Lemma 2.5, we have that
∫ t [
]
[
]
2
2
2
′
2
2
E |u(t)|H ≤ |ξ|H + (γ + 2∥σ(0)∥ )t − 2(κ − γ − 2L )E ∥u(s)∥V ds, t ≥ 0.
0

Since we assumed thatκ > γ + 2L2 ,by taking ξ = 0 in the above inequality and considering ν = δ0 , we have thatµT (∥ · ∥2V ) =

(3.6)

1
T

∫ T
E[∥u(s)∥2V ]ds ≤0

γ ′ + 2∥σ(0)∥2
2(κ − γ − 2L2 )holds for all T > 0. Since V is compactly embedded in H by Rellich-Kondrachov’s
theorem, with the help of (3.6) and Chebyshev’s inequality, we know that the family{µT }T >0 is tight. Consequently, the existence of invariant measures is proved.
Let us now state the proof of (3.1). Recalling the representation (3.2) of |un (t)|2H
and then using the Lipschitz condition on σ, we have
⟨

(3.7)

⟩

|un |2H t =4

K ∫ t
∑
k=1

∫ t
≤8

⟨un (s), σk (un (s))⟩2H ds0(∥σ(0)∥2 + L2 |un (s)|2H )ds,0⟨⟩
where |un |2H t denotes the quadratic variation process of |un (t)|2H .
In the sequel, let us choose δ > 0 being small enough such that
γ1 := κπ 2 − γ − 4δL2 > 0.
By Itô’s formula, (3.3), (3.4) and (3.7), we have that
(3.8)

exp(δ|un (t)|2H )∫ t= exp(δ|ξ|2H ) + δ
2 ∫ t

exp(δ|un (s)|2H )d|un (s)|2H0⟨⟩
exp(δ|un (s)|2H )d |un |2H s
0
∫ t
{
}
2
≤ exp(δ|ξ|H ) + δ
exp(δ|un (s)|2H ) −κ∥un (s)∥2V + ⟨un (s), b(un (s))⟩H ds
+

δ
2

0

210

B. Xie

+ 2δ

K ∫ t
∑

K ∫ t
∑
k=1

exp(δ|u

+ 2δ

2 ∫ t

2

0

⟨
⟩
exp(δ|un (s)|2H )d |un |2H sexp(δ|un (s)|2H )⟨un (s), σk (un (s))⟩H dBk (s)0≤ exp(δ|ξ|2H ) + 2∫ tδ(s)|2H )|σk (un (s))|2H ds +

0

k=1

+2

n

K ∫ t
∑
k=1

exp(δ|un (s)|2H )⟨un (s), σk (un (s))⟩H dBk (s)0

exp(δ|un (s)|2H ){γ ′ + 4δ∥σ(0)∥2 − γ1 |un (s)|2H }ds0≤ exp(δ|ξ|2H ) + 2

K ∫ t
∑
k=1

∫ t

exp(δ|un (s)|2H )⟨un (s), σk (un (s))⟩H dBk (s)0{C1 − C2 exp(δ|un (s)|2H )}ds,+ 2δ
0

where ∥ · ∥V ≥ π| · |H has been used for the second inequality, C1 and C2 are two positive
constants.
Taking expectations and letting n → ∞ in (3.8) with the help of Lemma 2.5, we
have
∫ t
[
]
[
]
2
2
E exp(δ|u(t)|H ) ≤ exp(δ|ξ|H ) + 2δ{C1 − C2 E exp(δ|u(s)|2H ) }ds.
0

Therefore, letting ξ = 0, we have that for any T > 0µT(

exp(δ| · |2H

)

1
=
T

∫ T
0

[
(
)]
1 + 2δC1 T.
E exp δ|u(s)|2H ds ≤2δC2 TAs a consequence, we proved that the invariant measure µ of (Pt )t≥0 has the desired
property (3.1).
Remark. Instead of the condition κ > γ + 2L2 , if we assume that κπ 2 > L andσ(u) is uniformly bounded in u ∈ Rd , then we can also obtain the existence of invariant
measures for (1.1). In fact, under such assumption, using Theorem 6.1.2 [3], it is enoughfor us to show that u(t) is bounded in probability, that is,lim sup P(|u(t)|H ≥ R) = 0.

R→∞ t≥0

However, this can be shown similarly to the proof of Theorem 11.2.2 [3]. On the other
hand, it is valuable to know that under the conditions of Theorem 3.2, the concentration
property (3.1) of the invariant measure is established, which is very important to thestudy on the contractivity of Markov semigroups, for instance, see [7].

A note on random strings

211

In the end, we give a suﬃcient condition for the uniqueness of the invariant measures.
Theorem 3.3.
(3.9)

If the coeﬃcient σ is independent of u ∈ Rd , then()|u(t; ξ1 ) − u(t; ξ2 )|2H ≤ |ξ1 − ξ2 |2H exp 2(L − κπ 2 )t , ξ1 , ξ2 ∈ L2O .

In particular, if κπ 2 > L, then there exists at most one invariant measure for (1.1).
Proof. Let us use the same notations as those in the proof of Theorem 2.3. It issuﬃcient to show (3.9) holds for ξi ∈ V ∩ L2O , i = 1, 2. Since σ is independent of u ∈ Rd ,similarly to the estimate for (2.6), we have
(3.10)

fn (t)|2H
d|u
⟨
⟩
⟨
⟩
fn (t), κ∆ufn (t) dt + 2 u
fn (t), b(un (t; ξ1 )) − b(un (t; ξ2 )) dt
=2 u
H
H
⟨
⟩
n
n
n
n
n
f (t), −{u (t; ξ1 ) − Π(u (t; ξ1 ))} + {u (t; ξ2 ) − Π(u (t; ξ2 ))} dt+ 2n uHfn (t), ∆ufn (t)⟩H dt + 2⟨ufn (t), b(un (t; ξ1 )) − b(un (t; ξ2 ))⟩H dt≤2κ⟨u
fn (t), ∆ufn (t)⟩H dt + 2L|u
fn (s)|H dt≤2κ⟨u
fn (t), ∆ufn (t)⟩H dt.
≤2(L − κπ 2 )⟨u

As a consequence of (3.10), we can easily obtain that()
fn (t)|2H ≤ |ξ1 − ξ2 |2H exp 2(L − κπ 2 )t ,|uwhich gives (3.9) by using Lemma 2.5 and letting n → ∞.
Finally, let µ be an arbitrary invariant measure of (Pt )t≥0 . By (3.9) and the dominated convergence theorem, we have for any ϕ ∈ Cb (L2O ) and ξ1 ∈ L2O ,(3.11)

lim |Pt ϕ(ξ1 ) − µ(ϕ)| = lim |E[ϕ(u(t; ξ1 ))] − µ(Pt (ϕ))|
t→∞
∫
≤ lim
|E [ϕ(u(t; ξ1 ))] − E[ϕ(u(t; ξ2 ))]| µ(dξ2 )

t→∞

t→∞

L2O

=0,
which clearly implies the uniqueness of invariant measures of (1.1). Consequently, theproof of this theorem is completed.
Remark. (i) Under the assumptions of Theorem 3.3, for any Lipschitz continuousfunction ϕ with its Lipschitz constant Γ, we also have()()
|Pt ϕ(ξ1 ) − µ(ϕ)| ≤ Γ |ξ|H + µ(| · |H ) exp 2(L − κπ 2 )t , t > 0.

212

B. XieIn fact, it can be easily obtained similarly to (3.11) by Theorem 2.3. In particular, it
implies that transition semigroup Pt converges weakly to its unique invariant measureµ with an exponential rate.
(ii) Suppose ⟨x − y, b(x) − b(y)⟩ ≤ L|x − y|2 , x, y ∈ Rd holds for a constant L ∈ R
satisfying κπ 2 > L. Then all of the results in Theorem 3.3 are still satisfied. Moreover,2
under this assumption, the assumption (A) holds with γ = max{L+ϵ, ϵ} and γ ′ = |b(0)|4ϵfor any ϵ > 0. Therefore, under this framework, the existence and uniqueness of invariant
measures can be also established.
Acknowledgements
This research was partially supported by Grant-in-Aid for Scientific Research (C) 16K05197
from Japan Society for the Promotion of Science(JSPS). The author is indebted to theanonymous referee for constructive comments and suggestions.

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