# The Small Time Asymptotics of SPDEs with Reflection

Authors: Juan Yang; Jianliang Zhai; Qing Zhou
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study stochastic partial differential equations with singular drifts and with reflection, driven by space-time white noise with nonconstant diffusion coefficients under periodic boundary conditions. The existence and uniqueness of invariant measures is established under appropriate conditions. As a byproduct, the Hölder continuity of the solution is obtained. The strong Feller property is also obtained. Moreover, we show large deviation principle.


---

## Body

## 1. Introduction

Stochastic partial differential equations (SPDEs in short) with reflection can be used to model the evolution of random interfaces near a hard wall. Nualart and Pardoux [1] and Donati-Martin and Pardoux [2] introduced “reflection” to prevent stochastic heat equations from becoming negative, which could be viewed as an extension of one-dimensional stochastic differential equations reflected at 0. Funaki and Olla [3] considered fluctuations around the hydrodynamical limit of a Ginzburg-Landau ∇
ϕ interface model on a wall. When the interface touches the wall, it will be repulsed. They proved that the fluctuations of a ∇
ϕ interface model near a hard wall converge in law to the stationary solution of a SPDE with reflection. They also showed that SPDEs with reflection have natural and meaningful bases in statistical mechanics, like entropic repulsion phenomena in Deuschel and Giacomin [4] and in Lebowitz and Maes [5]. There are also various models using SPDEs with reflection, such as stochastic Cahn-Hilliard equations with reflection (see da Prato and Zabczyk [6] and Debussche and Zambotti [7]) and stochastic generalized porous media equations with reflection (see Röckner et al. [8]).In the case considered here, two reflections have been added to our model (1), and it is believed that it is a natural extension of one reflection. There are some works on this topic recently; see [9–13].Through introducing the driftu
-
α, Mueller [14] gave a proof that solutions of SPDEs u
t
=
u
x
x
+
g
(
u
)
W
˙
(
t
,
x
), where g
(
u
) is non-Lipschitz, do not blow up in finite time. As a byproduct, it was proved that if α
>
3, this drift forces solutions to stay positive with probability 1, and it was believed that this result “may be of interest.” Then Mueller and Pardoux [15] concentrated on the case when 0
<
α
<
3 and showed that the solutions hit 0 in finite time with positive probability. Thus, the case α
=
3 is the critical case for u to hit zero and has been showed that it has essential relationship with reflections; for details, see Mueller [14], Mueller and Pardoux [15], and Zambotti [16, 17].Inspired by these, our interest stays in studying SPDEs which have double smooth reflecting wallsh
1 and h
2 and two singular drifts c
1
/
(
u
-
h
1
)
α and c
2
/
(
h
2
-
u
)
α, for all α
>
0.For the caseα
=
3, it is a quite interesting topic and needs more detailed studies. As an extension from one reflection (see Zambotti [16, 17]) to two-reflection case, one interesting problem is to find the explicit invariant probability measure and then study the detailed hitting properties of the solutions (see Dalang et al. [18]).In this paper, we consider the following SPDEs:(1)
∂
u
(
x
,
t
)
∂
t
=
∂
2
u
(
x
,
t
)
∂
x
2
+
f
(
u
(
x
,
t
)
)
+
c
1
(
u
(
x
,
t
)
-
h
1
(
x
)
)
α
-
c
2
(
h
2
(
x
)
-
u
(
x
,
t
)
)
α
+
σ
(
u
(
x
,
t
)
)
W
˙
(
x
,
t
)
+
η
(
x
,
t
)
-
ξ
(
x
,
t
)
;
u
(
x
,
0
)
=
u
0
(
x
)
∈
C
(
S
1
)
;
h
1
(
x
)
≤
u
(
x
,
t
)
≤
h
2
(
x
)
,
for
(
x
,
t
)
∈
Q
,
where Q
:
=
S
1
×
R
+, α
≥
0, c
1
>
0, c
2
>
0, S
1
:
=
R
(
mod
⁡
2
π
), or {
e
i
θ
;
θ
∈
R
} denotes a circular ring and the random field W
(
x
,
t
)
:
=
W
(
{
e
i
θ
;
0
≤
θ
≤
x
}
×
[
0
,
t
]
) is a regular Brownian sheet defined on a filtered probability space (
Ω
,
P
,
F
;
{
F
t
}
t
≥
0
). The random forces ξ and η are added to (1) to prevent the solution from leaving the interval [
h
1
,
h
2
].We assume that the reflecting wallsh
1
(
·
)
,
h
2
(
·
) are continuous functions satisfying the following.(H1)
h
1
(
x
)
<
h
2
(
x
) for x
∈
S
1.
(H2)
(
∂
2
h
i
(
x
)
/
∂
x
2
)
∈
L
2
(
S
1
), where ∂
2
/
∂
x
2 is interpreted in a distributional sense.We also assume that the coefficients f
,
σ
:
R
→
R satisfy the following:
(F1)
there existsL
>
0 such that
(2)
|
f
(
z
1
)
-
f
(
z
2
)
|
+
|
σ
(
z
1
)
-
σ
(
z
2
)
|
≤
L
|
z
1
-
z
2
|
,
z
1
,
z
2
∈
R
;
(F2)
there existsC
>
0 such that
(3)
|
f
(
z
)
|
+
|
σ
(
z
)
|
≤
C
(
1
+
|
z
|
)
,
z
∈
R
.
The initial conditionu
0
(
x
) satisfies the following:
(F3)
u
0
(
x
)
∈
C
(
S
1
) satisfy h
1
(
x
)
≤
u
0
(
x
)
≤
h
2
(
x
), for x
∈
S
1.The following is the definition of a solution of a SPDE with two reflecting wallsh
1
(
x
)
and
h
2
(
x
)
,
x
∈
S
1.Definition 1.
A triplet(
u
,
η
,
ξ
) is a solution to the SPDE (1) if(i)
u
=
{
u
(
x
,
t
)
;
(
x
,
t
)
∈
Q
} is a continuous, adapted random field (i.e., u
(
x
,
t
) is F
t-measurable ∀
t
≥
0
,
x
∈
S
1) satisfying h
1
(
x
)
≤
u
(
x
,
t
)
≤
h
2
(
x
) a.s.;
(ii)
η
(
d
x
,
d
t
) and ξ
(
d
x
,
d
t
) are positive and adapted (i.e., η
(
B
) and ξ
(
B
) are F
t-measurable if B
⊂
S
1
×
[
0
,
t
]) random measures on Q satisfying
(4)
η
(
S
1
×
[
0
,
T
]
)
<
∞
,
ξ
(
S
1
×
[
0
,
T
]
)
<
∞
,
for T
>
0;
(iii)
for allt
≥
0 and ϕ
∈
C
∞
(
S
1
), one has
(5)
(
u
(
t
)
,
ϕ
)
-
(
u
0
,
ϕ
)
-
∫
0
t
‍
(
u
(
s
)
,
ϕ
′′
)
d
s
-
∫
0
t
‍
(
f
(
u
(
s
)
)
,
ϕ
)
d
s
=
∫
0
t
‍
(
c
1
(
u
(
s
)
-
h
1
)
α
-
c
2
(
h
2
-
u
(
s
)
)
α
,
ϕ
)
d
s
+
∫
0
t
‍
∫
S
1
‍
ϕ
(
x
)
σ
(
u
(
x
,
s
)
)
W
(
d
x
,
d
s
)
+
∫
0
t
‍
∫
S
1
‍
ϕ
(
x
)
η
(
d
x
,
d
s
)
-
∫
0
t
‍
∫
S
1
‍
ϕ
(
x
)
ξ
(
d
x
,
d
s
)
,
a
.
s
.
,
where (
,
) denotes the inner product in L
2
(
S
1
) and u
(
t
) denotes u
(
·
,
t
);
(iv)
we consider the following:(6)
∫
Q
‍
(
u
(
x
,
t
)
-
h
1
(
x
)
)
η
(
d
x
,
d
t
)
=
∫
Q
‍
(
h
2
(
x
)
-
u
(
x
,
t
)
)
ξ
(
d
x
,
d
t
)
=
0
.For SPDEs without reflection, the existence and uniqueness of invariant measures has been studied by many people; see Sowers [19], Mueller [20], Peszat and Zabczyk [21], and da Prato and Zabczyk [6]. For SPDEs with reflection, when the diffusion coefficient σ is a constant, existence and uniqueness of invariant measures was obtained by Zambotti [16] and Otobe [10, 22], while the result was also obtained in [12] for the SPDE with a nonlinear diffusion coefficient by using coupling method. The strong Feller property of SPDEs has been studied by several authors; see Peszat and Zabczyk [21] and da Prato and Zabczyk [6]. The strong Feller property of SPDEs with reflection was proved in Zhang [23] and Yang and Zhang [12]. Moreover, with regard to the large deviations for the solution of the small noise perturbation of the equation, there exists a large amount of literature; see Dembo and Zeitouni [24] and references therein. For white noise-driven SPDEs, Sowers [19] and Cerrai and Röckner [25] set up some exponential estimates for proving large derivation principle. For SPDE with reflection, it is more efficient through the weak convergence approach; see Xu and Zhang [26] and, for the detail of this approach, the readers are referred to [27, 28].All of the results mentioned above are devoted to the case of the Lipschitz coefficient. The purpose of this paper is to deal with SPDEs with reflection and singular drifts (1). The existence and uniqueness of the solution of (1) is established in [11]. We show in this paper the existence and uniqueness of invariant measures and the strong Feller property, as well as large deviation principle of (1). For the existence of invariant measures, our approach is to use the Krylov-Bogolyubov theorem. For the uniqueness, we adapted a coupling method used in [20]; also see [12]. The strong Feller property will be obtained by introducing a sequence of approximating solutions with the uniform strong Feller property and passing to the limit; also see [23]. For large deviation principle, we adopted weak convergence approach as in [26].The rest of the paper is organized as follows. The existence and uniqueness of invariant measures and the Hölder continuity will be solved in Section2. Section 3 establishes the strong Feller property. In Section 4, we deal with large deviation principle.


## 2. Existence and Uniqueness of Invariant Measures

In the beginning of this section, we present the Hölder continuity of the solution of (1) which will be used in the proof of the existence of invariant measures.Consider the penalized problem as follows:(7)
∂
u
ɛ
,
δ
∂
t
=
∂
2
u
ɛ
,
δ
∂
x
2
+
f
(
u
ɛ
,
δ
)
+
σ
(
u
ɛ
,
δ
)
·
W
˙
+
f
ɛ
,
δ
(
u
ɛ
,
δ
)
;
u
ɛ
,
δ
(
x
,
0
)
=
u
0
(
x
)
∈
C
(
S
1
)
,
where
(8)
f
ɛ
,
δ
(
·
)
≔
(
{
c
1
[
δ
+
[
(
·
-
h
1
(
x
)
)
∨
0
]
2
]
α
/
2
}
arctan
⁡
(
[
(
·
-
h
1
(
x
)
)
∧
0
]
2
)
δ
h
h
h
+
c
1
[
δ
+
[
(
·
-
h
1
(
x
)
)
∨
0
]
2
]
α
/
2
)
-
(
{
c
2
[
ɛ
+
[
(
h
2
(
x
)
-
·
)
∨
0
]
2
]
α
/
2
}
arctan
⁡
(
[
(
h
2
(
x
)
-
·
)
∧
0
]
2
)
ɛ
h
h
h
+
c
2
[
ɛ
+
[
(
h
2
(
x
)
-
·
)
∨
0
]
2
]
α
/
2
)
,
with δ
,
ɛ
>
0. Notice that f
ɛ
,
δ is differentiable and f
ɛ
,
δ
′
≤
0.We need three lemmas before we present the regularity result.Lemma 2.
Suppose the hypotheses (H1), (H2), and (F1)–(F3) hold. For anyp
≥
1 and T
>
0, there exists ϱ
>
0 such that sup
⁡
0
<
ɛ
,
δ
<
ϱ
⁡
E
(
∥
u
ɛ
,
δ
∥
∞
T
)
p
<
∞
(
∥
ω
∥
∞
T
:
=
sup
⁡
x
∈
S
1
,
t
∈
[
0
,
T
]
⁡
|
ω
(
x
,
t
)
|
) and u
ɛ
,
δ converges uniformly on S
1
×
[
0
,
T
] to u as ɛ
,
δ
→
0 a.s., where u and u
ɛ
,
δ are the solutions of (1) and the penalized SPDEs (7), respectively.Proof.
Without loss of generality, assumec
1
=
c
2
=
1. By (H1), there exists δ
0
∈
(
0,1
) such that h
2
(
x
)
-
h
1
(
x
)
≥
δ
0
,
∀
x
∈
S
1. Set 0
<
ɛ
,
δ
<
δ
0
2
/
2.
Letv
ɛ
,
δ be the solution of equation
(9)
∂
v
ɛ
,
δ
(
x
,
t
)
∂
t
=
∂
2
v
ɛ
,
δ
(
x
,
t
)
∂
x
2
+
f
(
u
ɛ
,
δ
(
x
,
t
)
)
+
σ
(
u
ɛ
,
δ
(
x
,
t
)
)
W
˙
(
x
,
t
)
;
v
ɛ
,
δ
(
x
,
0
)
=
u
0
(
x
)
.
Set Φ
¯
ɛ
,
δ
(
t
)
=
sup
⁡
s
≤
t
,
y
∈
S
1
⁡
(
v
ɛ
,
δ
(
y
,
s
)
-
(
h
1
(
y
)
+
h
2
(
y
)
-
δ
0
)
/
2
). Note that Φ
¯
ɛ
,
δ
(
t
) is increasing with respect to t and v
ɛ
,
δ
(
x
,
t
)
-
Φ
¯
ɛ
,
δ
(
t
)
≤
(
h
1
(
x
)
+
h
2
(
x
)
-
δ
0
)
/
2. From analogue method as the proof in Proposition 3.1 in [11], z
¯
ɛ
,
δ
(
x
,
t
)
:
=
v
ɛ
,
δ
(
x
,
t
)
-
Φ
¯
ɛ
,
δ
(
t
)
-
u
ɛ
,
δ
(
x
,
t
) is a solution of equation
(10)
∂
z
¯
ɛ
,
δ
∂
t
+
∂
Φ
¯
ɛ
,
δ
∂
t
=
∂
2
z
¯
ɛ
,
δ
∂
x
2
-
f
ɛ
,
δ
(
u
ɛ
,
δ
)
;
z
¯
ɛ
,
δ
(
x
,
0
)
=
0
.
Assume(
z
¯
ɛ
,
δ
)
+
(
x
,
t
)
>
0; we have
(11)
0
<
(
z
¯
ɛ
,
δ
)
+
(
x
,
t
)
⟺
0
<
v
ɛ
,
δ
(
x
,
t
)
-
Φ
¯
ɛ
,
δ
(
t
)
-
u
ɛ
,
δ
(
x
,
t
)
≤
v
ɛ
,
δ
(
x
,
t
)
-
(
v
ɛ
,
δ
(
x
,
t
)
-
h
1
(
x
)
+
h
2
(
x
)
-
δ
0
2
)
-
u
ɛ
,
δ
(
x
,
t
)
=
h
1
(
x
)
+
h
2
(
x
)
-
δ
0
2
-
u
ɛ
,
δ
(
x
,
t
)
;
hence
(12)
u
ɛ
,
δ
(
x
,
t
)
≤
h
2
(
x
)
.
If (
z
¯
ɛ
,
δ
)
+
(
x
,
t
)
>
0 and u
ɛ
,
δ
(
x
,
t
)
≤
h
1
(
x
), then, by (12),
(13)
δ
+
[
(
u
ɛ
,
δ
(
x
,
t
)
-
h
1
(
x
)
)
∨
0
]
2
=
δ
≤
ɛ
+
δ
0
2
≤
ɛ
+
[
(
h
2
(
x
)
-
u
ɛ
,
δ
(
x
,
t
)
)
∨
0
]
2
.
On the other hand, if (
z
¯
ɛ
,
δ
)
+
(
x
,
t
)
>
0 and u
ɛ
,
δ
(
x
,
t
)
>
h
1
(
x
), then, by (11) and (12),
(14)
ɛ
+
[
(
h
2
(
x
)
-
u
ɛ
,
δ
(
x
,
t
)
)
∨
0
]
2
-
(
δ
+
[
(
u
ɛ
,
δ
(
x
,
t
)
-
h
1
(
x
)
)
∨
0
]
2
)
=
ɛ
-
δ
+
[
h
2
(
x
)
-
u
ɛ
,
δ
(
x
,
t
)
]
2
-
[
u
ɛ
,
δ
(
x
,
t
)
-
h
1
(
x
)
]
2
=
ɛ
-
δ
+
[
h
2
(
x
)
-
h
1
(
x
)
]
×
[
h
2
(
x
)
+
h
1
(
x
)
-
2
u
ɛ
,
δ
(
x
,
t
)
]
≥
ɛ
-
δ
+
δ
0
2
≥
δ
0
2
2
.
Hence combining (13) and (14), (
z
¯
ɛ
,
δ
)
+
(
x
,
t
)
>
0 implies
(15)
δ
+
[
(
u
ɛ
,
δ
(
x
,
t
)
-
h
1
(
x
)
)
∨
0
]
2
≤
ɛ
+
[
(
h
2
(
x
)
-
u
ɛ
,
δ
(
x
,
t
)
)
∨
0
]
2
.
By (12) and (15), multiplying (10) by (
z
¯
ɛ
,
δ
)
+, we have
(16)
arctan
(
[
(
h
2
(
x
)
-
u
ɛ
,
δ
(
x
,
t
)
)
∧
0
]
2
)
ɛ
(
z
¯
ɛ
,
δ
)
+
(
x
,
t
)
d
t
=
0
,
[
1
[
δ
+
[
(
u
ɛ
,
δ
(
x
,
t
)
-
h
1
(
x
)
)
∨
0
]
2
]
α
/
2
hl
-
1
[
ɛ
+
[
(
h
2
(
x
)
-
u
ɛ
,
δ
(
x
,
t
)
)
∨
0
]
2
]
α
/
2
]
×
(
z
¯
ɛ
,
δ
)
+
(
x
,
t
)
d
t
≥
0
.
Combining the fact that Φ
¯
ɛ
,
δ
(
t
) is increasing with respect to t, we have (
z
¯
ɛ
,
δ
)
+
=
0. Hence
(17)
u
ɛ
,
δ
(
x
,
t
)
≥
v
ɛ
,
δ
(
x
,
t
)
-
Φ
¯
ɛ
,
δ
(
x
,
t
)
.
Similarly, settingz
¯
ɛ
,
δ
(
x
,
t
)
=
u
ɛ
,
δ
(
x
,
t
)
-
v
ɛ
,
δ
(
x
,
t
)
-
sup
⁡
s
≤
t
,
y
∈
S
1
⁡
(
(
h
1
(
y
)
+
h
2
(
y
)
-
δ
0
)
/
2
-
v
ɛ
,
δ
(
y
,
s
)
), we can show that
(18)
u
ɛ
,
δ
(
x
,
t
)
≤
v
ɛ
,
δ
(
x
,
t
)
+
sup
⁡
s
≤
t
,
y
∈
S
1
(
h
1
(
y
)
+
h
2
(
y
)
-
δ
0
2
-
v
ɛ
,
δ
(
y
,
s
)
)
.
Using a similar proof in Donati-Martin and Pardoux [2], it can be shown that sup
⁡
ɛ
,
δ
⁡
E
(
∥
v
ɛ
,
δ
∥
∞
T
)
p
<
∞; hence the inequalities (17) and (18) imply
(19)
sup
⁡
ɛ
,
δ
E
(
∥
u
ɛ
,
δ
∥
∞
T
)
p
<
∞
.
Since u
ɛ
,
δ is increasing as δ
↓
0 by the comparison theorem of SPDEs (see Theorem 2.1 in [2]), we can show u
ɛ
:
=
lim
⁡
δ
↓
0
⁡
u
ɛ
,
δ exists a.s. and u
ɛ solves
(20)
∂
u
ɛ
(
x
,
t
)
∂
t
=
∂
2
u
ɛ
(
x
,
t
)
∂
x
2
+
f
(
u
ɛ
(
x
,
t
)
)
+
σ
(
u
ɛ
(
x
,
t
)
)
W
˙
(
x
,
t
)
+
(
η
ɛ
(
x
,
t
)
+
1
(
u
ɛ
(
x
,
t
)
-
h
1
(
x
)
)
α
)
-
(
arctan
(
[
(
h
2
(
x
)
-
u
ɛ
(
x
,
t
)
)
∧
0
]
2
)
ɛ
1
[
ɛ
+
[
(
h
2
(
x
)
-
u
ɛ
(
x
,
t
)
)
∨
0
]
2
]
α
/
2
h
h
h
h
arctan
(
[
(
h
2
(
x
)
-
u
ɛ
(
x
,
t
)
)
∧
0
]
2
)
ɛ
+
1
[
ɛ
+
[
(
h
2
(
x
)
-
u
ɛ
(
x
,
t
)
)
∨
0
]
2
]
α
/
2
)
;
u
ɛ
(
x
,
0
)
=
u
0
(
x
)
,
where η
ɛ
(
d
x
,
d
t
)
:
=
lim
⁡
δ
↓
0
⁡
(
(
u
ɛ
,
δ
(
x
,
t
)
-
h
1
(
x
)
)
+
/
δ
)
d
x
d
t. By Lemma 3.1 and Remark 3.1 of [11], we know that u
ɛ is decreasing as ɛ
↓
0. Let v
ɛ be the solution of (9) replacing u
ɛ
,
δ by u
ɛ. Setting z
¯
ɛ
(
x
,
t
)
=
u
ɛ
(
x
,
t
)
-
v
ɛ
(
x
,
t
)
-
sup
⁡
s
≤
t
,
y
∈
S
1
⁡
(
(
h
1
(
y
)
+
h
2
(
y
)
-
δ
0
)
/
2
-
v
ɛ
(
y
,
s
)
), similar to (18), we can show
(21)
u
ɛ
(
x
,
t
)
≤
v
ɛ
(
x
,
t
)
+
sup
⁡
s
≤
t
,
y
∈
S
1
(
h
1
(
y
)
+
h
2
(
y
)
-
δ
0
2
-
v
ɛ
(
y
,
s
)
)
.
In addition, by the fact that u
ɛ
(
x
,
t
)
≥
h
1
(
x
), u
:
=
lim
⁡
ɛ
↓
0
u
ɛ
=
lim
⁡
ɛ
↓
0
lim
⁡
δ
↓
0
u
ɛ
,
δ exists a.s., and we can show u is the solution of (1); see [11].
The continuity ofu can be proved similarly as in Theorem 4.1 of  [2]. The uniform convergence of u
ɛ
,
δ with respect to (
x
,
t
) follows from Dini's theorem.Recall the following lemma from [29].Lemma 3.
LetS
∈
C
α
,
β
(
S
1
×
[
0
,
T
]
) satisfying
(22)
|
S
(
x
,
t
)
-
S
(
y
,
s
)
|
≤
C
S
(
|
x
-
y
|
α
+
|
t
-
s
|
β
)
.
Then, for ρ
1
,
ρ
2
>
0, there exists S
ρ
1
,
ρ
2
∈
C
∞
(
S
1
×
[
0
,
T
]
) such that
(23)
∥
S
ρ
1
,
ρ
2
-
S
∥
∞
T
≤
C
α
,
β
C
S
(
ρ
1
α
+
ρ
2
β
)
,
∥
∂
S
ρ
1
,
ρ
2
∂
x
∥
∞
T
≤
C
α
,
β
C
S
ρ
1
α
-
1
,
∥
∂
S
ρ
1
,
ρ
2
∂
t
∥
∞
T
≤
C
α
,
β
C
S
ρ
2
β
-
1
,
where C
S
,
C
α
,
β are constants only depending on S and α
,
β, respectively.The following result is stated in [29].Lemma 4.
LetV
∈
C
1,2
(
S
1
×
[
0
,
T
]
) and Ψ
,
F
∈
C
(
S
1
×
[
0
,
T
]
) with Ψ
≤
0; V solves the equation with the homogeneous Dirichlet or Neumann boundary as follows:
(24)
∂
V
∂
t
=
∂
2
V
∂
x
2
+
Ψ
V
+
Ψ
F
;
V
(
x
,
0
)
=
0
;
then ∥
V
∥
∞
T
≤
∥
F
∥
∞
T.Theorem 5.
Letu be the solution of (1). Denote
(25)
u
¯
(
x
,
t
)
=
u
(
x
,
t
)
-
∫
S
1
G
t
(
x
,
y
)
u
0
(
y
)
d
y
‍
.
Here G
t
(
x
,
y
) is the fundamental solution of the heat equation on S
1. Then, for any ϑ
∈
(
0,1
), there exists a finite random variable γ
ϑ, which is independent of u
0
(
x
), such that, for x
,
y
∈
S
1
,
t
,
s
∈
[
0
,
T
],
(26)
|
u
¯
(
x
,
t
)
-
u
¯
(
y
,
s
)
|
≤
γ
ϑ
(
|
x
-
y
|
ϑ
/
2
+
|
t
-
s
|
ϑ
/
4
)
,
and, for any p
≥
1,
(27)
E
γ
ϑ
p
<
∞
.
Moreover, ifu
0
∈
C
β
(
S
1
)
,
β
∈
(
0,1
/
2
), then, for 0
<
ϖ
<
β, there exists a finite random variable γ
u
ϖ, which is dependent on u
0
(
x
), such that
(28)
|
u
(
x
,
t
)
-
u
(
y
,
s
)
|
≤
γ
u
ϖ
(
|
x
-
y
|
ϖ
+
|
t
-
s
|
ϖ
/
2
)
,
for x
,
y
∈
S
1
,
t
,
s
∈
[
0
,
T
], and for any p
≥
1,
(29)
E
[
γ
u
ϖ
]
p
<
∞
.Note. C
β
(
S
1
) denote the space of Hölder continuous functions on S
1 with Hölder exponent β, equipped with the norm
(30)
∥
u
∥
β
:
=
sup
⁡
x
∈
S
1
|
u
(
x
)
|
+
sup
⁡
x
,
y
∈
S
1
;
x
≠
y
|
u
(
x
)
-
u
(
y
)
|
|
x
-
y
|
β
.Proof.
The scheme of verifying inequality (26) is similar to that in [29]. For reader's convenience, we write the proof in detail.
Define the stochastic convolution as follows:(31)
S
t
ɛ
,
δ
(
x
)
:
=
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
f
(
u
ɛ
,
δ
(
y
,
s
)
)
d
y
d
s
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
σ
(
u
ɛ
,
δ
(
y
,
s
)
)
W
(
d
y
,
d
s
)
.
Here u
ɛ
,
δ is the solution of (7).
By Lemma2 and using similar arguments as Lemma 2.1 of [29], we have that, for any ϑ
∈
(
0,1
), there exists a random variable C
ɛ
,
δ such that, for any s
,
t
∈
[
0
,
T
]
,
x
,
y
∈
S
1,
(32)
|
S
t
ɛ
,
δ
(
x
)
-
S
s
ɛ
,
δ
(
y
)
|
≤
C
ɛ
,
δ
(
|
x
-
y
|
ϑ
/
2
+
|
t
-
s
|
ϑ
/
4
)
,
and for any p
≥
1
(33)
sup
⁡
ɛ
,
δ
E
(
C
ɛ
,
δ
)
p
<
∞
.
Letu
¯
ɛ
,
δ
(
x
,
t
)
=
u
ɛ
,
δ
(
x
,
t
)
-
∫
S
1
‍
G
t
(
x
,
y
)
u
0
(
y
)
d
y, and resolve u
¯
ɛ
,
δ into determine part and stochastic part; that is,
(34)
u
¯
ɛ
,
δ
(
x
,
t
)
=
z
S
ɛ
,
δ
(
x
,
t
)
+
S
t
ɛ
,
δ
(
x
)
,
where z
S
ɛ
,
δ is the unique solution of the following PDE:
(35)
∂
z
S
ɛ
,
δ
∂
t
=
∂
2
z
S
ɛ
,
δ
∂
x
2
+
f
¯
ɛ
,
δ
(
z
S
ɛ
,
δ
+
S
ɛ
,
δ
)
;
z
S
ɛ
,
δ
(
x
,
0
)
=
0
,
and S
ɛ
,
δ is defined in (31), f
¯
ɛ
,
δ
(
·
)
=
f
ɛ
,
δ
(
·
+
∫
S
1
‍
G
t
(
x
,
y
)
u
0
(
y
)
d
y
). Let S
ρ
1
,
ρ
2
∈
C
∞
(
S
1
×
[
0
,
T
]
), which depends on ρ
1
,
ρ
2 and will be determined later. Denote z
S
ρ
1
,
ρ
2 by the unique solution of (35) replacing S
ɛ
,
δ by S
ρ
1
,
ρ
2.
As similar proof in Proposition 3.1 in [11], we obtain
(36)
∥
z
S
ɛ
,
δ
-
z
S
ρ
1
,
ρ
2
∥
∞
T
≤
∥
S
ɛ
,
δ
-
S
ρ
1
,
ρ
2
∥
∞
T
.
Also, Lemma2 implies that
(37)
∥
S
ɛ
,
δ
-
S
ρ
1
,
ρ
2
∥
∞
T
≤
C
ϑ
/
2
C
ɛ
,
δ
(
ρ
1
ϑ
/
2
+
ρ
2
ϑ
/
4
)
.
And so
(38)
∥
z
S
-
z
S
ρ
1
,
ρ
2
∥
∞
T
≤
C
ϑ
/
2
C
ɛ
,
δ
(
ρ
1
ϑ
/
2
+
ρ
2
ϑ
/
4
)
.
Differentiatingz in (35), z
~
(
:
=
∂
x
z
S
ρ
1
,
ρ
2
) and z
¯
(
:
=
∂
t
z
S
ρ
1
,
ρ
2
) satisfy, respectively,
(39)
∂
z
~
∂
t
=
∂
2
z
~
∂
x
2
+
f
¯
ɛ
,
δ
′
(
z
S
ρ
1
,
ρ
2
+
S
ρ
1
,
ρ
2
)
·
(
z
~
+
∂
x
S
ρ
1
,
ρ
2
)
;
z
~
(
x
,
0
)
=
0
,
∂
z
¯
∂
t
=
∂
2
z
¯
∂
x
2
+
f
¯
ɛ
,
δ
′
(
z
S
ρ
1
,
ρ
2
+
S
ρ
1
,
ρ
2
)
·
(
z
¯
+
∂
t
S
ρ
1
,
ρ
2
)
;
z
¯
(
x
,
0
)
=
0
.
Therefore, from Lemmas3 and 4, we get
(40)
∥
z
~
∥
∞
T
≤
∥
∂
x
S
ρ
1
,
ρ
2
∥
∞
T
≤
C
ϑ
/
2
C
ɛ
,
δ
·
ρ
1
ϑ
/
2
-
1
;
∥
z
¯
∥
∞
T
≤
∥
∂
t
S
ρ
1
,
ρ
2
∥
∞
T
≤
C
ϑ
/
2
C
ɛ
,
δ
·
ρ
2
ϑ
/
4
-
1
.
In view of (34), (36), (38), and (40), one has
(41)
|
u
¯
ɛ
,
δ
(
x
,
t
)
-
u
¯
ɛ
,
δ
(
y
,
s
)
|
≤
|
z
S
ɛ
,
δ
(
x
,
t
)
-
z
S
ɛ
,
δ
(
y
,
s
)
|
+
|
S
t
ɛ
,
δ
(
x
)
-
S
s
ɛ
,
δ
(
y
)
|
≤
|
z
S
ɛ
,
δ
(
x
,
t
)
-
z
S
ρ
1
,
ρ
2
(
x
,
t
)
|
+
|
z
S
ρ
1
,
ρ
2
(
x
,
t
)
-
z
S
ρ
1
,
ρ
2
(
y
,
s
)
|
+
|
z
S
ρ
1
,
ρ
2
(
y
,
s
)
-
z
S
ɛ
,
δ
(
y
,
s
)
|
+
C
ɛ
,
δ
(
|
x
-
y
|
ϑ
/
2
+
|
t
-
s
|
ϑ
/
4
)
≤
2
∥
S
ɛ
,
δ
-
S
ρ
1
,
ρ
2
∥
∞
T
+
∥
z
~
∥
∞
T
·
|
x
-
y
|
+
∥
z
¯
∥
∞
T
·
|
t
-
s
|
+
C
ɛ
,
δ
(
|
x
-
y
|
ϑ
/
2
+
|
t
-
s
|
ϑ
/
4
)
≤
2
C
ϑ
/
2
C
ɛ
,
δ
(
ρ
1
ϑ
/
2
+
ρ
2
ϑ
/
4
)
+
C
ϑ
/
2
C
ɛ
,
δ
(
ρ
1
ϑ
/
2
-
1
·
|
x
-
y
|
+
ρ
2
ϑ
/
4
-
1
·
|
t
-
s
|
)
+
C
ɛ
,
δ
(
|
x
-
y
|
ϑ
/
2
+
|
t
-
s
|
β
/
2
)
.
Set ρ
1
:
=
|
x
-
y
| and ρ
2
:
=
|
t
-
s
|; then
(42)
|
u
¯
ɛ
,
δ
(
x
,
t
)
-
u
¯
ɛ
,
δ
(
y
,
s
)
|
≤
γ
ɛ
,
δ
(
|
x
-
y
|
ϑ
/
2
+
|
t
-
s
|
ϑ
/
4
)
,
where γ
ɛ
,
δ
:
=
(
3
C
ϑ
/
2
+
1
)
C
ɛ
,
δ and, for p
≥
1, sup
⁡
ɛ
,
δ
E
γ
ɛ
,
δ
p
<
∞. This yields, for p
≥
1,
(43)
sup
⁡
ɛ
,
δ
E
|
u
¯
ɛ
,
δ
(
x
,
t
)
-
u
¯
ɛ
,
δ
(
y
,
s
)
|
p
≤
C
M
p
(
|
x
-
y
|
(
ϑ
/
2
)
p
+
|
t
-
s
|
(
ϑ
/
4
)
p
)
.
By Lemma 2, we obtain from (43) that
(44)
E
|
u
¯
(
x
,
t
)
-
u
¯
(
y
,
s
)
|
p
≤
C
~
p
(
|
x
-
y
|
(
ϑ
/
2
)
p
+
|
t
-
s
|
(
ϑ
/
4
)
p
)
.
Applying a variant of Garsia's lemma (see PropositionA
.
1 and Corollary A
.
3 in [30]), we conclude that
(45)
|
u
¯
(
x
,
t
)
-
u
¯
(
y
,
s
)
|
≤
C
p
(
ω
)
(
|
x
-
y
|
ϑ
/
2
-
4
/
p
+
|
t
-
s
|
ϑ
/
4
-
2
/
p
)
.
Since p can be chosen to be arbitrarily large and ϑ to be as close to 1 as one wants, we have (26).
By Lemma 2.2 of [31], for any 0
<
ϖ
<
β,
(46)
|
∫
S
1
‍
G
t
(
x
,
y
)
u
0
(
y
)
d
y
-
∫
S
1
‍
G
t
(
z
,
y
)
u
0
(
y
)
d
y
|
≤
C
ϖ
∥
u
0
∥
ϖ
(
|
t
-
s
|
ϖ
/
2
+
|
x
-
z
|
ϖ
)
,
and combining (26), we have (28).
The proof is complete.Denote byB
(
C
(
S
1
)
) the σ-field of all Borel subsets of C
(
S
1
) and by M
(
C
(
S
1
)
) the set of all probability measures defined on (
C
(
S
1
)
,
B
(
C
(
S
1
)
)
). We denote by u
(
x
,
t
,
u
0
) the solution of (1) and by P
t
(
u
0
,
·
) the corresponding transition function
(47)
P
t
(
u
0
,
Γ
)
=
P
(
u
(
·
,
t
,
u
0
)
∈
Γ
)
,
Γ
∈
B
(
C
(
S
1
)
)
,
t
>
0
,
where u
0 is the initial condition. For μ
∈
M
(
C
(
S
1
)
), we set
(48)
P
t
*
μ
(
Γ
)
=
∫
C
(
S
1
)
‍
P
t
(
x
,
Γ
)
μ
(
d
x
)
,
where t
≥
0
,
Γ
∈
B
(
C
(
S
1
)
).Theorem 6.
Suppose the hypotheses (H1), (H2), and (F1)–(F3) hold. Then there exists an invariant measure to (1) on C
(
S
1
) for all α
≥
0, c
1
>
0, and c
2
>
0.Proof.
According to the Krylov-Bogolyubov theorem (see [6]), if there exists u
0
∈
C
(
S
1
) such that the family P
t
(
u
0
,
·
)
,
t
≥
1 is uniformly tight, then there exists an invariant measure for (1). We need to show that for any ɛ
>
0 there is a compact set K
⊂
C
(
S
1
) such that
(49)
P
(
u
(
t
)
∈
K
)
≥
1
-
ɛ
,
for
any
t
≥
1
,
where u
(
t
)
=
u
(
t
,
u
0
)
=
u
(
·
,
t
,
u
0
). On the other hand, for any t
≥
1, we have by the Markov property
(50)
P
(
u
(
t
)
∈
K
)
=
E
(
P
1
(
u
(
t
-
1
)
,
K
)
)
.
Thus it is enough to showP
(
u
(
1
,
u
(
t
-
1
)
)
∈
K
)
≥
1
-
ɛ, for any t
≥
1. As h
1
(
·
)
≤
u
(
t
-
1
)
(
·
)
≤
h
2
(
·
), it suffices to find a compact subset K
⊂
C
(
S
1
) such that
(51)
P
1
(
g
,
K
)
≥
1
-
ɛ
,
∀
g
∈
C
(
S
1
)
with
h
1
≤
g
≤
h
2
.
Put(52)
u
¯
(
x
,
t
,
g
)
=
u
(
x
,
t
,
g
)
-
∫
S
1
‍
G
t
(
x
,
y
)
g
(
y
)
d
y
.
Then (
u
¯
,
η
,
ξ
) solves a random obstacle problem.
Define(53)
∥
f
∥
β
=
sup
⁡
{
|
f
(
x
,
t
)
-
f
(
y
,
s
)
|
(
|
x
-
y
|
2
+
(
t
-
s
)
2
)
1
/
2
:
hhhhh
|
f
(
x
,
t
)
-
f
(
y
,
s
)
|
(
|
x
-
y
|
2
+
(
t
-
s
)
2
)
1
/
2
(
x
,
t
)
,
(
y
,
s
)
∈
S
1
×
[
0,1
]
,
(
x
,
t
)
≠
(
y
,
s
)
}
.
By the Arzela-Ascoli theorem, for allr
>
0, K
r
:
=
{
f
;
∥
f
∥
β
≤
r
} is a compact subset of C
(
S
1
×
[
0,1
]
). In view of (26), we see that, for given ɛ
>
0, there exists r
0 such that
(54)
P
(
u
¯
(
·
,
·
,
g
)
∈
K
r
0
c
)
≤
ɛ
,
∀
g
with
h
1
≤
g
≤
h
2
.
Let K
′
=
{
φ
(
·
,
1
)
:
φ
(
·
,
·
)
∈
K
r
0
} which is a compact subset in C
(
S
1
); we obtain P
(
u
¯
(
·
,
1
,
g
)
∈
K
′
)
≥
1
-
ɛ, for all g with h
1
≤
g
≤
h
2.
Also, it is easy to see that there is a compact subsetK
0
⊂
C
(
S
1
) such that
(55)
{
∫
S
1
‍
G
1
(
x
,
y
)
g
(
y
)
d
y
;
h
1
≤
g
≤
h
2
}
⊂
K
0
.
Define K
=
K
′
+
K
0. We have
(56)
P
1
(
g
,
K
)
=
P
(
u
(
·
,
1
,
g
)
∈
K
)
≥
P
(
u
¯
(
·
,
1
,
g
)
∈
K
′
)
≥
1
-
ɛ
,
for all g
∈
C
(
S
1
) with h
1
≤
g
≤
h
2. This finishes the proof.The following result is the uniqueness of invariant measures.Theorem 7.
Under the assumptions in Theorem6 and the fact that |
σ
(
·
)
|
≥
L
0 for some constant L
0
>
0, there is a unique invariant measure for (1) for all α
≥
0, c
1
>
0, and c
2
>
0.Proof of Theorem7.
To prove the uniqueness, we apply coupling method to SPDEs with reflection and singular terms; see in [20] or in [12]. Suppose there are two invariant probabilities μ
1 and μ
2 which are distributions of initial values u
1
(
x
,
0
) and u
2
(
x
,
0
), respectively. And so u
1
(
x
,
t
) and u
2
(
x
,
t
) have these distributions for any t
>
0. Then the total variation distance between μ
1 and μ
2 satisfies
(57)
Var
⁡
(
μ
1
-
μ
2
)
≤
P
(
sup
⁡
x
∈
S
1
|
u
1
(
x
,
t
)
-
u
2
(
x
,
t
)
|
≠
0
)
.
Consider the following SPDEs:(58)
∂
u
(
x
,
t
)
∂
t
=
∂
2
u
(
x
,
t
)
∂
x
2
+
f
(
u
(
x
,
t
)
)
+
σ
(
u
(
x
,
t
)
)
W
˙
1
(
x
,
t
)
+
η
1
(
x
,
t
)
-
ξ
1
(
x
,
t
)
+
c
1
(
u
(
x
,
t
)
-
h
1
(
x
)
)
α
-
c
2
(
h
2
(
x
)
-
u
(
x
,
t
)
)
α
,
∂
v
(
x
,
t
)
∂
t
=
∂
2
v
(
x
,
t
)
∂
x
2
+
f
(
v
(
x
,
t
)
)
+
σ
(
v
(
x
,
t
)
)
×
[
(
1
-
|
u
(
x
,
t
)
-
v
(
x
,
t
)
|
∧
1
)
1
/
2
W
˙
1
(
x
,
t
)
h
h
h
+
(
|
u
(
x
,
t
)
-
v
(
x
,
t
)
|
∧
1
)
1
/
2
W
˙
2
(
x
,
t
)
]
+
η
2
(
x
,
t
)
-
ξ
2
(
x
,
t
)
+
c
1
(
v
(
x
,
t
)
-
h
1
(
x
)
)
α
-
c
2
(
h
2
(
x
)
-
v
(
x
,
t
)
)
α
,
u
(
x
,
0
)
=
u
1
(
x
,
0
)
,
v
(
x
,
0
)
=
u
2
(
x
,
0
)
,
where W
1
(
x
,
t
)
,
W
2
(
x
,
t
) are two independent space-time white noises defined on the same probability space (
Ω
,
F
,
P
). The existence of solutions of (58) can be obtained by a similar method as that in the paper [11]. Indeed, (
u
,
v
) is the limit of the following SPDEs with Lipschitz diffusion coefficients:
(59)
∂
u
(
x
,
t
)
∂
t
=
∂
2
u
(
x
,
t
)
∂
x
2
+
f
(
u
(
x
,
t
)
)
+
σ
(
u
(
x
,
t
)
)
W
˙
1
(
x
,
t
)
+
η
1
(
x
,
t
)
-
ξ
1
(
x
,
t
)
+
c
1
(
u
(
x
,
t
)
-
h
1
(
x
)
)
α
-
c
2
(
h
2
(
x
)
-
u
(
x
,
t
)
)
α
,
∂
v
n
(
x
,
t
)
∂
t
=
∂
2
v
n
(
x
,
t
)
∂
x
2
+
f
(
v
n
(
x
,
t
)
)
+
σ
(
v
n
(
x
,
t
)
)
×
[
g
n
(
|
u
(
x
,
t
)
-
v
n
(
x
,
t
)
|
∧
1
)
W
˙
1
(
x
,
t
)
hhh
+
f
n
(
|
u
(
x
,
t
)
-
v
n
(
x
,
t
)
|
∧
1
)
W
˙
2
(
x
,
t
)
]
+
η
¯
2
n
(
x
,
t
)
-
ξ
¯
2
n
(
x
,
t
)
+
c
1
(
v
n
(
x
,
t
)
-
h
1
(
x
)
)
α
-
c
2
(
h
2
(
x
)
-
v
n
(
x
,
t
)
)
α
,
u
(
x
,
0
)
=
u
1
(
x
,
0
)
,
v
(
x
,
0
)
=
u
2
(
x
,
0
)
,
where f
n
(
·
)
=
(
·
+
1
/
n
)
1
/
2
-
(
1
/
n
)
1
/
2 and g
n
(
·
)
=
(
1
-
f
n
(
·
)
2
)
1
/
2.
Therefore, in order to get the uniqueness of invariant measure, it is enough to show(60)
lim
⁡
t
→
∞
P
(
sup
⁡
x
∈
S
1
|
u
1
(
x
,
t
)
-
u
2
(
x
,
t
)
|
≠
0
)
=
0
,
for given two initial functions u
1
(
x
,
0
) and u
2
(
x
,
0
).
Using a similar argument as in the proof of Theorem 2.2 in [12], the details of which we omit, it suffices to establish the following equation when u
1
(
x
,
0
)
≥
u
2
(
x
,
0
), x
∈
S
1:
(61)
∫
S
1
‍
(
u
(
x
,
t
)
-
v
(
x
,
t
)
)
d
x
=
∫
S
1
‍
(
u
1
(
x
,
0
)
-
u
2
(
x
,
0
)
)
d
x
+
C
(
t
)
+
M
(
t
)
,
where
(62)
M
(
t
)
=
∫
0
t
‍
∫
S
1
‍
σ
(
u
(
x
,
s
)
)
W
1
(
d
x
,
d
s
)
-
∫
0
t
‍
∫
S
1
‍
σ
(
v
(
x
,
s
)
)
hhhhhh
×
(
1
-
|
u
(
x
,
s
)
-
v
(
x
,
s
)
|
∧
1
)
1
/
2
hhhhhh
×
W
˙
1
(
d
x
,
d
s
)
-
∫
0
t
‍
∫
S
1
‍
σ
(
v
(
x
,
s
)
)
(
|
u
(
x
,
s
)
-
v
(
x
,
s
)
|
∧
1
)
1
/
2
hhhhhh
×
W
˙
2
(
d
x
,
d
s
)
,
and C
(
t
) is a continuous, adapted, and nonincreasing process.
In view of the reflection and singular terms, we will construct approximating solutions of uniform convergence limits. Consider the following approximating SPDEs:(63)
∂
u
ɛ
,
δ
∂
t
=
∂
2
u
ɛ
,
δ
∂
x
2
+
f
(
u
ɛ
,
δ
)
+
f
ɛ
,
δ
(
u
ɛ
,
δ
)
+
σ
(
u
ɛ
,
δ
)
W
˙
1
;
∂
v
n
,
ɛ
,
δ
∂
t
=
∂
2
v
n
,
ɛ
,
δ
∂
x
2
+
f
(
v
n
,
ɛ
,
δ
)
+
f
ɛ
,
δ
(
v
n
,
ɛ
,
δ
)
+
σ
(
v
n
,
ɛ
,
δ
)
×
[
g
n
(
|
u
ɛ
,
δ
-
v
n
,
ɛ
,
δ
|
∧
1
)
W
˙
1
(
x
,
t
)
hhh
+
f
n
(
|
u
ɛ
,
δ
-
v
n
,
ɛ
,
δ
|
∧
1
)
W
˙
2
(
x
,
t
)
]
;
u
ɛ
,
δ
(
x
,
0
)
=
u
1
(
x
,
0
)
,
v
n
,
ɛ
,
δ
(
x
,
0
)
=
u
2
(
x
,
0
)
,
where f
ɛ
,
δ is the same as in (7).
Whenu
1
(
x
,
0
)
≥
u
2
(
x
,
0
), x
∈
S
1, we have u
ɛ
,
δ
≥
v
n
,
ɛ
,
δ which can be shown as Lemma 3.1 in [20]. Also, we will assume that f
(
u
) is nonincreasing. Otherwise, consider u
~
:
=
e
-
L
t
u and v
~
:
=
e
-
L
t
v, where L is the Lipschitz constant in (F1). Then the new drift term e
-
L
t
f
(
e
L
t
x
)
-
L
x is nonincreasing. Thus,
(64)
C
n
,
ɛ
,
δ
(
t
)
=
∫
S
1
‍
{
f
(
u
ɛ
,
δ
)
-
f
(
v
n
,
ɛ
,
δ
)
h
h
h
h
+
[
f
ɛ
,
δ
(
u
ɛ
,
δ
)
-
f
ɛ
,
δ
(
v
n
,
ɛ
,
δ
)
]
}
d
x
≤
0
.
Now we only needv
n
,
ɛ
,
δ
(
x
,
t
)
→
v
n
(
x
,
t
) as ɛ
,
δ
→
0 which will be shown in the lemma below. Letting ɛ
,
δ
→
0 in the relation of ∫
S
1
‍
(
u
ɛ
,
δ
(
x
,
t
)
-
v
n
,
ɛ
,
δ
(
x
,
t
)
)
d
x from (63) and then sending n to ∞, we obtain (61). The proof is complete.Lemma 8.
v
n
,
ɛ
,
δ
(
x
,
t
)
→
v
n
(
x
,
t
) in probability on S
1
×
[
0
,
T
], for any T
>
0, as ɛ
,
δ
→
0.Proof.
Consider the following SPDE:(65)
∂
v
~
n
,
ɛ
,
δ
∂
t
=
∂
2
v
~
n
,
ɛ
,
δ
∂
x
2
+
f
(
v
~
n
,
ɛ
,
δ
)
+
f
ɛ
,
δ
(
v
~
n
,
ɛ
,
δ
)
+
σ
(
v
~
n
,
ɛ
,
δ
)
×
[
g
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
W
˙
1
(
x
,
t
)
hhh
+
f
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
W
˙
2
(
x
,
t
)
]
;
v
~
n
,
ɛ
,
δ
(
x
,
0
)
=
u
2
(
x
,
0
)
.
Denote the solution by v
~
n
,
ɛ
,
δ
(
x
,
t
). Let v
~
n
,
ɛ
,
δ
=
z
~
n
,
ɛ
,
δ
+
w
~
n
,
ɛ
,
δ; here w
~
n
,
ɛ
,
δ
(
x
,
t
) and z
~
n
,
ɛ
,
δ
(
x
,
t
) satisfy, respectively,
(66)
∂
w
~
n
,
ɛ
,
δ
∂
t
=
∂
2
w
~
n
,
ɛ
,
δ
∂
x
2
+
f
(
v
n
,
ɛ
,
δ
)
+
σ
(
v
n
,
ɛ
,
δ
)
×
[
g
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
W
˙
1
(
x
,
t
)
hhh
+
f
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
W
˙
2
(
x
,
t
)
]
;
w
~
n
,
ɛ
,
δ
(
x
,
0
)
=
u
2
(
x
,
0
)
,
∂
z
~
n
,
ɛ
,
δ
∂
t
=
∂
2
z
~
n
,
ɛ
,
δ
∂
x
2
+
f
ɛ
,
δ
(
z
~
n
,
ɛ
,
δ
+
w
~
n
,
ɛ
,
δ
)
;
z
~
n
,
ɛ
,
δ
(
x
,
0
)
=
0
.
Similarly, define w
n
,
ɛ
,
δ
(
x
,
t
) and z
n
,
ɛ
,
δ
(
x
,
t
) satisfy, respectively,
(67)
∂
w
n
,
ɛ
,
δ
∂
t
=
∂
2
w
n
,
ɛ
,
δ
∂
x
2
+
f
(
v
n
,
ɛ
,
δ
)
+
σ
(
v
n
,
ɛ
,
δ
)
×
[
g
n
(
|
u
ɛ
,
δ
-
v
n
,
ɛ
,
δ
|
∧
1
)
W
˙
1
(
x
,
t
)
hhh
+
f
n
(
|
u
ɛ
,
δ
-
v
n
,
ɛ
,
δ
|
∧
1
)
W
˙
2
(
x
,
t
)
]
;
w
n
,
ɛ
,
δ
(
x
,
0
)
=
u
2
(
x
,
0
)
,
∂
z
n
,
ɛ
,
δ
∂
t
=
∂
2
z
n
,
ɛ
,
δ
∂
x
2
+
f
ɛ
,
δ
(
z
n
,
ɛ
,
δ
+
w
n
,
ɛ
,
δ
)
;
z
n
,
ɛ
,
δ
(
x
,
0
)
=
0
.
Then v
n
,
ɛ
,
δ
=
z
n
,
ɛ
,
δ
+
w
n
,
ɛ
,
δ.
From similar proof in Proposition 3.1 in [11], one has
(68)
∥
z
n
,
ɛ
,
δ
-
z
~
n
,
ɛ
,
δ
∥
∞
T
≤
∥
w
n
,
ɛ
,
δ
-
w
~
n
,
ɛ
,
δ
∥
∞
T
.
Thus,
(69)
∥
v
n
,
ɛ
,
δ
-
v
~
n
,
ɛ
,
δ
∥
∞
T
≤
2
∥
w
n
,
ɛ
,
δ
-
w
~
n
,
ɛ
,
δ
∥
∞
T
.
Now we will show that ∥
v
n
,
ɛ
,
δ
-
v
~
n
,
ɛ
,
δ
∥
∞
T
→
0 in probability as ɛ
,
δ
→
0.
As the similar proof in Donati-Martin and Pardoux [2], it can be shown that sup
⁡
ɛ
,
δ
⁡
E
(
∥
v
n
,
ɛ
,
δ
∥
∞
T
)
p
<
∞, for arbitrarily large p and any T
>
0. Set the stopping time τ
M
:
=
inf
⁡
{
t
;
sup
⁡
x
∈
S
1
⁡
|
v
n
,
ɛ
,
δ
(
x
,
t
)
|
>
M
}. For fixed M
>
0, we have
(70)
E
(
∥
w
n
,
ɛ
,
δ
(
t
∧
τ
M
)
-
w
~
n
,
ɛ
,
δ
(
t
∧
τ
M
)
∥
∞
T
)
2
≤
C
1
∫
0
T
‍
E
(
∥
f
(
v
n
,
ɛ
,
δ
)
-
f
(
v
~
n
,
ɛ
,
δ
)
∥
∞
t
∧
τ
M
)
2
d
s
+
C
2
∫
0
T
‍
E
(
{
×
g
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
∥
∞
t
∧
τ
M
}
∥
σ
(
v
n
,
ɛ
,
δ
)
g
n
(
|
u
ɛ
,
δ
-
v
n
,
ɛ
,
δ
|
∧
1
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
-
σ
(
v
~
n
,
ɛ
,
δ
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
g
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
∥
∞
t
∧
τ
M
)
2
d
s
+
C
2
∫
0
T
‍
E
(
{
×
g
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
∥
∞
t
∧
τ
M
}
∥
σ
(
v
n
,
ɛ
,
δ
)
f
n
(
|
u
ɛ
,
δ
-
v
n
,
ɛ
,
δ
|
∧
1
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
-
σ
(
v
~
n
,
ɛ
,
δ
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
f
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
∥
∞
t
∧
τ
M
)
2
d
s
≤
C
1
∫
0
T
‍
E
(
∥
w
n
,
ɛ
,
δ
-
w
~
n
,
ɛ
,
δ
∥
∞
t
∧
τ
M
)
2
d
s
+
C
2
∫
0
T
‍
E
(
{
×
g
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
∥
∞
t
∧
τ
M
}
∥
σ
(
v
n
,
ɛ
,
δ
)
∥
∞
t
∧
τ
M
hhhhhhhhhh
·
[
{
×
g
n
(
|
u
-
v
~
n
,
ɛ
,
δ
|
∧
1
)
∥
∞
t
∧
τ
M
}
∥
u
ɛ
,
δ
-
u
∥
∞
t
∧
τ
M
hhhhhhhhhhhh
+
∥
w
n
,
ɛ
,
δ
-
w
~
n
,
ɛ
,
δ
∥
∞
t
∧
τ
M
]
)
2
d
s
≤
C
1
ϵ
+
C
2
∫
0
T
‍
E
(
∥
w
n
,
ɛ
,
δ
-
w
~
n
,
ɛ
,
δ
∥
∞
t
∧
τ
M
)
2
d
s
,
where the constants change in each line and ϵ
→
0 as ɛ
,
δ
→
0 because of Lemma 2. By the Gronwall inequality, we get E
(
∥
w
n
,
ɛ
,
δ
(
t
∧
τ
M
)
-
w
~
n
,
ɛ
,
δ
(
t
∧
τ
M
)
∥
∞
T
)
2
→
0 as ɛ
,
δ
→
0. Hence, E
(
∥
v
n
,
ɛ
,
δ
(
t
∧
τ
M
)
-
v
~
n
,
ɛ
,
δ
(
t
∧
τ
M
)
∥
∞
T
)
2
→
0. So ∥
v
n
,
ɛ
,
δ
-
v
~
n
,
ɛ
,
δ
∥
∞
T
→
0 in probability. In fact, for every ϵ
0
>
0, set the event A
n
,
ɛ
,
δ
:
=
{
∥
v
n
,
ɛ
,
δ
-
v
~
n
,
ɛ
,
δ
∥
∞
T
>
ϵ
0
},
(71)
P
(
A
n
,
ɛ
,
δ
)
=
P
(
A
n
,
ɛ
,
δ
,
τ
M
≤
T
)
+
P
(
A
n
,
ɛ
,
δ
,
τ
M
>
T
)
≤
P
(
τ
M
≤
T
)
+
E
(
∥
w
n
,
ɛ
,
δ
(
t
∧
τ
M
)
-
w
~
n
,
ɛ
,
δ
(
t
∧
τ
M
)
∥
∞
T
)
2
ϵ
0
2
≤
E
(
∥
v
n
,
ɛ
,
δ
∥
∞
T
)
2
M
2
+
E
(
∥
w
n
,
ɛ
,
δ
(
t
∧
τ
M
)
-
w
~
n
,
ɛ
,
δ
(
t
∧
τ
M
)
∥
∞
T
)
2
ϵ
0
2
⟶
0
,
as
ɛ
,
δ
⟶
0
,
M
⟶
∞
.
Since we can apply Lemma2 to conclude that u
ɛ
,
δ
(
x
,
t
)
→
u
(
x
,
t
), v
~
n
,
ɛ
,
δ
(
x
,
t
)
→
v
n
(
x
,
t
) uniformly on S
1
×
[
0
,
T
] as ɛ
,
δ
→
0. Therefore, v
n
,
ɛ
,
δ
(
x
,
t
)
→
v
n
(
x
,
t
) in probability on S
1
×
[
0
,
T
] as ɛ
,
δ
→
0.


## 3. Strong Feller Property

In this section, we consider the strong Feller property of the solution of (1). Let H
=
L
2
(
S
1
). If φ
∈
B
b
(
H
) (the Banach space of all real bounded Borel functions, endowed with the sup norm), we define, for x
∈
S
1, 0
≤
t
≤
T, and g
∈
H, P
t
φ
(
g
)
=
E
φ
(
u
(
x
,
t
,
g
)
).Theorem 9.
Under the hypotheses (H1), (H2), and (F1)–(F3) and the fact thatp
1
≤
|
σ
(
·
)
|
≤
p
2, for some constants p
1
,
p
2
>
0, for any T
>
0, there exists a constant C
T
′ such that, for all φ
∈
B
b
(
H
) and t
∈
(
0
,
T
],
(72)
|
P
t
φ
(
u
0
1
)
-
P
t
φ
(
u
0
2
)
|
≤
C
T
′
t
∥
φ
∥
∞
|
u
0
1
-
u
0
2
|
H
,
for u
0
1
,
u
0
2
∈
H with h
1
(
x
)
≤
u
0
1
(
x
)
,
u
0
2
(
x
)
≤
h
2
(
x
), where ∥
φ
∥
∞
=
sup
⁡
u
0
∈
H
⁡
|
φ
(
u
0
)
|. In particular, P
t
,
t
>
0 is strong Feller.Proof.
Choose a nonnegative functionϕ
∈
C
0
∞
(
R
) with ∫
R
‍
ϕ
(
x
)
=
1 and denote
(73)
f
n
(
ζ
)
≔
n
∫
R
‍
ϕ
(
n
(
ζ
-
y
)
)
f
(
y
)
d
y
,
σ
n
(
ζ
)
:
=
n
∫
R
‍
ϕ
(
n
(
ζ
-
y
)
)
σ
(
y
)
d
y
,
(74)
k
n
,
ɛ
,
δ
(
ζ
,
x
)
:
=
n
∫
R
‍
ϕ
(
n
(
ζ
-
y
)
)
k
n
,
ɛ
,
δ
(
ζ
,
x
)
:
=
n
∫
R
‍
×
[
1
[
δ
+
[
(
y
-
h
1
(
x
)
)
∨
0
]
2
]
α
/
2
k
n
,
ɛ
,
δ
(
ζ
,
x
)
:
=
n
h
h
h
∫
R
‍
-
1
[
ɛ
+
[
(
h
2
(
x
)
-
y
)
∨
0
]
2
]
α
/
2
{
1
[
δ
+
[
(
y
-
h
1
(
x
)
)
∨
0
]
2
]
α
2
}
]
d
y
,
l
n
,
ɛ
,
δ
(
ζ
,
x
)
:
=
k
n
,
ɛ
,
δ
(
ζ
,
x
)
l
n
,
ɛ
,
δ
(
ζ
,
x
)
≔
+
(
arctan
⁡
(
[
(
ζ
-
h
1
(
x
)
)
∧
0
]
2
)
δ
l
n
,
ɛ
,
δ
(
ζ
,
hh
.
hh
x
)
≔
-
arctan
(
[
(
h
2
(
x
)
-
ζ
)
∧
0
]
2
)
ɛ
{
arctan
⁡
(
[
(
ζ
-
h
1
(
x
)
)
∧
0
]
2
)
δ
}
)
.
Sof
n
,
σ
n
,
l
n
,
ɛ
,
δ are smooth with respect to ζ and f
n
(
ζ
)
→
f
(
ζ
), σ
n
(
ζ
)
→
σ
(
ζ
), and l
n
,
ɛ
,
δ
(
ζ
,
x
)
→
f
ɛ
,
δ
(
ζ
) as n
→
∞.
Let(75)
u
n
ɛ
,
δ
(
x
,
t
,
u
0
)
=
∫
S
1
‍
G
t
(
x
,
y
)
u
0
(
y
)
d
y
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
f
n
(
u
n
ɛ
,
δ
(
y
,
s
,
u
0
)
)
d
y
d
s
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
σ
n
(
u
n
ɛ
,
δ
(
y
,
s
,
u
0
)
)
W
(
d
y
,
d
s
)
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
l
n
,
ɛ
,
δ
(
u
n
ɛ
,
δ
(
y
,
s
,
u
0
)
,
y
)
d
y
d
s
.
Using similar arguments as in [23], we can show that, for any fixed ɛ
,
δ and p
≥
1,
(76)
lim
⁡
n
→
∞
sup
⁡
t
∈
[
0
,
T
]
E
(
|
u
n
ɛ
,
δ
(
t
,
·
,
u
0
)
-
u
ɛ
,
δ
(
t
,
·
,
u
0
)
|
H
p
)
=
0
.
Furthermore, in view of Lemma7.1
.
5 in [6] and Lemma 2, it is enough to prove that there exists a constant C
T
′, independent of ɛ
,
δ and n, such that
(77)
|
P
t
n
,
ɛ
,
δ
φ
(
u
0
1
)
-
P
t
n
,
ɛ
,
δ
φ
(
u
0
2
)
|
≤
C
T
′
t
∥
φ
∥
∞
|
u
0
1
-
u
0
2
|
H
,
where P
t
n
,
ɛ
,
δ
φ
(
u
0
)
:
=
E
(
φ
(
u
n
ɛ
,
δ
(
·
,
·
,
u
0
)
)
) and u
0
1
,
u
0
2
∈
H.
From Theorem5.4
.
1 in [6], u
n
ɛ
,
δ
(
·
,
·
,
u
0
) is continuously differentiable with respect to u
0. Denote by X
n
ɛ
,
δ
(
x
,
t
)
:
=
(
D
u
n
ɛ
,
δ
(
·
,
·
,
u
0
)
(
u
¯
0
)
)
(
x
,
t
) the directional derivative of u
n
ɛ
,
δ
(
·
,
·
,
u
0
) at u
0 in the direction of u
¯
0 and it satisfies the mild form of a SPDE as follows:
(78)
X
n
ɛ
,
δ
(
x
,
t
)
=
∫
S
1
‍
G
t
(
x
,
y
)
u
¯
0
(
y
)
d
y
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
f
n
′
(
u
n
ɛ
,
δ
(
y
,
s
,
u
0
)
)
h
h
h
h
h
h
×
X
n
ɛ
,
δ
(
y
,
s
)
d
y
d
s
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
σ
n
′
(
u
n
ɛ
,
δ
(
y
,
s
,
u
0
)
)
h
h
h
h
h
h
×
X
n
ɛ
,
δ
(
y
,
s
)
W
(
d
y
,
d
s
)
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
∂
∂
ζ
l
n
,
ɛ
,
δ
(
u
n
ɛ
,
δ
(
y
,
s
,
u
0
)
,
y
)
h
h
h
h
h
h
×
X
n
ɛ
,
δ
(
y
,
s
)
d
y
d
s
.
Since (
∂
/
∂
ζ
)
l
n
,
ɛ
,
δ
(
u
n
ɛ
,
δ
(
y
,
s
,
u
0
)
,
y
)
≤
0, we use similar arguments as that in [23] to get
(79)
sup
⁡
ɛ
,
δ
≥
0
,
t
∈
[
0
,
T
]
E
(
∫
S
1
‍
(
X
n
ɛ
,
δ
(
y
,
t
)
)
2
d
y
)
≤
C
|
u
¯
0
|
H
2
,
where C is a constant. By Elworthy-Li formula (Lemma 7.1
.
3 in [6]), we obtain
(80)
|
〈
D
P
t
φ
(
u
0
)
,
u
¯
0
〉
|
2
≤
C
p
1
2
t
∥
φ
∥
∞
2
|
u
¯
0
|
H
2
.
This implies inequality (77) which completes the proof.


## 4. Large Deviation Principle

We consider a small perturbation of the equation of (1); that is,
(81)
∂
u
ϵ
∂
t
=
∂
2
u
ϵ
∂
x
2
+
f
(
u
ϵ
)
+
c
1
(
u
ϵ
-
h
1
)
α
-
c
2
(
h
2
-
u
ϵ
)
α
+
ϵ
σ
(
u
)
W
˙
+
η
ϵ
-
ξ
ϵ
;
u
ϵ
(
x
,
0
)
=
u
0
(
x
)
.
As a similar discussion in [11], for any ϵ
>
0, there exists a continuous solution u
ϵ satisfying
(82)
u
ϵ
(
x
,
t
)
=
∫
S
1
‍
G
t
(
x
,
y
)
u
0
(
y
)
d
y
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
hhhhhh
×
(
f
(
u
ϵ
(
y
,
s
)
)
+
c
1
(
u
ϵ
(
y
,
s
)
-
h
1
(
y
)
)
α
hhhhhhhhhh
-
c
2
(
h
2
(
y
)
-
u
ϵ
(
y
,
s
)
)
α
)
d
y
d
s
+
ϵ
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
σ
(
u
ϵ
(
y
,
s
)
)
W
(
d
y
,
d
s
)
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
(
η
ϵ
(
d
y
d
s
)
-
ξ
ϵ
(
d
y
d
s
)
)
.
The Cameron-Martin space associated with the white noise {
W
(
x
,
t
)
,
(
x
,
t
)
∈
S
1
×
[
0
,
T
]
} is given by
(83)
H
=
{
h
(
x
,
t
)
=
∫
0
x
‍
∫
0
t
‍
h
˙
(
y
,
s
)
d
y
d
s
;
l
h
∫
S
1
‍
∫
0
T
‍
(
h
˙
(
y
,
s
)
)
2
d
y
d
s
<
∞
}
,
endowed with norm ∥
h
∥
H
=
(
∫
S
1
‍
∫
0
T
‍
(
h
˙
(
y
,
s
)
)
2
d
y
d
s
)
1
/
2.Consider the following PDE (the skeleton equation):(84)
∂
u
h
∂
t
=
∂
2
u
h
∂
x
2
+
f
(
u
h
)
+
c
1
(
u
h
-
h
1
)
α
-
c
2
(
h
2
-
u
h
)
α
+
σ
(
u
h
)
h
˙
+
η
h
-
ξ
h
;
u
h
(
x
,
0
)
=
u
0
(
x
)
;
h
1
(
x
)
≤
u
h
(
x
,
t
)
≤
h
2
(
x
)
.
As the analogous discussion in [11], for any h
∈
H, there exists a continuous solution u
h satisfying
(85)
u
h
(
x
,
t
)
=
∫
S
1
‍
G
t
(
x
,
y
)
u
0
(
y
)
d
y
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
hhhhhh
×
(
f
(
u
h
(
y
,
s
)
)
+
c
1
(
u
h
(
y
,
s
)
-
h
1
(
y
)
)
α
hhhhhhhhh
-
c
2
(
h
2
(
y
)
-
u
h
(
y
,
s
)
)
α
)
d
y
d
s
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
σ
(
u
h
(
y
,
s
)
)
h
˙
(
y
,
s
)
d
y
d
s
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
(
η
h
(
d
y
d
s
)
-
ξ
h
(
d
y
d
s
)
)
.We state our main result in this section.Theorem 10.
The lawsμ
ϵ of the {
u
ϵ
}
ϵ
>
0 satisfy a large deviation principle on D
:
=
{
u
(
x
,
t
)
∈
C
(
S
1
×
[
0
,
T
]
)
;
h
1
(
x
)
≤
u
(
x
,
t
)
≤
h
2
(
x
)
} with the rate function
(86)
I
(
f
)
=
inf
⁡
{
1
2
∥
h
∥
H
2
;
u
h
=
f
}
,
with the convention inf
⁡
{
∅
}
=
∞; that is,(i)
for any closed subsetA
⊂
D,
(87)
lim
⁡
sup
⁡
ϵ
→
0
ϵ
2
log
⁡
μ
ϵ
(
A
)
≤
-
inf
⁡
f
∈
A
I
(
f
)
;
(ii)
for any open setB
⊂
D,
(88)
lim
⁡
inf
⁡
ϵ
→
0
ϵ
2
log
⁡
μ
ϵ
(
B
)
≥
-
inf
⁡
f
∈
B
I
(
f
)
.Proof.
In order to prove Theorem10, by Theorem 4.4 in [27], it suffices to check the following.(i)
For anya
>
0
,
u
h is a continuous mapping from {
h
:
∥
h
∥
H
≤
a
} into C
(
S
1
×
[
0
,
T
]
).
(ii)
For a family{
h
ϵ
}
⊂
A
N, where A
N
:
=
{
h
∈
H
,
;
h
(
x
,
t
)
i
s
F
t
-
m
e
a
s
u
r
a
b
l
e
,
∥
h
∥
H
≤
N
P
-
a
.
s
.
}, 0
<
N
<
∞, that converges in distribution to h, u
ϵ
,
h
ϵ converges in distribution to u
h, where u
ϵ
,
h
ϵ solves the following SPDE:
(89)
∂
u
ϵ
,
h
ϵ
∂
t
=
∂
2
u
ϵ
,
h
ϵ
∂
x
2
+
f
(
u
ϵ
,
h
ϵ
)
+
c
1
(
u
ϵ
,
h
ϵ
-
h
1
)
α
-
c
2
(
h
2
-
u
ϵ
,
h
ϵ
)
α
+
ϵ
σ
(
u
ϵ
,
h
ϵ
)
W
˙
+
σ
(
u
ϵ
,
h
ϵ
)
h
˙
ϵ
+
η
ϵ
,
h
ϵ
-
ξ
ϵ
,
h
ϵ
;
u
ϵ
,
h
ϵ
(
x
,
0
)
=
u
0
(
x
)
.
Let(90)
v
h
i
(
x
,
t
)
=
∫
S
1
‍
G
t
(
x
,
y
)
u
0
(
y
)
d
y
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
f
(
u
h
i
(
y
,
s
)
)
d
y
d
s
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
σ
(
u
h
i
(
y
,
s
)
)
h
˙
i
(
y
,
s
)
d
y
d
s
,
with i
=
1,2. In terms of (36), we have
(91)
∥
u
h
1
-
u
h
2
∥
∞
T
≤
2
∥
v
h
1
-
v
h
2
∥
∞
T
.
Proof of (i). We need to prove that, for fixed a
>
0
,
h
1
,
h
2
∈
{
h
∈
H
:
∥
h
∥
H
≤
a
},
(92)
∥
u
h
1
-
u
h
2
∥
∞
T
≤
C
∥
h
1
-
h
2
∥
H
,
where u
h
1
,
u
h
2 are the solution of (84) associated with h
1
,
h
2, respectively. Let
(93)
v
h
i
(
x
,
t
)
=
∫
S
1
‍
G
t
(
x
,
y
)
u
0
(
y
)
d
y
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
f
(
u
h
i
(
y
,
s
)
)
d
y
d
s
+
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
σ
(
u
h
i
(
y
,
s
)
)
h
˙
i
(
y
,
s
)
d
y
d
s
with i
=
1,2. In terms of (36), we have
(94)
∥
u
h
1
-
u
h
2
∥
∞
T
≤
2
∥
v
h
1
-
v
h
2
∥
∞
T
≤
2
∥
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
h
h
h
h
h
h
h
h
h
h
h
×
(
f
(
u
h
1
(
y
,
s
)
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∫
0
t
‍
∫
S
1
‍
-
f
(
u
h
2
(
y
,
s
)
)
)
d
y
d
s
∥
∞
T
+
2
∥
∫
0
t
‍
∫
S
1
‍
G
t
-
s
(
x
,
y
)
h
h
h
h
h
h
h
h
h
h
h
h
×
(
σ
(
u
h
1
(
y
,
s
)
)
h
˙
1
(
y
,
s
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∫
0
t
‍
∫
S
1
‍
-
σ
(
u
h
2
(
y
,
s
)
)
h
˙
2
(
y
,
s
)
)
d
y
d
s
∥
∞
T
:
=
I
1
+
I
2
.
We know
(95)
I
1
≤
C
(
∫
0
T
‍
(
∥
u
h
1
-
u
h
2
∥
∞
t
)
2
d
t
)
1
/
2
,
I
2
≤
C
∥
(
∫
0
t
‍
∫
S
1
‍
|
G
t
-
s
(
x
,
y
)
|
2
d
y
d
s
)
1
/
2
hhh
×
(
∫
0
t
‍
∫
S
1
‍
|
σ
(
u
h
1
(
y
,
s
)
)
h
˙
1
(
y
,
s
)
h
h
h
h
h
h
h
h
h
h
h
h
∫
0
t
‍
∫
S
1
‍
-
σ
(
u
h
2
(
y
,
s
)
)
h
˙
2
(
y
,
s
)
|
2
d
y
d
s
)
1
/
2
∥
∞
T
.
≤
C
∥
h
1
-
h
2
∥
H
.
Then using Gronwall's inequality, we obtain(96)
∥
u
h
1
-
u
h
2
∥
∞
T
≤
C
∥
h
1
-
h
2
∥
H
.
Proof of (ii). It is similar to that in [26] with the help of inequality (91).
Thus the proof of Theorem10 is completed.



---

*Source: 264263-2014-05-20.xml*