The Small Time Asymptotics of SPDEs with Reflection

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.

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 drift 𝑢^−𝛼, Mueller [14] gave a proof that solutions of SPDEs 𝑢_𝑡=𝑢_𝑥𝑥+𝑔(𝑢)𝑊̇(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 walls ℎ¹ and ℎ² and two singular drifts 𝑐₁/(𝑢−ℎ¹)^𝛼 and 𝑐₂/(ℎ²−𝑢)^𝛼, for all 𝛼>0.

For the case 𝛼=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: 𝜕𝑢(𝑥,𝑡)=𝜕𝜕𝑡²𝑢(𝑥,𝑡)𝜕𝑥²(𝑢(+𝑐+𝑓𝑥,𝑡))₁𝑢(𝑥,𝑡)−ℎ¹(𝑥)^𝛼−𝑐₂ℎ²(𝑥)−𝑢(𝑥,𝑡)^𝛼(𝑢(𝑊̇()()();𝑢()+𝜎𝑥,𝑡))𝑥,𝑡+𝜂𝑥,𝑡−𝜉𝑥,𝑡𝑥,0=𝑢₀(𝑥)∈𝐶𝑆¹;ℎ¹(𝑥)≤𝑢(𝑥,𝑡)≤ℎ²(𝑥),for(𝑥,𝑡)∈𝑄, (1) where 𝑄∶=𝑆¹×ℝ₊, 𝛼≥0, 𝑐₁>0, 𝑐₂>0, 𝑆¹∶=ℝ(mod2𝜋), or {𝑒^𝑖𝜃;𝜃∈ℝ} denotes a circular ring and the random field 𝑊(𝑥,𝑡)∶=𝑊({𝑒^𝑖𝜃;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 [ℎ¹,ℎ²].

We assume that the reflecting walls ℎ¹(⋅),ℎ²(⋅) are continuous functions satisfying the following.(H1)ℎ¹(𝑥)<ℎ²(𝑥) for 𝑥∈𝑆¹.(H2) (𝜕²ℎ^𝑖(𝑥)/𝜕𝑥²)∈𝐿²(𝑆¹), where 𝜕²/𝜕𝑥² is interpreted in a distributional sense.We also assume that the coefficients 𝑓,𝜎∶ℝ→ℝ satisfy the following:(F1)there exists 𝐿>0 such that 𝑓𝑧₁−𝑓𝑧₂+𝜎𝑧₁−𝜎𝑧₂≤𝐿𝑧₁−𝑧₂,𝑧₁,𝑧₂∈ℝ; (2)(F2)there exists 𝐶>0 such that 𝑓(𝑧)+|𝜎(𝑧)|≤𝐶(1+|𝑧|),𝑧∈ℝ. (3) The initial condition 𝑢₀(𝑥) satisfies the following:(F3)𝑢₀(𝑥)∈𝐶(𝑆¹) satisfy ℎ¹(𝑥)≤𝑢₀(𝑥)≤ℎ²(𝑥), for 𝑥∈𝑆¹.

The following is the definition of a solution of a SPDE with two reflecting walls ℎ¹(𝑥)andℎ²(𝑥),𝑥∈𝑆¹.

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∈𝑆¹) satisfying ℎ¹(𝑥)≤𝑢(𝑥,𝑡)≤ℎ²(𝑥) a.s.;(ii)𝜂(𝑑𝑥,𝑑𝑡) and 𝜉(𝑑𝑥,𝑑𝑡) are positive and adapted (i.e., 𝜂(𝐵) and 𝜉(𝐵) are ℱ_𝑡-measurable if 𝐵⊂𝑆¹×[0,𝑡]) random measures on 𝑄 satisfying 𝜂𝑆¹×[0,𝑇]<∞,𝜉𝑆¹×[0,𝑇]<∞, (4) for 𝑇>0;(iii)for all 𝑡≥0 and 𝜙∈𝐶^∞(𝑆¹), one has 𝑢(𝑡),𝜙−𝑢₀,𝜙−^𝑡₀𝑢(𝑠),𝜙^′′−𝑑𝑠_𝑡0𝑓(𝑢(𝑠)),𝜙𝑑𝑠=^𝑡₀𝑐₁𝑢(𝑠)−ℎ^1𝛼−𝑐₂ℎ²−𝑢(𝑠)^𝛼+,𝜙𝑑𝑠_𝑡0𝑆1𝜙(𝑥)𝜎(𝑢(𝑊()+𝑥,𝑠))𝑑𝑥,𝑑𝑠_𝑡0𝑆1𝜙(𝑥)𝜂()−𝑑𝑥,𝑑𝑠_𝑡0𝑆1𝜙(𝑥)𝜉(𝑑𝑥,𝑑𝑠),a.s., (5) where (,) denotes the inner product in 𝐿²(𝑆¹) and 𝑢(𝑡) denotes 𝑢(⋅,𝑡);(iv)we consider the following: _𝑄𝑢(𝑥,𝑡)−ℎ¹(𝑥)𝜂()=𝑑𝑥,𝑑𝑡_𝑄ℎ²(𝑥)−𝑢(𝑥,𝑡)𝜉(𝑑𝑥,𝑑𝑡)=0. (6)

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 Section 2. 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: 𝜕𝑢^𝜀,𝛿=𝜕𝜕𝑡²𝑢^𝜀,𝛿𝜕𝑥²+𝑓𝑢^𝜀,𝛿+𝜎𝑢^𝜀,𝛿⋅𝑊̇+𝑓_𝜀,𝛿𝑢^𝜀,𝛿;𝑢^𝜀,𝛿(𝑥,0)=𝑢₀(𝑥)∈𝐶𝑆¹, (7) where 𝑓_𝜀,𝛿(⋅)∶=arctan⋅−ℎ¹(𝑥)∧0²𝛿+𝑐₁𝛿+⋅−ℎ¹(𝑥)∨0^2𝛼/2−arctanℎ²(𝑥)−⋅∧0²𝜀+𝑐₂𝜀+ℎ²(𝑥)−⋅∨0^2𝛼/2, (8) with 𝛿,𝜀>0. Notice that 𝑓_𝜀,𝛿 is differentiable and 𝑓^′_𝜀,𝛿≤0.

We need three lemmas before we present the regularity result.

Lemma 2. Suppose the hypotheses (H1), (H2), and (F1)–(F3) hold. For any 𝑝≥1 and 𝑇>0, there exists 𝜚>0 such that sup_{0<𝜀,𝛿<𝜚}𝔼(‖𝑢^𝜀,𝛿‖^𝑇_∞)^𝑝<∞ (‖𝜔‖^𝑇_∞∶=sup_𝑥∈𝑆¹_{,𝑡∈[0,𝑇]}|𝜔(𝑥,𝑡)|) and 𝑢^𝜀,𝛿 converges uniformly on 𝑆¹×[0,𝑇] to 𝑢 as 𝜀,𝛿→0 a.s., where 𝑢 and 𝑢^𝜀,𝛿 are the solutions of (1) and the penalized SPDEs (7), respectively.

Proof. Without loss of generality, assume 𝑐₁=𝑐₂=1. By (H1), there exists 𝛿₀∈(0,1) such that ℎ²(𝑥)−ℎ¹(𝑥)≥𝛿₀,∀𝑥∈𝑆¹. Set 0<𝜀,𝛿<𝛿²₀/2.
Let 𝑣^𝜀,𝛿 be the solution of equation 𝜕𝑣^𝜀,𝛿(𝑥,𝑡)=𝜕𝜕𝑡²𝑣^𝜀,𝛿(𝑥,𝑡)𝜕𝑥²+𝑓𝑢^𝜀,𝛿(𝑢𝑥,𝑡)+𝜎^𝜀,𝛿(𝑊̇();𝑣𝑥,𝑡)𝑥,𝑡_𝜀,𝛿(𝑥,0)=𝑢₀(𝑥). (9) Set Φ^𝜀,𝛿(𝑡)=sup_{𝑠≤𝑡,𝑦∈𝑆}¹(𝑣^𝜀,𝛿(𝑦,𝑠)−(ℎ¹(𝑦)+ℎ²(𝑦)−𝛿₀)/2). Note that Φ^𝜀,𝛿(𝑡) is increasing with respect to 𝑡 and 𝑣^𝜀,𝛿(𝑥,𝑡)−Φ^𝜀,𝛿(𝑡)≤(ℎ¹(𝑥)+ℎ²(𝑥)−𝛿₀)/2. From analogue method as the proof in Proposition 3.1 in [11], 𝑧^𝜀,𝛿(𝑥,𝑡)∶=𝑣^𝜀,𝛿(𝑥,𝑡)−Φ^𝜀,𝛿(𝑡)−𝑢^𝜀,𝛿(𝑥,𝑡) is a solution of equation 𝜕𝑧^𝜀,𝛿+𝜕𝜕𝑡Φ^𝜀,𝛿=𝜕𝜕𝑡²𝑧^𝜀,𝛿𝜕𝑥²−𝑓_𝜀,𝛿𝑢^𝜀,𝛿;𝑧^𝜀,𝛿(𝑥,0)=0. (10)
Assume (𝑧^𝜀,𝛿)⁺(𝑥,𝑡)>0; we have 0<𝑧^𝜀,𝛿+(𝑥,𝑡)⟺0<𝑣^𝜀,𝛿(𝑥,𝑡)−Φ^𝜀,𝛿(𝑡)−𝑢^𝜀,𝛿(𝑥,𝑡)≤𝑣^𝜀,𝛿(𝑥,𝑡)−𝑣^𝜀,𝛿()−ℎ𝑥,𝑡¹(𝑥)+ℎ²(𝑥)−𝛿₀2−𝑢^𝜀,𝛿()=ℎ𝑥,𝑡₁(𝑥)+ℎ²(𝑥)−𝛿₀2−𝑢^𝜀,𝛿(𝑥,𝑡); (11) hence 𝑢^𝜀,𝛿(𝑥,𝑡)≤ℎ²(𝑥). (12) If (𝑧^𝜀,𝛿)⁺(𝑥,𝑡)>0 and 𝑢^𝜀,𝛿(𝑥,𝑡)≤ℎ¹(𝑥), then, by (12), 𝛿+𝑢^𝜀,𝛿(𝑥,𝑡)−ℎ¹(𝑥)∨0²=𝛿≤𝜀+𝛿²₀≤𝜀+ℎ²(𝑥)−𝑢^𝜀,𝛿(𝑥,𝑡)∨0². (13) On the other hand, if (𝑧^𝜀,𝛿)⁺(𝑥,𝑡)>0 and 𝑢^𝜀,𝛿(𝑥,𝑡)>ℎ¹(𝑥), then, by (11) and (12), 𝜀+ℎ²(𝑥)−𝑢^𝜀,𝛿(𝑥,𝑡)∨0²−𝛿+𝑢^𝜀,𝛿(𝑥,𝑡)−ℎ¹(𝑥)∨0²=𝜀−𝛿+ℎ²(𝑥)−𝑢^𝜀,𝛿(𝑥,𝑡)²−𝑢^𝜀,𝛿(𝑥,𝑡)−ℎ¹(𝑥)²=𝜀−𝛿+ℎ²(𝑥)−ℎ¹(𝑥×ℎ₂)(𝑥)+ℎ¹(𝑥)−2𝑢^𝜀,𝛿(𝑥,𝑡)≥𝜀−𝛿+𝛿²₀≥𝛿²₀2. (14) Hence combining (13) and (14), (𝑧^𝜀,𝛿)⁺(𝑥,𝑡)>0 implies 𝛿+𝑢^𝜀,𝛿(𝑥,𝑡)−ℎ¹(𝑥)∨0²≤𝜀+ℎ²(𝑥)−𝑢^𝜀,𝛿(𝑥,𝑡)∨0². (15)
By (12) and (15), multiplying (10) by (𝑧^𝜀,𝛿)⁺, we have arctanℎ²(𝑥)−𝑢^𝜀,𝛿(𝑥,𝑡)∧0²𝜀𝑧^𝜀,𝛿+()1𝑥,𝑡𝑑𝑡=0,𝛿+𝑢^𝜀,𝛿(𝑥,𝑡)−ℎ¹(𝑥)∨0^2𝛼/2−1𝜀+ℎ²(𝑥)−𝑢^𝜀,𝛿(𝑥,𝑡)∨0^2𝛼/2×𝑧^𝜀,𝛿+(𝑥,𝑡)𝑑𝑡≥0. (16) Combining the fact that Φ^𝜀,𝛿(𝑡) is increasing with respect to 𝑡, we have (𝑧^𝜀,𝛿)⁺=0. Hence 𝑢^𝜀,𝛿(𝑥,𝑡)≥𝑣^𝜀,𝛿(𝑥,𝑡)−Φ^𝜀,𝛿(𝑥,𝑡). (17)
Similarly, setting 𝑧^𝜀,𝛿(𝑥,𝑡)=𝑢^𝜀,𝛿(𝑥,𝑡)−𝑣^𝜀,𝛿(𝑥,𝑡)−sup_{𝑠≤𝑡,𝑦∈𝑆}¹((ℎ¹(𝑦)+ℎ²(𝑦)−𝛿₀)/2−𝑣^𝜀,𝛿(𝑦,𝑠)), we can show that 𝑢^𝜀,𝛿(𝑥,𝑡)≤𝑣^𝜀,𝛿(𝑥,𝑡)+sup_{𝑠≤𝑡,𝑦∈𝑆1}ℎ¹𝑦+ℎ²𝑦−𝛿₀2−𝑣^𝜀,𝛿𝑦,𝑠. (18)
Using a similar proof in Donati-Martin and Pardoux [2], it can be shown that sup_𝜀,𝛿𝔼(‖𝑣^𝜀,𝛿‖^𝑇_∞)^𝑝<∞; hence the inequalities (17) and (18) imply sup_𝜀,𝛿𝔼𝑢^{𝜀,𝛿𝑇}_∞^𝑝<∞. (19) Since 𝑢^𝜀,𝛿 is increasing as 𝛿↓0 by the comparison theorem of SPDEs (see Theorem 2.1 in [2]), we can show 𝑢^𝜀∶=lim_𝛿↓0𝑢^𝜀,𝛿 exists a.s. and 𝑢^𝜀 solves 𝜕𝑢^𝜀(𝑥,𝑡)=𝜕𝜕𝑡²𝑢^𝜀(𝑥,𝑡)𝜕𝑥²+𝑓𝑢^𝜀()𝑢𝑥,𝑡+𝜎^𝜀(𝑊̇()+𝜂𝑥,𝑡)𝑥,𝑡_𝜀()+1𝑥,𝑡𝑢^𝜀(𝑥,𝑡)−ℎ¹(𝑥)^𝛼−arctanℎ²(𝑥)−𝑢^𝜀(𝑥,𝑡)∧0²𝜀+1𝜀+ℎ²(𝑥)−𝑢^𝜀(𝑥,𝑡)∨0^2𝛼/2;𝑢^𝜀(𝑥,0)=𝑢₀(𝑥), (20) where 𝜂^𝜀(𝑑𝑥,𝑑𝑡)∶=lim_𝛿↓0((𝑢^𝜀,𝛿(𝑥,𝑡)−ℎ¹(𝑥))⁺/𝛿)𝑑𝑥𝑑𝑡. By Lemma 3.1 and Remark 3.1 of [11], we know that 𝑢^𝜀 is decreasing as 𝜀↓0. Let 𝑣^𝜀 be the solution of (9) replacing 𝑢^𝜀,𝛿 by 𝑢^𝜀. Setting 𝑧^𝜀(𝑥,𝑡)=𝑢^𝜀(𝑥,𝑡)−𝑣^𝜀(𝑥,𝑡)−sup_{𝑠≤𝑡,𝑦∈𝑆}¹((ℎ¹(𝑦)+ℎ²(𝑦)−𝛿₀)/2−𝑣^𝜀(𝑦,𝑠)), similar to (18), we can show 𝑢^𝜀(𝑥,𝑡)≤𝑣^𝜀(𝑥,𝑡)+sup_{𝑠≤𝑡,𝑦∈𝑆1}ℎ¹𝑦+ℎ²𝑦−𝛿₀2−𝑣^𝜀𝑦,𝑠. (21) In addition, by the fact that 𝑢^𝜀(𝑥,𝑡)≥ℎ¹(𝑥), 𝑢∶=lim_𝜀↓0𝑢^𝜀=lim_𝜀↓0lim_𝛿↓0𝑢^𝜀,𝛿 exists a.s., and we can show 𝑢 is the solution of (1); see [11].
The continuity of 𝑢 can be proved similarly as in Theorem 4.1 of  [2]. The uniform convergence of 𝑢^𝜀,𝛿 with respect to (𝑥,𝑡) follows from Dini's theorem.

Recall the following lemma from [29].

Lemma 3. Let 𝑆∈𝐶^𝛼,𝛽(𝑆¹×[0,𝑇]) satisfying 𝑆(𝑥,𝑡)−𝑆𝑦,𝑠≤𝐶_𝑆𝑥−𝑦^𝛼+|𝑡−𝑠|^𝛽. (22) Then, for 𝜌₁,𝜌₂>0, there exists 𝑆^𝜌1,𝜌2∈𝐶^∞(𝑆¹×[0,𝑇]) such that 𝑆^𝜌1,𝜌2−𝑆^𝑇_∞≤𝐶_𝛼,𝛽𝐶_𝑆𝜌^𝛼₁+𝜌^𝛽₂,𝜕𝑆^𝜌1,𝜌2𝜕𝑥^𝑇_∞≤𝐶_𝛼,𝛽𝐶_𝑆𝜌₁^𝛼−1,𝜕𝑆^𝜌1,𝜌2𝜕𝑡^𝑇_∞≤𝐶_𝛼,𝛽𝐶_𝑆𝜌₂^𝛽−1, (23) 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. Let 𝑉∈𝐶^1,2(𝑆¹×[0,𝑇]) and Ψ,𝐹∈𝐶(𝑆¹×[0,𝑇]) with Ψ≤0; 𝑉 solves the equation with the homogeneous Dirichlet or Neumann boundary as follows: 𝜕𝑉=𝜕𝜕𝑡²𝑉𝜕𝑥²𝑉()+Ψ𝑉+Ψ𝐹;𝑥,0=0; (24) then ‖𝑉‖^𝑇_∞≤‖𝐹‖^𝑇_∞.

Theorem 5. Let 𝑢 be the solution of (1). Denote 𝑢(𝑥,𝑡)=𝑢(𝑥,𝑡)−_𝑆1𝐺_𝑡𝑥,𝑦𝑢₀𝑦𝑑𝑦. (25) Here 𝐺_𝑡(𝑥,𝑦) is the fundamental solution of the heat equation on 𝑆¹. Then, for any 𝜗∈(0,1), there exists a finite random variable 𝛾_𝜗, which is independent of 𝑢₀(𝑥), such that, for 𝑥,𝑦∈𝑆¹,𝑡,𝑠∈[0,𝑇], 𝑢(𝑥,𝑡)−𝑢𝑦,𝑠≤𝛾_𝜗𝑥−𝑦^𝜗/2+|𝑡−𝑠|^𝜗/4, (26) and, for any 𝑝≥1, 𝔼𝛾^𝑝_𝜗<∞. (27)
Moreover, if 𝑢₀∈𝐶^𝛽(𝑆¹),𝛽∈(0,1/2), then, for 0<𝜛<𝛽, there exists a finite random variable 𝛾^𝜛_𝑢, which is dependent on 𝑢₀(𝑥), such that 𝑢(𝑥,𝑡)−𝑢𝑦,𝑠≤𝛾^𝜛_𝑢𝑥−𝑦^𝜛+|𝑡−𝑠|^𝜛/2, (28) for 𝑥,𝑦∈𝑆¹,𝑡,𝑠∈[0,𝑇], and for any 𝑝≥1, 𝔼𝛾^𝜛_𝑢^𝑝<∞. (29)

Note. 𝐶^𝛽(𝑆¹) denote the space of Hölder continuous functions on 𝑆¹ with Hölder exponent 𝛽, equipped with the norm ‖𝑢‖_𝛽∶=sup_𝑥∈𝑆1|𝑢(𝑥)|+sup_{𝑥,𝑦∈𝑆1;𝑥≠𝑦}𝑢(𝑥)−𝑢𝑦𝑥−𝑦^𝛽. (30)

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: 𝑆_𝑡^𝜀,𝛿(𝑥)∶=^𝑡_0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝑓𝑢^𝜀,𝛿+𝑦,𝑠𝑑𝑦𝑑𝑠_𝑡0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝜎𝑢^𝜀,𝛿𝑦,𝑠𝑊𝑑𝑦,𝑑𝑠. (31) Here 𝑢^𝜀,𝛿 is the solution of (7).
By Lemma 2 and using similar arguments as Lemma 2.1 of [29], we have that, for any 𝜗∈(0,1), there exists a random variable 𝐶_𝜀,𝛿 such that, for any 𝑠,𝑡∈[0,𝑇],𝑥,𝑦∈𝑆¹, 𝑆_𝑡^𝜀,𝛿(𝑥)−𝑆_𝑠^𝜀,𝛿𝑦≤𝐶_𝜀,𝛿𝑥−𝑦^𝜗/2+|𝑡−𝑠|^𝜗/4, (32) and for any 𝑝≥1sup_𝜀,𝛿𝔼𝐶_𝜀,𝛿^𝑝<∞. (33)
Let 𝑢^𝜀,𝛿(𝑥,𝑡)=𝑢^𝜀,𝛿(𝑥,𝑡)−∫_𝑆1𝐺_𝑡(𝑥,𝑦)𝑢₀(𝑦)𝑑𝑦, and resolve 𝑢^𝜀,𝛿 into determine part and stochastic part; that is, 𝑢^𝜀,𝛿(𝑥,𝑡)=𝑧^{𝑆𝜀,𝛿}(𝑥,𝑡)+𝑆_𝑡^𝜀,𝛿(𝑥), (34) where 𝑧^{𝑆𝜀,𝛿} is the unique solution of the following PDE: 𝜕𝑧^{𝑆𝜀,𝛿}=𝜕𝜕𝑡²𝑧^{𝑆𝜀,𝛿}𝜕𝑥²+𝑓_𝜀,𝛿𝑧^{𝑆𝜀,𝛿}+𝑆^𝜀,𝛿;𝑧^{𝑆𝜀,𝛿}(𝑥,0)=0, (35) and 𝑆^𝜀,𝛿 is defined in (31), 𝑓_𝜀,𝛿(⋅)=𝑓_𝜀,𝛿(⋅+∫_𝑆1𝐺_𝑡(𝑥,𝑦)𝑢₀(𝑦)𝑑𝑦). Let 𝑆^𝜌1,𝜌2∈𝐶^∞(𝑆¹×[0,𝑇]), which depends on 𝜌₁,𝜌₂ and will be determined later. Denote 𝑧^{𝑆𝜌12,𝜌} by the unique solution of (35) replacing 𝑆^𝜀,𝛿 by 𝑆^𝜌1,𝜌2.
As similar proof in Proposition 3.1 in [11], we obtain 𝑧^{𝑆𝜀,𝛿}−𝑧^{𝑆𝜌12,𝜌𝑇}_∞≤𝑆^𝜀,𝛿−𝑆^{𝜌1,𝜌2𝑇}_∞. (36)
Also, Lemma 2 implies that 𝑆^𝜀,𝛿−𝑆^{𝜌1,𝜌2𝑇}_∞≤𝐶_𝜗/2𝐶_𝜀,𝛿𝜌₁^𝜗/2+𝜌₂^𝜗/4. (37) And so 𝑧^𝑆−𝑧^{𝑆𝜌12,𝜌𝑇}_∞≤𝐶_𝜗/2𝐶_𝜀,𝛿𝜌₁^𝜗/2+𝜌₂^𝜗/4. (38)
Differentiating 𝑧 in (35), 𝑧(∶=𝜕_𝑥𝑧^{𝑆𝜌12,𝜌}) and 𝑧(∶=𝜕_𝑡𝑧^{𝑆𝜌12,𝜌}) satisfy, respectively, 𝜕𝑧=𝜕𝜕𝑡²𝑧𝜕𝑥²+𝑓^′_𝜀,𝛿𝑧^{𝑆𝜌12,𝜌}+𝑆^𝜌1,𝜌2⋅𝑧+𝜕_𝑥𝑆^𝜌1,𝜌2;𝑧()𝜕𝑥,0=0,𝑧=𝜕𝜕𝑡²𝑧𝜕𝑥²+𝑓^′_𝜀,𝛿𝑧^{𝑆𝜌12,𝜌}+𝑆^𝜌1,𝜌2⋅𝑧+𝜕_𝑡𝑆^𝜌1,𝜌2;𝑧(𝑥,0)=0. (39)
Therefore, from Lemmas 3 and 4, we get ‖𝑧‖^𝑇_∞≤𝜕_𝑥𝑆^{𝜌1,𝜌2𝑇}_∞≤𝐶_𝜗/2𝐶_𝜀,𝛿⋅𝜌₁^𝜗/2−1;‖𝑧‖^𝑇_∞≤𝜕_𝑡𝑆^{𝜌1,𝜌2𝑇}_∞≤𝐶_𝜗/2𝐶_𝜀,𝛿⋅𝜌₂^𝜗/4−1. (40)
In view of (34), (36), (38), and (40), one has 𝑢^𝜀,𝛿(𝑥,𝑡)−𝑢^𝜀,𝛿≤𝑧𝑦,𝑠_{𝑆𝜀,𝛿}(𝑥,𝑡)−𝑧^{𝑆𝜀,𝛿}𝑦,𝑠+𝑆_𝑡^𝜀,𝛿(𝑥)−𝑆_𝑠^𝜀,𝛿𝑦≤𝑧^{𝑆𝜀,𝛿}(𝑥,𝑡)−𝑧^{𝑆𝜌12,𝜌}(+𝑧𝑥,𝑡)_{𝑆𝜌12,𝜌}(𝑥,𝑡)−𝑧^{𝑆𝜌12,𝜌}+𝑧𝑦,𝑠_{𝑆𝜌12,𝜌}𝑦,𝑠−𝑧^{𝑆𝜀,𝛿}𝑦,𝑠+𝐶_𝜀,𝛿𝑥−𝑦^𝜗/2+|𝑡−𝑠|^𝜗/4≤2𝑆^𝜀,𝛿−𝑆^{𝜌1,𝜌2𝑇}_∞+‖𝑧‖^𝑇_∞⋅+‖𝑥−𝑦𝑧‖^𝑇_∞⋅|𝑡−𝑠|+𝐶_𝜀,𝛿𝑥−𝑦^𝜗/2+|𝑡−𝑠|^𝜗/4≤2𝐶_𝜗/2𝐶_𝜀,𝛿𝜌₁^𝜗/2+𝜌₂^𝜗/4+𝐶_𝜗/2𝐶_𝜀,𝛿𝜌₁^𝜗/2−1⋅𝑥−𝑦+𝜌₂^𝜗/4−1⋅|𝑡−𝑠|+𝐶_𝜀,𝛿𝑥−𝑦^𝜗/2+|𝑡−𝑠|^𝛽/2. (41) Set 𝜌₁∶=|𝑥−𝑦| and 𝜌₂∶=|𝑡−𝑠|; then 𝑢^𝜀,𝛿(𝑥,𝑡)−𝑢^𝜀,𝛿𝑦,𝑠≤𝛾_𝜀,𝛿𝑥−𝑦^𝜗/2+|𝑡−𝑠|^𝜗/4, (42) where 𝛾_𝜀,𝛿∶=(3𝐶_𝜗/2+1)𝐶_𝜀,𝛿 and, for 𝑝≥1, sup_𝜀,𝛿𝔼𝛾^𝑝_𝜀,𝛿<∞. This yields, for 𝑝≥1, sup_𝜀,𝛿𝔼𝑢^𝜀,𝛿(𝑥,𝑡)−𝑢^𝜀,𝛿𝑦,𝑠^𝑝≤𝐶𝑀_𝑝𝑥−𝑦^(𝜗/2)𝑝+|𝑡−𝑠|^(𝜗/4)𝑝. (43) By Lemma 2, we obtain from (43) that 𝔼𝑢(𝑥,𝑡)−𝑢𝑦,𝑠^𝑝≤𝐶_𝑝𝑥−𝑦^(𝜗/2)𝑝+|𝑡−𝑠|^(𝜗/4)𝑝. (44)
Applying a variant of Garsia's lemma (see Proposition A.1 and Corollary A.3 in [30]), we conclude that 𝑢(𝑥,𝑡)−𝑢𝑦,𝑠≤𝐶_𝑝(𝜔)𝑥−𝑦^{𝜗/2−4/𝑝}+|𝑡−𝑠|^{𝜗/4−2/𝑝}. (45) 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, _𝑆1𝐺_𝑡𝑥,𝑦𝑢₀𝑦𝑑𝑦−_𝑆1𝐺_𝑡𝑧,𝑦𝑢₀𝑦𝑑𝑦≤𝐶_𝜛𝑢_0𝜛|𝑡−𝑠|^𝜛/2+|𝑥−𝑧|^𝜛, (46) and combining (26), we have (28).
The proof is complete.

Denote by ℬ(𝐶(𝑆¹)) the 𝜎-field of all Borel subsets of 𝐶(𝑆¹) and by ℳ(𝐶(𝑆¹)) the set of all probability measures defined on (𝐶(𝑆¹),ℬ(𝐶(𝑆¹))). We denote by 𝑢(𝑥,𝑡,𝑢₀) the solution of (1) and by 𝑃_𝑡(𝑢₀,⋅) the corresponding transition function 𝑃_𝑡𝑢₀,Γ=𝑃𝑢⋅,𝑡,𝑢₀∈Γ,Γ∈ℬ𝐶𝑆¹,𝑡>0, (47) where 𝑢₀ is the initial condition. For 𝜇∈ℳ(𝐶(𝑆¹)), we set 𝑃^∗_𝑡𝜇(Γ)=_𝐶(𝑆1)𝑃_𝑡(𝑥,Γ)𝜇(𝑑𝑥), (48) where 𝑡≥0,Γ∈ℬ(𝐶(𝑆¹)).

Theorem 6. Suppose the hypotheses (H1), (H2), and (F1)–(F3) hold. Then there exists an invariant measure to (1) on 𝐶(𝑆¹) for all 𝛼≥0, 𝑐₁>0, and 𝑐₂>0.

Proof. According to the Krylov-Bogolyubov theorem (see [6]), if there exists 𝑢₀∈𝐶(𝑆¹) such that the family 𝑃_𝑡(𝑢₀,⋅),𝑡≥1 is uniformly tight, then there exists an invariant measure for (1). We need to show that for any 𝜀>0 there is a compact set 𝐾⊂𝐶(𝑆¹) such that 𝑃(𝑢(𝑡)∈𝐾)≥1−𝜀,forany𝑡≥1, (49) where 𝑢(𝑡)=𝑢(𝑡,𝑢₀)=𝑢(⋅,𝑡,𝑢₀). On the other hand, for any 𝑡≥1, we have by the Markov property 𝑃(𝑢(𝑡)∈𝐾)=𝔼𝑃₁(𝑢(𝑡−1),𝐾). (50)
Thus it is enough to show 𝑃(𝑢(1,𝑢(𝑡−1))∈𝐾)≥1−𝜀, for any 𝑡≥1. As ℎ¹(⋅)≤𝑢(𝑡−1)(⋅)≤ℎ²(⋅), it suffices to find a compact subset 𝐾⊂𝐶(𝑆¹) such that 𝑃₁𝑔,𝐾≥1−𝜀,∀𝑔∈𝐶𝑆¹withℎ¹≤𝑔≤ℎ². (51)
Put 𝑢𝑥,𝑡,𝑔=𝑢𝑥,𝑡,𝑔−_𝑆1𝐺_𝑡𝑥,𝑦𝑔𝑦𝑑𝑦. (52) Then (𝑢,𝜂,𝜉) solves a random obstacle problem.
Define 𝑓_𝛽𝑓()=sup𝑥,𝑡−𝑓𝑦,𝑠𝑥−𝑦²+(𝑡−𝑠)^21/2∶(𝑥,𝑡),𝑦,𝑠∈𝑆¹×[0,1],(𝑥,𝑡)≠𝑦,𝑠. (53)
By the Arzela-Ascoli theorem, for all 𝑟>0, 𝐾_𝑟∶={𝑓;‖𝑓‖_𝛽≤𝑟} is a compact subset of 𝐶(𝑆¹×[0,1]). In view of (26), we see that, for given 𝜀>0, there exists 𝑟₀ such that 𝑃𝑢⋅,⋅,𝑔∈𝐾^𝑐_𝑟0≤𝜀,∀𝑔withℎ¹≤𝑔≤ℎ². (54) Let 𝐾^′={𝜑(⋅,1)∶𝜑(⋅,⋅)∈𝐾_𝑟0} which is a compact subset in 𝐶(𝑆¹); we obtain 𝑃(𝑢(⋅,1,𝑔)∈𝐾^′)≥1−𝜀, for all 𝑔 with ℎ¹≤𝑔≤ℎ².
Also, it is easy to see that there is a compact subset 𝐾₀⊂𝐶(𝑆¹) such that _𝑆1𝐺₁𝑥,𝑦𝑔𝑦𝑑𝑦;ℎ¹≤𝑔≤ℎ²⊂𝐾₀. (55) Define 𝐾=𝐾^′+𝐾₀. We have 𝑃₁𝑔,𝐾=𝑃𝑢⋅,1,𝑔∈𝐾≥𝑃𝑢⋅,1,𝑔∈𝐾^′≥1−𝜀, (56) for all 𝑔∈𝐶(𝑆¹) with ℎ¹≤𝑔≤ℎ². This finishes the proof.

The following result is the uniqueness of invariant measures.

Theorem 7. Under the assumptions in Theorem 6 and the fact that |𝜎(⋅)|≥𝐿₀ for some constant 𝐿₀>0, there is a unique invariant measure for (1) for all 𝛼≥0, 𝑐₁>0, and 𝑐₂>0.

Proof of Theorem 7. 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 𝜇₁ and 𝜇₂ which are distributions of initial values 𝑢¹(𝑥,0) and 𝑢²(𝑥,0), respectively. And so 𝑢¹(𝑥,𝑡) and 𝑢²(𝑥,𝑡) have these distributions for any 𝑡>0. Then the total variation distance between 𝜇₁ and 𝜇₂ satisfies Var𝜇₁−𝜇₂≤𝑃sup_𝑥∈𝑆1𝑢¹(𝑥,𝑡)−𝑢²(𝑥,𝑡)≠0. (57)
Consider the following SPDEs: 𝜕𝑢(𝑥,𝑡)=𝜕𝜕𝑡²𝑢(𝑥,𝑡)𝜕𝑥²(𝑢((𝑢(𝑊̇+𝑓𝑥,𝑡))+𝜎𝑥,𝑡))₁(𝑥,𝑡)+𝜂₁(𝑥,𝑡)−𝜉₁()+𝑐𝑥,𝑡¹𝑢(𝑥,𝑡)−ℎ¹(𝑥)^𝛼−𝑐₂ℎ²(𝑥)−𝑢(𝑥,𝑡)^𝛼,𝜕𝑣(𝑥,𝑡)=𝜕𝜕𝑡²𝑣(𝑥,𝑡)𝜕𝑥²(𝑣((𝑣(×|𝑢()()+𝑓𝑥,𝑡))+𝜎𝑥,𝑡))(1−𝑥,𝑡−𝑣𝑥,𝑡)|∧1^1/2𝑊̇₁()+𝑢()()𝑥,𝑡(|𝑥,𝑡−𝑣𝑥,𝑡)|∧1^1/2𝑊̇₂(𝑥,𝑡)+𝜂₂(𝑥,𝑡)−𝜉₂()+𝑐𝑥,𝑡¹𝑣(𝑥,𝑡)−ℎ¹(𝑥)^𝛼−𝑐₂ℎ²(𝑥)−𝑣(𝑥,𝑡)^𝛼,𝑢(𝑥,0)=𝑢¹(𝑥,0),𝑣(𝑥,0)=𝑢²(𝑥,0), (58) where 𝑊₁(𝑥,𝑡),𝑊₂(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: 𝜕𝑢(𝑥,𝑡)=𝜕𝜕𝑡²𝑢(𝑥,𝑡)𝜕𝑥²(𝑢((𝑢(𝑊̇+𝑓𝑥,𝑡))+𝜎𝑥,𝑡))₁(𝑥,𝑡)+𝜂₁(𝑥,𝑡)−𝜉₁()+𝑐𝑥,𝑡¹𝑢(𝑥,𝑡)−ℎ¹(𝑥)^𝛼−𝑐₂ℎ²(𝑥)−𝑢(𝑥,𝑡)^𝛼,𝜕𝑣^𝑛(𝑥,𝑡)=𝜕𝜕𝑡²𝑣^𝑛(𝑥,𝑡)𝜕𝑥²+𝑓𝑣^𝑛()𝑣𝑥,𝑡+𝜎^𝑛()×𝑔𝑥,𝑡_𝑛𝑢(𝑥,𝑡)−𝑣^𝑛(𝑥,𝑡)∧1𝑊̇₁(𝑥,𝑡)+𝑓_𝑛𝑢(𝑥,𝑡)−𝑣^𝑛(𝑥,𝑡)∧1𝑊̇₂(+𝑥,𝑡)𝜂^𝑛₂(𝑥,𝑡)−𝜉^𝑛₂()+𝑐𝑥,𝑡¹𝑣^𝑛(𝑥,𝑡)−ℎ¹(𝑥)^𝛼−c₂ℎ²(𝑥)−𝑣^𝑛(𝑥,𝑡)^𝛼,𝑢(𝑥,0)=𝑢¹(𝑥,0),𝑣(𝑥,0)=𝑢²(𝑥,0), (59) 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 lim_𝑡→∞𝑃sup_𝑥∈𝑆1𝑢¹(𝑥,𝑡)−𝑢²(𝑥,𝑡)≠0=0, (60) for given two initial functions 𝑢¹(𝑥,0) and 𝑢²(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 𝑢¹(𝑥,0)≥𝑢²(𝑥,0), 𝑥∈𝑆¹: _𝑆1(𝑢()(=𝑥,𝑡−𝑣𝑥,𝑡))𝑑𝑥_𝑆1𝑢₁(𝑥,0)−𝑢₂(𝑥,0)𝑑𝑥+𝐶(𝑡)+𝑀(𝑡), (61) where 𝑀(𝑡)=^𝑡_0𝑆1𝜎(𝑢(𝑥,𝑠))𝑊₁()−𝑑𝑥,𝑑𝑠_𝑡0𝑆1𝜎(𝑣(×(|𝑢()()𝑥,𝑠))1−𝑥,𝑠−𝑣𝑥,𝑠)|∧1^1/2×𝑊̇₁()−𝑑𝑥,𝑑𝑠_𝑡0𝑆1𝜎(𝑣(𝑥,𝑠))(|𝑢(𝑥,𝑠)−𝑣(𝑥,𝑠)|∧1)^1/2×𝑊̇₂(𝑑𝑥,𝑑𝑠), (62) 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: 𝜕𝑢^𝜀,𝛿=𝜕𝜕𝑡²𝑢^𝜀,𝛿𝜕𝑥²+𝑓𝑢^𝜀,𝛿+𝑓_𝜀,𝛿𝑢^𝜀,𝛿+𝜎𝑢^𝜀,𝛿𝑊̇₁;𝜕𝑣^{𝑛,𝜀,𝛿}=𝜕𝜕𝑡²𝑣^{𝑛,𝜀,𝛿}𝜕𝑥²+𝑓𝑣^{𝑛,𝜀,𝛿}+𝑓_𝜀,𝛿𝑣^{𝑛,𝜀,𝛿}+𝜎𝑣^{𝑛,𝜀,𝛿}×𝑔_𝑛𝑢^𝜀,𝛿−𝑣^{𝑛,𝜀,𝛿}∧1𝑊̇₁(𝑥,𝑡)+𝑓_𝑛𝑢^𝜀,𝛿−𝑣^{𝑛,𝜀,𝛿}∧1𝑊̇₂(;𝑢𝑥,𝑡)_𝜀,𝛿(𝑥,0)=𝑢¹(𝑥,0),𝑣^{𝑛,𝜀,𝛿}(𝑥,0)=𝑢²(𝑥,0), (63) where 𝑓_𝜀,𝛿 is the same as in (7).
When 𝑢¹(𝑥,0)≥𝑢²(𝑥,0), 𝑥∈𝑆¹, we have 𝑢^𝜀,𝛿≥𝑣^{𝑛,𝜀,𝛿} which can be shown as Lemma 3.1 in [20]. Also, we will assume that 𝑓(𝑢) is nonincreasing. Otherwise, consider 𝑢∶=𝑒^−𝐿𝑡𝑢 and 𝑣̃∶=𝑒^−𝐿𝑡𝑣, where 𝐿 is the Lipschitz constant in (F1). Then the new drift term 𝑒^−𝐿𝑡𝑓(𝑒^𝐿𝑡𝑥)−𝐿𝑥 is nonincreasing. Thus, 𝐶^{𝑛,𝜀,𝛿}(𝑡)=_𝑆1𝑓𝑢^𝜀,𝛿−𝑓𝑣^{𝑛,𝜀,𝛿}+𝑓_𝜀,𝛿𝑢^𝜀,𝛿−𝑓_𝜀,𝛿𝑣^{𝑛,𝜀,𝛿}𝑑𝑥≤0. (64)
Now we only need 𝑣^{𝑛,𝜀,𝛿}(𝑥,𝑡)→𝑣^𝑛(𝑥,𝑡) as 𝜀,𝛿→0 which will be shown in the lemma below. Letting 𝜀,𝛿→0 in the relation of ∫_𝑆1(𝑢^𝜀,𝛿(𝑥,𝑡)−𝑣^{𝑛,𝜀,𝛿}(𝑥,𝑡))𝑑𝑥 from (63) and then sending 𝑛 to ∞, we obtain (61). The proof is complete.

Lemma 8. 𝑣^{𝑛,𝜀,𝛿}(𝑥,𝑡)→𝑣^𝑛(𝑥,𝑡) in probability on 𝑆¹×[0,𝑇], for any 𝑇>0, as 𝜀,𝛿→0.

Proof. Consider the following SPDE: 𝜕𝑣̃^{𝑛,𝜀,𝛿}=𝜕𝜕𝑡²𝑣̃^{𝑛,𝜀,𝛿}𝜕𝑥²+𝑓𝑣̃^{𝑛,𝜀,𝛿}+𝑓_𝜀,𝛿𝑣̃^{𝑛,𝜀,𝛿}+𝜎𝑣̃^{𝑛,𝜀,𝛿}×𝑔_𝑛𝑢−𝑣̃^{𝑛,𝜀,𝛿}∧1𝑊̇₁(𝑥,𝑡)+𝑓_𝑛𝑢−𝑣̃^{𝑛,𝜀,𝛿}∧1𝑊̇₂(;𝑣̃𝑥,𝑡)_{𝑛,𝜀,𝛿}(𝑥,0)=𝑢²(𝑥,0). (65) Denote the solution by 𝑣̃^{𝑛,𝜀,𝛿}(𝑥,𝑡). Let 𝑣̃^{𝑛,𝜀,𝛿}=𝑧^{𝑛,𝜀,𝛿}+𝑤^{𝑛,𝜀,𝛿}; here 𝑤^{𝑛,𝜀,𝛿}(𝑥,𝑡) and 𝑧^{𝑛,𝜀,𝛿}(𝑥,𝑡) satisfy, respectively, 𝜕𝑤^{𝑛,𝜀,𝛿}=𝜕𝜕𝑡²𝑤^{𝑛,𝜀,𝛿}𝜕𝑥²+𝑓𝑣^{𝑛,𝜀,𝛿}+𝜎𝑣^{𝑛,𝜀,𝛿}×𝑔_𝑛𝑢−𝑣̃^{𝑛,𝜀,𝛿}∧1𝑊̇₁(𝑥,𝑡)+𝑓_𝑛𝑢−𝑣̃^{𝑛,𝜀,𝛿}∧1𝑊̇₂(;𝑤𝑥,𝑡)_{𝑛,𝜀,𝛿}(𝑥,0)=𝑢²(),𝜕𝑧𝑥,0_{𝑛,𝜀,𝛿}=𝜕𝜕𝑡²𝑧^{𝑛,𝜀,𝛿}𝜕𝑥²+𝑓_𝜀,𝛿𝑧^{𝑛,𝜀,𝛿}+𝑤^{𝑛,𝜀,𝛿};𝑧^{𝑛,𝜀,𝛿}(𝑥,0)=0. (66) Similarly, define 𝑤^{𝑛,𝜀,𝛿}(𝑥,𝑡) and 𝑧^{𝑛,𝜀,𝛿}(𝑥,𝑡) satisfy, respectively, 𝜕𝑤^{𝑛,𝜀,𝛿}=𝜕𝜕𝑡²𝑤^{𝑛,𝜀,𝛿}𝜕𝑥²+𝑓𝑣^{𝑛,𝜀,𝛿}+𝜎𝑣^{𝑛,𝜀,𝛿}×𝑔_𝑛𝑢^𝜀,𝛿−𝑣^{𝑛,𝜀,𝛿}∧1𝑊̇₁(𝑥,𝑡)+𝑓_𝑛𝑢^𝜀,𝛿−𝑣^{𝑛,𝜀,𝛿}∧1𝑊̇₂(;𝑤𝑥,𝑡)_{𝑛,𝜀,𝛿}(𝑥,0)=𝑢²(𝑥,0),𝜕𝑧^{𝑛,𝜀,𝛿}=𝜕𝜕𝑡²𝑧^{𝑛,𝜀,𝛿}𝜕𝑥²+𝑓_𝜀,𝛿𝑧^{𝑛,𝜀,𝛿}+𝑤^{𝑛,𝜀,𝛿};𝑧^{𝑛,𝜀,𝛿}(𝑥,0)=0. (67) Then 𝑣^{𝑛,𝜀,𝛿}=𝑧^{𝑛,𝜀,𝛿}+𝑤^{𝑛,𝜀,𝛿}.
From similar proof in Proposition 3.1 in [11], one has 𝑧^{𝑛,𝜀,𝛿}−𝑧^{𝑛,𝜀,𝛿𝑇}_∞≤𝑤^{𝑛,𝜀,𝛿}−𝑤^{𝑛,𝜀,𝛿𝑇}_∞. (68) Thus, 𝑣^{𝑛,𝜀,𝛿}−𝑣̃^{𝑛,𝜀,𝛿𝑇}_∞≤2𝑤^{𝑛,𝜀,𝛿}−𝑤^{𝑛,𝜀,𝛿𝑇}_∞. (69) Now we will show that ‖𝑣^{𝑛,𝜀,𝛿}−𝑣̃^{𝑛,𝜀,𝛿}‖^𝑇_∞→0 in probability as 𝜀,𝛿→0.
As the similar proof in Donati-Martin and Pardoux [2], it can be shown that sup_𝜀,𝛿𝔼(‖𝑣^{𝑛,𝜀,𝛿}‖^𝑇_∞)^𝑝<∞, for arbitrarily large 𝑝 and any 𝑇>0. Set the stopping time 𝜏_𝑀∶=inf{𝑡;sup_𝑥∈𝑆¹|𝑣^{𝑛,𝜀,𝛿}(𝑥,𝑡)|>𝑀}. For fixed 𝑀>0, we have 𝔼𝑤^{𝑛,𝜀,𝛿}𝑡∧𝜏_𝑀−𝑤^{𝑛,𝜀,𝛿}𝑡∧𝜏_𝑀^𝑇_∞²≤𝐶₁^𝑇₀𝔼𝑓𝑣^{𝑛,𝜀,𝛿}−𝑓𝑣̃^{𝑛,𝜀,𝛿𝑡∧𝜏𝑀}_∞²𝑑𝑠+𝐶₂^𝑇₀𝔼𝜎𝑣^{𝑛,𝜀,𝛿}𝑔_𝑛𝑢^𝜀,𝛿−𝑣^{𝑛,𝜀,𝛿}𝑣̃∧1−𝜎^{𝑛,𝜀,𝛿}×𝑔_𝑛𝑢−𝑣̃^{𝑛,𝜀,𝛿}∧1^{𝑡∧𝜏𝑀}_∞²𝑑𝑠+𝐶₂^𝑇₀𝔼𝜎𝑣^{𝑛,𝜀,𝛿}𝑓_𝑛𝑢^𝜀,𝛿−𝑣^{𝑛,𝜀,𝛿}𝑣̃∧1−𝜎^{𝑛,𝜀,𝛿}×𝑓_𝑛𝑢−𝑣̃^{𝑛,𝜀,𝛿}∧1^{𝑡∧𝜏𝑀}_∞²𝑑𝑠≤𝐶₁^𝑇₀𝔼𝑤^{𝑛,𝜀,𝛿}−𝑤^{𝑛,𝜀,𝛿𝑡∧𝜏𝑀}_∞²𝑑𝑠+𝐶₂^𝑇₀𝔼𝜎𝑣^{𝑛,𝜀,𝛿𝑡∧𝜏𝑀}_∞⋅𝑢^𝜀,𝛿−𝑢^{𝑡∧𝜏𝑀}_∞+𝑤^{𝑛,𝜀,𝛿}−𝑤^{𝑛,𝜀,𝛿𝑡∧𝜏𝑀}_∞²𝑑𝑠≤𝐶₁𝜖+𝐶₂^𝑇₀𝔼𝑤^{𝑛,𝜀,𝛿}−𝑤^{𝑛,𝜀,𝛿𝑡∧𝜏𝑀}_∞²𝑑𝑠, (70) where the constants change in each line and 𝜖→0 as 𝜀,𝛿→0 because of Lemma 2. By the Gronwall inequality, we get 𝔼(‖𝑤^{𝑛,𝜀,𝛿}(𝑡∧𝜏_𝑀)−𝑤^{𝑛,𝜀,𝛿}(𝑡∧𝜏_𝑀)‖^𝑇_∞)²→0 as 𝜀,𝛿→0. Hence, 𝔼(‖𝑣^{𝑛,𝜀,𝛿}(𝑡∧𝜏_𝑀)−𝑣̃^{𝑛,𝜀,𝛿}(𝑡∧𝜏_𝑀)‖^𝑇_∞)²→0. So ‖𝑣^{𝑛,𝜀,𝛿}−𝑣̃^{𝑛,𝜀,𝛿}‖^𝑇_∞→0 in probability. In fact, for every 𝜖₀>0, set the event 𝐴^{𝑛,𝜀,𝛿}∶={‖𝑣^{𝑛,𝜀,𝛿}−𝑣̃^{𝑛,𝜀,𝛿}‖^𝑇_∞>𝜖₀}, 𝑃𝐴^{𝑛,𝜀,𝛿}=𝑃𝐴^{𝑛,𝜀,𝛿},𝜏_𝑀≤𝑇+𝑃𝐴^{𝑛,𝜀,𝛿},𝜏_𝑀𝜏>𝑇≤𝑃_𝑀+𝔼𝑤≤𝑇_{𝑛,𝜀,𝛿}𝑡∧𝜏_𝑀−𝑤^{𝑛,𝜀,𝛿}𝑡∧𝜏_𝑀^𝑇_∞²𝜖₀²≤𝔼𝑣^{𝑛,𝜀,𝛿𝑇}_∞²𝑀²+𝔼𝑤^{𝑛,𝜀,𝛿}𝑡∧𝜏_𝑀−𝑤^{𝑛,𝜀,𝛿}𝑡∧𝜏_𝑀^𝑇_∞²𝜖₀²⟶0,as𝜀,𝛿⟶0,𝑀⟶∞. (71)
Since we can apply Lemma 2 to conclude that 𝑢^𝜀,𝛿(𝑥,𝑡)→𝑢(𝑥,𝑡), 𝑣̃^{𝑛,𝜀,𝛿}(𝑥,𝑡)→𝑣^𝑛(𝑥,𝑡) uniformly on 𝑆¹×[0,𝑇] as 𝜀,𝛿→0. Therefore, 𝑣^{𝑛,𝜀,𝛿}(𝑥,𝑡)→𝑣^𝑛(𝑥,𝑡) in probability on 𝑆¹×[0,𝑇] as 𝜀,𝛿→0.

3. Strong Feller Property

In this section, we consider the strong Feller property of the solution of (1). Let 𝐻=𝐿²(𝑆¹). If 𝜑∈𝐵_𝑏(𝐻) (the Banach space of all real bounded Borel functions, endowed with the sup norm), we define, for 𝑥∈𝑆¹, 0≤𝑡≤𝑇, and 𝑔∈𝐻, 𝑃_𝑡𝜑(𝑔)=𝔼𝜑(𝑢(𝑥,𝑡,𝑔)).

Theorem 9. Under the hypotheses (H1), (H2), and (F1)–(F3) and the fact that 𝑝₁≤|𝜎(⋅)|≤𝑝₂, for some constants 𝑝₁,𝑝₂>0, for any 𝑇>0, there exists a constant 𝐶^′_𝑇 such that, for all 𝜑∈𝐵_𝑏(𝐻) and 𝑡∈(0,𝑇], 𝑃_𝑡𝜑𝑢¹₀−𝑃_𝑡𝜑𝑢²₀≤𝐶^′_𝑇√𝑡𝜑_∞𝑢¹₀−𝑢²_0𝐻, (72) for 𝑢¹₀,𝑢²₀∈𝐻 with ℎ¹(𝑥)≤u¹₀(𝑥),𝑢²₀(𝑥)≤ℎ²(𝑥), where ‖𝜑‖_∞=sup_𝑢0∈𝐻|𝜑(𝑢₀)|. In particular, 𝑃_𝑡,𝑡>0 is strong Feller.

Proof. Choose a nonnegative function 𝜙∈𝐶^∞₀(𝑅) with ∫_𝑅𝜙(𝑥)=1 and denote 𝑓_𝑛(𝜁)∶=𝑛_𝑅𝜙𝑛𝑓𝑦𝜎𝜁−𝑦𝑑𝑦,_𝑛(𝜁)∶=𝑛_𝑅𝜙𝑛𝜁−𝑦𝜎𝑦𝑑𝑦,𝑘_{𝑛,𝜀,𝛿}(𝜁,𝑥)∶=𝑛_𝑅𝜙𝑛×1𝜁−𝑦𝛿+𝑦−ℎ¹(𝑥)∨0^2𝛼/2−1𝜀+ℎ²(𝑥)−𝑦∨0^2𝛼/2𝑙𝑑𝑦,_{𝑛,𝜀,𝛿}(𝜁,𝑥)∶=𝑘_{𝑛,𝜀,𝛿}()+𝜁,𝑥arctan𝜁−ℎ¹(𝑥)∧0²𝛿−arctanℎ²(𝑥)−𝜁∧0²𝜀. (73)(74)
So 𝑓_𝑛,𝜎_𝑛,𝑙_{𝑛,𝜀,𝛿} are smooth with respect to 𝜁 and 𝑓_𝑛(𝜁)→𝑓(𝜁), 𝜎_𝑛(𝜁)→𝜎(𝜁), and 𝑙_{𝑛,𝜀,𝛿}(𝜁,𝑥)→𝑓_𝜀,𝛿(𝜁) as 𝑛→∞.
Let 𝑢_𝑛^𝜀,𝛿𝑥,𝑡,𝑢₀=_𝑆1𝐺_t𝑥,𝑦𝑢₀𝑦+𝑑𝑦_𝑡0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝑓_𝑛𝑢_𝑛^𝜀,𝛿𝑦,𝑠,𝑢₀+𝑑𝑦𝑑𝑠_𝑡0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝜎_𝑛𝑢_𝑛^𝜀,𝛿𝑦,𝑠,𝑢₀𝑊+𝑑𝑦,𝑑𝑠_𝑡0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝑙_{𝑛,𝜀,𝛿}𝑢_𝑛^𝜀,𝛿𝑦,𝑠,𝑢₀,𝑦𝑑𝑦𝑑𝑠. (75)
Using similar arguments as in [23], we can show that, for any fixed 𝜀,𝛿 and 𝑝≥1, lim_𝑛→∞sup_{[]𝑡∈0,𝑇}𝔼𝑢_𝑛^𝜀,𝛿𝑡,⋅,𝑢₀−𝑢^𝜀,𝛿𝑡,⋅,𝑢₀^𝑝_𝐻=0. (76)
Furthermore, in view of Lemma 7.1.5 in [6] and Lemma 2, it is enough to prove that there exists a constant 𝐶^′_𝑇, independent of 𝜀,𝛿 and 𝑛, such that 𝑃_𝑡^{𝑛,𝜀,𝛿}𝜑𝑢¹₀−𝑃_𝑡^{𝑛,𝜀,𝛿}𝜑𝑢²₀≤𝐶^′_𝑇√𝑡𝜑_∞𝑢¹₀−𝑢²_0𝐻, (77) where 𝑃_𝑡^{𝑛,𝜀,𝛿}𝜑(𝑢₀)∶=𝔼(𝜑(𝑢_𝑛^𝜀,𝛿(⋅,⋅,𝑢₀))) and 𝑢¹₀,𝑢²₀∈𝐻.
From Theorem 5.4.1 in [6], 𝑢_𝑛^𝜀,𝛿(⋅,⋅,𝑢₀) is continuously differentiable with respect to 𝑢₀. Denote by 𝑋_𝑛^𝜀,𝛿(𝑥,𝑡)∶=(𝐷𝑢_𝑛^𝜀,𝛿(⋅,⋅,𝑢₀)(𝑢₀))(𝑥,𝑡) the directional derivative of 𝑢_𝑛^𝜀,𝛿(⋅,⋅,𝑢₀) at 𝑢₀ in the direction of 𝑢₀ and it satisfies the mild form of a SPDE as follows: 𝑋_𝑛^𝜀,𝛿(𝑥,𝑡)=_𝑆1𝐺_𝑡𝑥,𝑦𝑢₀𝑦+𝑑𝑦_𝑡0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝑓^′_𝑛𝑢_𝑛^𝜀,𝛿𝑦,𝑠,𝑢₀×𝑋_𝑛^𝜀,𝛿+𝑦,𝑠𝑑𝑦𝑑𝑠_𝑡0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝜎^′_𝑛𝑢_𝑛^𝜀,𝛿𝑦,𝑠,𝑢₀×𝑋_𝑛^𝜀,𝛿𝑊+𝑦,𝑠𝑑𝑦,𝑑𝑠_𝑡0𝑆1𝐺_𝑡−𝑠𝜕𝑥,𝑦𝑙𝜕𝜁^{𝑛,𝜀,𝛿}𝑢_𝑛^𝜀,𝛿𝑦,𝑠,𝑢₀,𝑦×𝑋_𝑛^𝜀,𝛿𝑦,𝑠𝑑𝑦𝑑𝑠. (78) Since (𝜕/𝜕𝜁)𝑙_{𝑛,𝜀,𝛿}(𝑢_𝑛^𝜀,𝛿(𝑦,𝑠,𝑢₀),𝑦)≤0, we use similar arguments as that in [23] to get sup_{[]𝜀,𝛿≥0,𝑡∈0,𝑇}𝔼_𝑆1𝑋_𝑛^𝜀,𝛿𝑦,𝑡²𝑑𝑦≤𝐶𝑢₀²_𝐻, (79) where 𝐶 is a constant. By Elworthy-Li formula (Lemma 7.1.3 in [6]), we obtain 𝐷𝑃_𝑡𝜑𝑢₀,𝑢₀²≤𝐶𝑝²₁𝑡𝜑²_∞𝑢₀²_𝐻. (80) This implies inequality (77) which completes the proof.

4. Large Deviation Principle

We consider a small perturbation of the equation of (1); that is, 𝜕𝑢^𝜖=𝜕𝜕𝑡²𝑢^𝜖𝜕𝑥²+𝑓𝑢^𝜖+𝑐₁𝑢^𝜖−ℎ^1𝛼−𝑐₂ℎ²−𝑢^𝜖𝛼+𝜖𝜎(𝑢)𝑊̇+𝜂^𝜖−𝜉^𝜖;𝑢^𝜖(𝑥,0)=𝑢₀(𝑥). (81) As a similar discussion in [11], for any 𝜖>0, there exists a continuous solution 𝑢^𝜖 satisfying 𝑢^𝜖(𝑥,𝑡)=_𝑆1𝐺_𝑡𝑥,𝑦𝑢₀𝑦+𝑑𝑦_𝑡0𝑆1𝐺_𝑡−𝑠×𝑓𝑢𝑥,𝑦_𝜖+𝑐𝑦,𝑠¹𝑢^𝜖𝑦,𝑠−ℎ¹𝑦^𝛼−𝑐₂ℎ²𝑦−𝑢^𝜖𝑦,𝑠^𝛼𝑑𝑦𝑑𝑠+𝜖^𝑡_0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝜎𝑢^𝜖𝑦,𝑠𝑊𝑑𝑦,𝑑𝑠+^𝑡_0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝜂^𝜖𝑑𝑦𝑑𝑠−𝜉^𝜖𝑑𝑦𝑑𝑠. (82) The Cameron-Martin space associated with the white noise {𝑊(𝑥,𝑡),(𝑥,𝑡)∈𝑆¹×[0,𝑇]} is given by ℋ=ℎ(𝑥,𝑡)=^𝑥₀^𝑡₀ℎ̇𝑦,𝑠𝑑𝑦𝑑𝑠;_𝑆1𝑇0ℎ̇𝑦,𝑠²𝑑𝑦𝑑𝑠<∞, (83) endowed with norm ‖ℎ‖_ℋ=(∫_𝑆1∫^𝑇₀(ℎ̇(𝑦,𝑠))²𝑑𝑦𝑑𝑠)^1/2.

Consider the following PDE (the skeleton equation): 𝜕𝑢^ℎ=𝜕𝜕𝑡²𝑢^ℎ𝜕𝑥²+𝑓𝑢^ℎ+𝑐₁𝑢^ℎ−ℎ^1𝛼−𝑐₂ℎ²−𝑢^ℎ𝛼+𝜎𝑢^ℎℎ̇+𝜂^ℎ−𝜉^ℎ;𝑢^ℎ(𝑥,0)=𝑢₀(𝑥);ℎ¹(𝑥)≤𝑢^ℎ(𝑥,𝑡)≤ℎ²(𝑥). (84) As the analogous discussion in [11], for any ℎ∈ℋ, there exists a continuous solution 𝑢^ℎ satisfying 𝑢^ℎ(𝑥,𝑡)=_𝑆1𝐺_𝑡𝑥,𝑦𝑢₀𝑦+𝑑𝑦_𝑡0𝑆1𝐺_𝑡−𝑠×𝑓𝑢𝑥,𝑦_ℎ+𝑐𝑦,𝑠¹𝑢^ℎ𝑦,𝑠−ℎ¹𝑦^𝛼−𝑐₂ℎ²𝑦−𝑢^ℎ𝑦,𝑠^𝛼+𝑑𝑦𝑑𝑠_𝑡0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝜎𝑢^ℎ𝑦,𝑠ℎ̇𝑦,𝑠𝑑𝑦𝑑𝑠+^𝑡_0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝜂^ℎ𝑑𝑦𝑑𝑠−𝜉^ℎ𝑑𝑦𝑑𝑠. (85)

We state our main result in this section.

Theorem 10. The laws 𝜇^𝜖 of the {𝑢^𝜖}_𝜖>0 satisfy a large deviation principle on 𝐷∶={𝑢(𝑥,𝑡)∈𝐶(𝑆¹×[0,𝑇]);ℎ¹(𝑥)≤𝑢(𝑥,𝑡)≤ℎ²(𝑥)} with the rate function 𝐼𝑓1=inf2‖ℎ‖²_ℋ;𝑢^ℎ=𝑓, (86) with the convention inf{0/}=∞; that is,(i)for any closed subset 𝐴⊂𝐷, limsup_𝜖→0𝜖²log𝜇^𝜖(𝐴)≤−inf_𝑓∈𝐴𝐼𝑓; (87)(ii)for any open set 𝐵⊂𝐷, liminf_𝜖→0𝜖²log𝜇^𝜖(𝐵)≥−inf_𝑓∈𝐵𝐼𝑓. (88)

Proof. In order to prove Theorem 10, by Theorem 4.4 in [27], it suffices to check the following.(i)For any 𝑎>0,𝑢^ℎ is a continuous mapping from {ℎ∶‖ℎ‖_ℋ≤𝑎} into 𝐶(𝑆¹×[0,𝑇]).(ii)For a family {ℎ^𝜖}⊂𝐴_𝑁, where 𝐴_𝑁∶={ℎ∈ℋ,;ℎ(𝑥,𝑡)𝑖𝑠ℱ_𝑡−𝑚𝑒𝑎𝑠𝑢𝑟𝑎𝑏𝑙𝑒,‖ℎ‖_ℋ≤𝑁𝑃−a.s.}, 0<𝑁<∞, that converges in distribution to ℎ, 𝑢^𝜖,ℎ𝜖 converges in distribution to 𝑢^ℎ, where 𝑢^𝜖,ℎ𝜖 solves the following SPDE: 𝜕𝑢^𝜖,ℎ𝜖=𝜕𝜕𝑡²𝑢^𝜖,ℎ𝜖𝜕𝑥²+𝑓𝑢^𝜖,ℎ𝜖+𝑐₁𝑢^𝜖,ℎ𝜖−ℎ^1𝛼−𝑐₂ℎ²−𝑢^{𝜖,ℎ𝜖𝛼}+𝜖𝜎𝑢^𝜖,ℎ𝜖𝑊̇+𝜎𝑢^𝜖,ℎ𝜖ℎ̇^𝜖+𝜂^𝜖,ℎ𝜖−𝜉^𝜖,ℎ𝜖;𝑢^𝜖,ℎ𝜖(𝑥,0)=𝑢₀(𝑥). (89)
Let 𝑣^ℎ𝑖(𝑥,𝑡)=_𝑆1𝐺_𝑡𝑥,𝑦𝑢₀𝑦+𝑑𝑦_𝑡0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝑓𝑢^ℎ𝑖𝑦,𝑠𝑑𝑦𝑑𝑠+^𝑡_0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝜎𝑢^ℎ𝑖𝑦,𝑠ℎ̇_i𝑦,𝑠𝑑𝑦𝑑𝑠, (90) with 𝑖=1,2. In terms of (36), we have 𝑢^ℎ1−𝑢^ℎ2𝑇_∞≤2𝑣^ℎ1−𝑣^ℎ2𝑇_∞. (91)
Proof of (i). We need to prove that, for fixed 𝑎>0,ℎ₁,ℎ₂∈{ℎ∈ℋ∶‖ℎ‖_ℋ≤𝑎}, 𝑢^ℎ1−𝑢^ℎ2𝑇_∞≤𝐶ℎ₁−ℎ_2ℋ, (92) where 𝑢^ℎ1,𝑢^ℎ2 are the solution of (84) associated with ℎ₁,ℎ₂, respectively. Let 𝑣^ℎ𝑖(𝑥,𝑡)=_𝑆1𝐺_𝑡𝑥,𝑦𝑢₀𝑦+𝑑𝑦_𝑡0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝑓𝑢^ℎ𝑖𝑦,𝑠𝑑𝑦𝑑𝑠+^𝑡_0𝑆1𝐺_𝑡−𝑠𝑥,𝑦𝜎𝑢^ℎ𝑖𝑦,𝑠ℎ̇_𝑖𝑦,𝑠𝑑𝑦𝑑𝑠 (93) with 𝑖=1,2. In terms of (36), we have 𝑢^ℎ1−𝑢^ℎ2𝑇_∞≤2𝑣^ℎ1−𝑣^ℎ2𝑇_∞≤2^𝑡_0𝑆1𝐺_𝑡−𝑠×𝑓𝑢𝑥,𝑦_ℎ1𝑢𝑦,𝑠−𝑓^ℎ2𝑦,𝑠𝑑𝑦𝑑𝑠^𝑇_∞+2^𝑡_0𝑆1𝐺_𝑡−𝑠×𝜎𝑢𝑥,𝑦_ℎ1𝑦,𝑠ℎ̇₁𝑢𝑦,𝑠−𝜎^ℎ2𝑦,𝑠ℎ̇₂𝑦,𝑠𝑑𝑦𝑑𝑠^𝑇_∞∶=𝐼₁+𝐼₂. (94) We know 𝐼₁≤𝐶^𝑇₀𝑢^ℎ1−𝑢^ℎ2𝑡_∞²𝑑𝑡^1/2,𝐼₂≤𝐶^𝑡_0𝑆1𝐺_𝑡−𝑠𝑥,𝑦²𝑑𝑦𝑑𝑠^1/2×^𝑡_0𝑆1𝜎𝑢^ℎ1𝑦,𝑠ℎ̇₁𝑢𝑦,𝑠−𝜎^ℎ2𝑦,𝑠ℎ̇₂𝑦,𝑠²𝑑𝑦𝑑𝑠^1/2𝑇_∞.≤𝐶ℎ₁−ℎ_2ℋ. (95)
Then using Gronwall's inequality, we obtain 𝑢^ℎ1−𝑢^ℎ2𝑇_∞≤𝐶ℎ₁−ℎ_2ℋ. (96)Proof of (ii). It is similar to that in [26] with the help of inequality (91).
Thus the proof of Theorem 10 is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments and suggestions. This work was supported by the Fundamental Research Funds for the Central Universities, no. 2013RC0906, and by NSFC, nos. 11101419 and 11371362.