We obtain a large deviation principle for the stochastic differential equations on the sphere Sd associated with the critical Sobolev Brownian vector fields.

1. Introduction

The purpose of our paper is to prove a large deviation principle on the asymptotic behavior of the stochastic differential equations on the sphere Sd associated with a critical Sobolev Brownian vector field which was constructed by Fang and Zhang [1].

Recall that Schilder theorem states that if B is the real Brownian motion and C0[0,1] is the space of real continuous functions defined on [0,1], null at 0, which endowed with the uniform norm, then for any open set G⊂C0[0,1] and closed set F⊂C0[0,1],liminfɛ→0ɛ2logP(ɛB∈G)≥-inff∈GI0(f),limsupɛ→0ɛ2logP(ɛB∈F)≤-inff∈FI0(f),
withI0(f)={12∫01|ḟ|2ds,fabsolutelycontinuous,∞,otherwise.

This result was then generalized by Freidlin and Wentzell in their famous paper [2] by considering the Itö equationdxtɛ=ɛσ(xtɛ)dW(t)+b(xtɛ)dt,x0ɛ=x.
They proved a large deviation principle for the above equation under usual Lipschitz conditions.

Recently, Ren and Zhang in [3] proved a large deviation principle for flows associated with differential equations with non-Lipschitz coefficients by using the weak convergence approach which is systematically developed in [4], and as an application, they established a Schilder Theorem for Brownian motion on the group of diffeomorphisms of the circle.

In this paper, we consider the large deviation principle of the critical Sobolev isotropic Brownian flows on the sphere Sd which is defined by the following SDE:
dxt=∑l=1∞{dalDl,1∑k=1Dl,1Al,k1(xt)∘dBl,k1(t)+dblDl,2∑k=1Dl,2Al,k2(xt)∘dBl,k2(t)},x0=x,
where Aℓ,ki are eigenvector fields of Laplace operator Δ on the sphere Sd with respect to the metric H(d+2)/2. Dℓ,1=dim𝒢ℓ, Dℓ,2=dim𝒟ℓ, 𝒢ℓ, and 𝒟ℓ are the eigenspaces of eigenvalues -cℓ,d and -cℓ,δ, respectively.

The authors in [1] consider the stochastic differential equations on Sddxtn=∑l=12n{dalDl,1∑k=1Dl,1Al,k1(xtn)∘dBl,k1(t)+dblDl,2∑k=1Dl,2Al,k2(xtn)∘dBl,k2(t)},x0n=x.
Let θn(t)=d(xtn,xtn+1), and there exists a real-valued Brownian motion Wn(t) such that dθn(t)=-σn(t)dWn(t)-Bn(t)dt,
therefore, the coefficients of SDEs which defined the Brownian motion on Sd with respect to the metric Hd+2/2 are non-Lipschitz (see Lemma 4.2 or page 582–585 [1] and Theorem 2.3 in [1]).

Because of the complex structure of this equation, it seems hard to prove the large deviation principle for the small perturbation of the equation by using its recursive approximating system as Ren and Zhang did in [3]. We will adopt a different approach which is similar to those of Fang and Zhang [1] and Liang [5]. We first work with the solution xn,ɛ of (5.1) (below) driven by finitely many Brownian motions, and this equation has smooth coefficients, so the large deviation principle for this equation is well known. Next, we show that xn,ɛ→xɛ is exponentially fast, which together with the special relation of rate functions guaranties that the large deviation estimate of xn,ɛ can be transferred to xɛ, where xɛ is the solution of the small perturbed system (3.1).

The rest of the paper is organized as follows. In Section 2, we recall the critical Sobolev isotropic Brownian flows on the sphere Sd. In Section 3, we introduce the main result. Section 4 is devoted to the study of the rate function. The large deviation principle is proved in Section 5.

2. Framework

Let Δ be the Laplace operator on Sd, acting on vector fields. The spectrum of Δ is given by spectrum(Δ)={-cℓ,d;ℓ≥1}∪{-cℓ,δ;ℓ≥1}, where cℓ,d=ℓ(ℓ+d-1),cℓ,δ=(ℓ+1)(ℓ+d-2). Let 𝒢ℓ be the eigenspace associated to cℓ,d and 𝒟ℓ the eigenspace associated to cℓ,δ. Their dimensions will be denoted by Dℓ,1=dim𝒢ℓ, Dℓ,2=dim𝒟ℓ. It is known (see [6]) that Dl,1=O(ld-1),Dl,2=O(ld-1)asl⟶+∞.

Denote by {Aℓ,ki;k=1,…,Dℓ,i,ℓ≥1} for i=1,2 the orthonormal basis of 𝒢ℓ and 𝒟ℓ in L2; that is, ∫Sd〈Al,ki(x),Aα,βj(x)〉dx=δijδlαδkβ,
where δij is the Kronecker symbol and dx is the normalized Riemannian measure on Sd, which is the unique one invariant by actions of g∈SO(d+1). By Weyl theorem, the vector fields {Aℓ,ki} are smooth. For more detailed properties of the eigenvector fields, we refer the reader to Appendix A in [1].

Let s>0 and Hs(Sd) be the Sobolev space of vector fields on Sd, which is the completion of smooth vector fields with respect to the norm ‖V‖Hs2=∫Sd〈(-Δ+1)sV,V〉dx.

Then, {Aℓ,k1/(1+cℓ,d)s/2,Aℓ,β2/(1+cℓ,δ)s/2;ℓ≥1,1≤k≤Dℓ,1,1≤β≤Dℓ,2} is an orthonormal basis of Hs. If we consider al=a(l-1)1+α,bl=b(l-1)1+α,α>0,a,b>0,l≥2,
then alDl,1=O(1l(α+d)/2),blDl,2=O(1l(α+d)/2).

Let {Bℓ,ki(t);ℓ≥1,1≤k≤Dℓ,i} for i=1,2 be two family of independent standard Brownian motions defined on a probability space (Ω,ℱ,P). Consider the series Wt(ω)=∑l≥1{dalDl,1∑k=1Dl,1Bl,k1(t)Al,k1+dblDl,2∑k=1Dl,2Bl,k2(t)Al,k2},
which converges in L2, uniformly with respect to t in any compact subset of [0,+∞[. According to (2.5), (Wt)t≥0 is a cylindrical Brownian motion in the Sobolev space H(α+d)/2. Moreover, Wt takes values in the space Hs(Sd) for any 0<s<α/2. By Sobolev embedding theorem, in order to ensure that Wt takes values in the space of C2 vector fields, α must be larger than d+2. In this later case, the classical Kunita's framework [7] can be applied to integrate the vector field Wt so that we obtain a flow of diffeomorphisms. For the case of small α, the notion of statistical solutions was introduced in [6], and the phenomenon of phase transition appears. It was also shown in [6] that the statistical solutions give rise to a flow of maps if α>2 and the solution is not a flow of maps if 0<α<2 The critical case α=2 was studied in [1]. Instead of introducing (Wt)t≥0 as in (2.6), the authors in [1] consider first the stochastic differential equations on Sddxtn=∑l=12n{dalDl,1∑k=1Dl,1Al,k1(xtn)∘dBl,k1(t)+dblDl,2∑k=1Dl,2Al,k2(xtn)∘dBl,k2(t)},x0n=x.

Using the specific properties of eigenvector fields, it was proved that xtn(x) converges uniformly in (t,x)∈[0,T]×Sd to a solution of the sde (2.8) below. We quote the following result from [1].

Theorem A (see [<xref ref-type="bibr" rid="B5">1</xref>]).

Let α=2 in definition (2.4). Then, the stochastic differential equation on Sddxt=∑l=1∞{dalDl,1∑k=1Dl,1Al,k1(xt)∘dBl,k1(t)+dblDl,2∑k=1Dl,2Al,k2(xt)∘dBl,k2(t)},x0=x
admits a unique strong solution (xt(x))t≥0, which gives rise to a flow of homeomorphisms.

In the case of the circle S1, this property of flows of homeomorphisms was discovered in [8] then studied in [9, 10].

3. Statement of the Result

Consider the small perturbation of (2.8) dxtɛ=ɛ∑l=1∞{dalDl,1∑k=1Dl,1Al,k1(xtɛ)∘dBl,k1(t)+dblDl,2∑k=1Dl,2Al,k2(xtɛ)∘dBl,k2(t)},x0ɛ=x
Equation (3.1) has a unique strong solution (xtɛ(x))t≥0 according to Theorem A, denoted by xtɛ.

We consider the abstract Wiener space (Ω,ℋ,ℱ,P) associated with Wiener processes W(s)={Bℓ,ki(t);ℓ≥1,1≤k≤Dℓ,i,i=1,2}. P is the Wiener measure and H={h={hl,ki(t)},l≥1,1≤k≤Dl,i,i=1,2,‖h‖H2<∞}
is the Cameron-Martin space associated with W, where ‖h‖H2=∑l≥1{∑k=1Dl,1∫0T|ḣl,k1(t)|2dt+∑k=1Dl,2∫0T|ḣl,k2(t)|2dt}.
The purpose of this paper is to prove a large deviation principle for the family {xɛ,ɛ>0} in the space Cx([0,T],Sd) and the collection of continuous functions f from [0,T] into Sd with f(0)=x. To state the result, let us introduce the rate function. For any h∈ℋ, let {Sh(t),t∈[0,T]} be the solution of dSh(t)=∑l=1∞{dalDl,1∑k=1Dl,1Al,k1(Sh(t))ḣl,k1(t)dt+dblDl,2∑k=1Dl,2Al,k2(Sh(t))ḣl,k2(t)dt},Sh(0)=x.
And for any f∈Cx([0,T],Sd), let I(f)=inf{12‖h‖H2:f=Sh,h∈H}.

We recall the definition of the good rate function.

Definition 3.1.

A function I mapping a metric space E into [0,∞] is called a good rate function if for each a<∞, the level set {f∈E:I(f)≤a} is compact.

Our main result reads as follows.

Theorem 3.2.

Let xtɛ be the solution of (3.1) on Cx([0,T],Sd), then {xtɛ,ɛ>0} satisfies a large deviation principle with a good rate function I(f), f∈Cx([0,T],Sd); that is,

for any closed subset C⊂Cx([0,T],Sd),
limsupɛ→0ɛ2logP(xɛ∈C)≤-inff∈CI(f),

for any open set G⊂Cx([0,T],Sd),
liminfɛ→0ɛ2logP(xɛ∈G)≥-inff∈GI(f).

4. Skeleton Equation and the Rate FunctionTheorem 4.1.

For any h∈ℋ, (3.4) has a unique solution, denoted by Sh(t).

In order to prove Theorem 4.1, we now introduce the following estimates which is Theorem 2.3 in [1].

Lemma 4.2.

Let
σn(θ)=-Un(θ)sinθ,Bn(θ)=Vn(θ)sinθ+12cosθsin3θUn(θ).Un,Vn is defined respectively, by (2.14) and (2.13) in [1]. Then, there exist some constants N>0, c>0 such that for any n>N,
σn2(θ)≤Cθ2log2πθ+2-n,-Bn(θ)≤Cθlog2πθ+2-n.

Proof of Theorem <xref ref-type="statement" rid="thm4.1">4.1</xref>.

Let Sn,h be the solution of the following system:
dSn,h(t)=∑l=12n{dalDl,1∑k=1Dl,1Al,k1(Sn,h(t))ḣl,k1(t)dt+dblDl,2∑k=1Dl,2Al,k2(Sn,h(t))ḣl,k2(t)dt},Sh(0)=x.
Since Aℓ,ki are smooth, the solution of (4.3) exists.

For x,y∈Sd, consider the Riemannian distance d(x,y) defined by
cosd(x,y)=〈x,y〉,
where 〈·,·〉 denotes the inner product in Rd+1. Let |·| denote the Euclidean distance. We have the relation
|x-y|≤d(x,y)≤π2|x-y|.

Our aim is to show that Sn,h converges to a solution of (3.4). By the chain rule, d〈Sn,h(t),Sn+1,h(t)〉=〈dSn,h(t),Sn+1,h(t)〉+〈Sn,h(t),dSn+1,h(t)〉=∑l=12n[dalDl,1∑k=1Dl,1〈Sn+1,h(t),Al,k1(Sn,h(t))〉ḣl,k1(t)+dblDl,2∑k=1Dl,2〈Sn+1,h(t),Al,k2(Sn,h(t))〉ḣl,k2(t)]dt+∑l=12n+1[dalDl,1∑k=1Dl,1〈Sn,h(t),Al,k1(Sn+1,h(t))〉ḣl,k1(t)+dblDl,2∑k=1Dl,2〈Sn,h(t),Al,k2(Sn+1,h(t))〉ḣl,k2(t)]dt.Let θtn=d(Sn,h(t),Sn+1,h(t)), then
dθtn=-1sinθtnd〈Sn,h(t),Sn+1,h(t)〉.
Let
I1(t)=-∫0t1sinθsn×∑l=12n[dalDl,1∑k=1Dl,1(〈Sn+1,h(s),Al,k1(Sn,h(s))〉+〈Sn,h(s),Al,k1(Sn+1,h(s))〉)ḣl,k1(s)+dblDl,2∑k=1Dl,2(〈Sn+1,h(s),Al,k2(Sn,h(s))〉+〈Sn,h(s),Al,k2(Sn+1,h(s))〉)ḣl,k2(s)]ds,I2(t)=-∫0t1sinθsn∑l=2n+12n+1[dalDl,1∑k=1Dl,1〈Sn,h(s),Al,k1(Sn+1,h(s))〉ḣl,k1(s)+dblDl,2∑k=1Dl,2〈Sn,h(s),Al,k2(Sn+1,h(s))〉ḣl,k2(s)]ds.
We have
I12(t)≤(((∑l=12n[∑k=1Dl,1|ḣl,k1(s)|2+∑k=1Dl,2|ḣl,k2(s)|2])1)∫0t1sinθsn×(∑l=12n[dalDl,1∑k=1Dl,1(〈Sn+1,h(s),Al,k1(Sn,h(s))〉+〈Sn,h(s),Al,k1(Sn+1,h(s))〉)2+dblDl,2∑k=1Dl,2(〈Sn+1,h(s),Al,k2(Sn,h(s))〉+〈Sn,h(s),Al,k2(Sn+1,h(s))〉)2])1/2×(∑l=12n[∑k=1Dl,1|ḣl,k1(s)|2+∑k=1Dl,2|ḣl,k2(s)|2])1/2ds)2.
Using Proposition A.4 in [1] and Lemma 4.2, we see that
I12(t)≤(∫0t(2∑l=12n{al[1-cosθsnγl(cosθsn)+sin2θsnγl′(cosθsn)]+bl[1-γl(cosθsn)]})1/2×(∑l=12n[∑k=1Dl,1|ḣl,k1(s)|2+∑k=1Dl,2|ḣl,k2(s)|2])1/2ds)2≤2∫0t(∑l=12n{al[1-cosθsnγl(cosθsn)+γl′(cosθsn)]+bl[1-γl(cosθsn)]})ds×∫0t∑l=12n[∑k=1Dl,1|ḣl,k1(s)|2+∑k=1Dl,2|ḣl,k2(s)|2]ds≤2‖h‖H2∫0t(∑l=12n{al[1-cosθsnγl(cosθsn)+γl′(cosθsn)]+bl[1-γl(cosθsn)]})ds.
Similarly, we have
I22(t)≤(∫0t1sinθsn(∑l=2n+12n+1[dalDl,1∑k=1Dl,1〈Sn,h(s),Al,k1(Sn+1,h(s))〉2+dblDl,2∑k=1Dl,2〈Sn,h(s),Al,k2(Sn+1,h(s))〉2])1/2×(∑l=2n+12n+1[∑k=1Dl,1|ḣl,k1(s)|2+∑k=1Dl,2|ḣl,k2(s)|2])1/2ds)2≤2‖h‖H2∫0t∑l=2n2n+1(al+bl)ds.
Therefore,
|θtn|2≤2I12(t)+2I22(t)≤4‖h‖H2[∫0t∑l=12n{al[1-cosθsnγl(cosθsn)+γl′(cosθsn)]+bl[1-γl(cosθsn)]}ds+∫0t∑l=2n+12n+1(al+bl)ds]≤4∫0Tσn2(θsn)ds≤4∫0t((θsn)2log2πθsn+2-n)ds.
Using the similar arguments as that in [1], the above inequality implies that there exist constants C1,C2 such that
|θtn|2≤C12-ne-C2t,
and C1,C2 are independent of n,t. Hence,
|Sn,h(t)-Sn+1,h(t)|≤|θtn|≤C12-ne-C2t.
Thus, Sn,h(t) uniformly converges to some function Sh in Cx([0,T],Sd).

Next, we show that {Sh(t),t≥0} satisfies (3.4).

It suffices to show that for any u∈Sd,
d〈u,Sh(t)〉=∑l=1∞{dalDl,1∑k=1Dl,1〈u,Al,k1(Sh(t))〉ḣl,k1(t)dt+dblDl,2∑k=1Dl,2〈u,Al,k2(Sh(t))〉ḣl,k2(t)dt}.
Set ηt=〈u,Sh(t)〉, ηtn=〈u,Sn,h(t)〉, θtn=d(u,Sn,h(t)), and θt=d(u,Sh(t)).

Fix N0>0, and by Proposition A.4 in [1] and Lemma 4.2, we have
|∑l=N02n∫0t{dalDl,1∑k=1Dl,1〈u,Al,k1(Sn,h(s))〉ḣl,k1(s)ds+dblDl,2∑k=1Dl,2〈u,Al,k2(Sn,h(s))〉ḣl,k2(s)ds}|2≤∫0t∑l=N02n(dalDl,1∑k=1Dl,1〈u,Al,k1(Sn,h(s))〉2+dblDl,2∑k=1Dl,2〈u,Al,k2(Sn,h(s))〉2)ds×∫0t∑l=N02n(∑k=1Dl,1|ḣl,k1(s)|2+∑k=1Dl,2|ḣl,k2(s)|2ds)=∫0t∑l=N02n(al+bl)sin2θsnds×∫0t∑l=N02n(∑k=1Dl,1|ḣl,k1(s)|2+∑k=1Dl,2|ḣl,k2(s)|2)ds≤t‖h‖H2∑N02n(al+bl)≤t‖h‖H2∑N0∞(al+bl).
Thus, for any ε>0, there exists N0>0, for 2n>N0,
|∑l=N02n∫0t{dalDl,1∑k=1Dl,1〈u,Al,k1(Sn,h(s))〉ḣl,k1(s)ds+dblDl,2∑k=1Dl,2〈u,Al,k2(Sn,h(s))〉ḣl,k2(s)ds}|<ɛ2.
By similar reasons, we also have
|∑l=N0∞∫0t{dalDl,1∑k=1Dl,1〈u,Al,k1(Sh(s))〉ḣl,k1(s)ds+dblDl,2∑k=1Dl,2〈u,Al,k2(Sh(s))〉ḣl,k2(s)}ds|≤t‖h‖H2∑N0∞(al+bl)<ɛ2.
On the other hand, because Sn,h⇒Sh in Cx([0,T],Sd), for any ɛ>0, one can find N1>0 such that for n>N1,
|∑l=1N0-1∫0t{dalDl,1∑k=1Dl,1〈u,Al,k1(Sn,h(s))-Al,k1(Sh(s))〉ḣl,k1(s)ds+dblDl,2∑k=1Dl,2〈u,Al,k2(Sn,h(s))-Al,k2(Sh(s))〉ḣl,k2(s)ds}|<ɛ2.
Therefore, for any ɛ>0, one can find N2>0 such that for n>N2,
|∑l=1N0-1∫0t{dalDl,1∑k=1Dl,1〈u,Al,k1(Sn,h(s))-Al,k1(Sh(s))〉ḣl,k1(s)ds+dblDl,2∑k=1Dl,2〈u,Al,k2(Sn,h(s))-Al,k2(Sh(s))〉ḣl,k2(s)ds}|+|∑l=N02n∫0t{dalDl,1∑k=1Dl,1〈u,Al,k1(Sn,h(s))〉ḣl,k1(s)ds+dblDl,2∑k=1Dl,2〈u,Al,k2(Sn,h(s))〉ḣl,k2(s)ds}|+|∑l=N0∞∫0t{dalDl,1∑k=1Dl,1〈u,Al,k1(Sh(s))〉ḣl,k1(s)ds+dblDl,2∑k=1Dl,2〈u,Al,k2(Sh(s))〉ḣl,k2(s)ds}|<ɛ.
Since ɛ is arbitrary, we obtain that
d〈u,Sh(t)〉=∑l=1∞{dalDl,1∑k=1Dl,1〈u,Al,k1(Sh(t))〉ḣl,k1(t)dt+dblDl,2∑k=1Dl,2〈u,Al,k2(Sh(t))〉ḣl,k2(t)dt}.
The uniqueness is deduced from similar estimates.

Lemma 4.3.

For any N>0, the set {Sh:∥h∥ℋ≤N} is relatively compact in Cx([0,T],Sd).

Proof.

By the Ascoli-Arzela lemma, we need to show that {Sh:∥h∥ℋ≤N} is uniformly bounded and equicontinuous. The first fact is obvious, because ∥Sh∥=1 for any h∈ℋ. Next, we will show that {Sh:∥h∥ℋ≤N} is equicontinuous.

Let {ui,i=1,…,d+1} be an orthonormal basis of Rd+1, and by Proposition A.4 in [1] and Lemma 4.2, we have
|〈Sh(t)-Sh(s),ui〉|2=|∫st∑l=1∞[dalDl,1∑k=1Dl,1〈Al,k1(Sh(u)),ui〉ḣl,k1(u)du+dblDl,2∑k=1Dl,2〈Al,k2(Sh(u)),ui〉ḣl,k2(u)du]|2≤∫st∑l=1∞(dalDl,1∑k=1Dl,1〈Al,k1(Sh(u)),ui〉2+dblDl,2∑k=1Dl,2〈Al,k2(Sh(u)),ui〉2)du×∫st∑l=1∞(∑k=1Dl,1|ḣl,k1(u)|2+∑k=1Dl,2|ḣl,k2(u)|2)du=∫st∑l=1∞(al+bl)sin2θudu×∫st∑l=1∞(∑k=1Dl,1|ḣl,k1(u)|2+∑k=1Dl,2|ḣl,k2(u)|2)du≤∑l=1∞(al+bl)‖h‖H2|t-s|,
where θt=d(Sh(t),ui). Thus,
|Sh(t)-Sh(s)|2=∑i=1d+1|〈Sh(t)-Sh(s),ui〉|2≤(d+1)∑l=1∞(al+bl)‖h‖H2|t-s|,
which finishes the proof.

Lemma 4.4.

The mapping h→Sh is continuous from {h:∥h∥ℋ≤N} with respect to the topology on Ω into Cx([0,T],Sd).

Proof.

Let hn∈ℋ with ∥hn∥ℋ≤N and assume that hn converges to h in Ω, then hn→h weakly in ℋ. By Lemma 4.2, {Shn,n≥1} is relatively compact. Let g∈Cx([0,T],Sd) be a limit of any convergent subsequence of {Shn,n≥1}. We will finish the proof the lemma by showing that g=Sh. Now, for simplicity, we drop the subindex kShn(t)=x+∫0t∑l=1∞[dalDl,1∑k=1Dl,1Al,k1(Shn(u))ḣn,l,k1(u)+dblDl,2∑k=1Dl,2Al,k2(Shn(u))ḣn,l,k2(u)]du,Sh(t)=x+∫0t∑l=1∞[dalDl,1∑k=1Dl,1Al,k1(Sh(u))ḣl,k1(u)+dblDl,2∑k=1Dl,2Al,k2(Sh(u))ḣl,k2(u)]du.
It is sufficient to show that Shn⇒Sh in Cx([0,T],Sd).

Write Shn(t)-Sh(t)=I3-I4 with I3,I4 being given by
I3=∫0t∑l=1∞[dalDl,1∑k=1Dl,1(Al,k1(Shn(u))-Al,k1(Sh(u)))ḣn,l,k1(u)+dblDl,2∑k=1Dl,2(Al,k2(Shn(u))-Al,k2(Sh(u)))ḣn,l,k2(u)]du,I4=∫0t∑l=1∞[dalDl,1∑k=1Dl,1Al,k1(Sh(u))(ḣn,l,k1(u)-ḣl,k1(u))+dblDl,2∑k=1Dl,2Al,k2(Sh(u))(ḣn,l,k2(u)-ḣl,k2(u))]du.
Let θt=d(Sh(t),Shn(t)), and by Proposition A.4 in [1] and Lemma 4.2, we haveI3≤∫0t[∑l=1∞(dalDl,1∑k=1Dl,1|Al,k1(Shn(u))-Al,k1(Sh(u))|2+dblDl,2∑k=1Dl,2|Al,k2(Shn(u))-Al,k2(Sh(u))|2)]1/2×[∑l=1∞(∑k=1Dl,1|ḣn,l,k1(u)|2+∑k=1Dl,2|ḣn,l,k2(u)|2)]1/2du≤‖hn‖H(∫0t∑l=1∞[(dcosθuγl(cosθu)-sin2θuγ′l(cosθu))2dal-2al(d-1+cosθu)γl(cosθu)-cosθusin2θuγl′(cosθu)+2dbl-2bl(dcosθuγl(cosθu)-sin2θuγl′(cosθu))]du∑l=1∞)1/2≤‖hn‖H(∫2t2[daG(0)-a((d-1+cos2θu)G(θu)+cosθusinθuG′(θ))+dbG(0)-b(dcosθuG(θu)+sinθuG′(θu))]du∫2t2)1/2≤(∫0tCθu2log2πθudu)1/2.Let
fl,k1(v)=∫0vdalDl,1Al,k1(Sh(u))I[0,t](u)du,fl,k2(v)=∫0vdblDl,2Al,k2(Sh(u))I[0,t](u)du.
Then, f=((fℓ,k1(v))ℓ≥1,1≤k≤Dℓ,1,(fℓ,k2(v))ℓ≥1,1≤k≤Dℓ,2)∈ℋ, because of
‖f‖H=∑l=1∞[∫0tdalDl,1∑k=1Dl,1|Al,k1(Sh(u))|2du+∫0tdalDl,2∑k=1Dl,2|Al,k2(Sh(u))|2du]=∑l=1∞∫0t(al+bl)du<∞.
Therefore,
I4=〈f,hn-h〉H=∫0t∑l=1∞[dalDl,1∑k=1Dl,1Al,k1(Sh(u))(ḣnl,k1(u)-ḣl,k1(u))du+dblDl,2∑k=1Dl,2Al,k2(Sh(u))(ḣnl,k2(u)-ḣl,k2(u))dudblDl,2∑k=1Dl,2]⟶0asn⟶∞.
Combining above estimates,
θt=d(Sh(t),Shn(t))≤π2[(∫0tCθu2log2πθudu)1/2+|〈f,hn-h〉H|].
Hence,
θt2≤2C∫0tθu2log2πθudu+2|〈f,hn-h〉H|2.
This implies
θt≤C4|〈f,hn-h〉H|e-C5t,
which yields
Shn⟹Shasn⟹∞.

Lemma 4.5.

I(f) is a good rate function.

Proof.

For any a>0,
{I(f)≤a}={Sh,12‖h‖H2≤a}=Sh(‖h‖H≤2a).
The subset {∥h∥ℋ≤2a} is a compact set in Ω and h→Sh is a continuous map for any a. Therefore, {I(f)≤a} is a compact set for any a. So, I(f) is a good rate function.

5. The Proof of Theorem <xref ref-type="statement" rid="thm3.2">3.2</xref>

Let xtn,ɛ be the solution to dxtn,ɛ=ɛ∑l=12n{dalDl,1∑k=1Dl,1Al,k1(xtn,ɛ)∘dBl,k1(t)+dblDl,2∑k=1Dl,2Al,k2(xtn,ɛ)∘dBl,k2(t)},x0n,ɛ=x.

We first have the following proposition.

Proposition 5.1.

For any δ>0,
limn→∞limsupɛ→0ɛ2logP(sup0≤t≤T|xtɛ-xtn,ɛ|>δ)=-∞.

Proof.

Let θnɛ(t)=d(xtn,ɛ,xtɛ). Using the similar estimates as that in [1] (see pages 582–585), there exists a real-valued Brownian motion Wn(t) such that
dθnɛ(t)=-ɛσn(t)dWn(t)-ɛ2Bn(t)dt,
where σn(t)=σn(θnɛ(t)), Bn(t)=Bn(θnɛ(t)) are defined as in Lemma 4.2.

Let ξn(t)=(θnɛ)2(t), we have
dξn(t)=2θnɛ(t)dθnɛ(t)+dθnɛ(t)dθnɛ(t)=-2ɛθnɛ(t)σn(θnɛ(t))dWn(t)+ɛ2(σn2(θnɛ(t))-2θnɛ(t)Bn(θnɛ(t)))dt,dξn(t)dξn(t)=4ɛ2θnɛ(t)2σn2(θnɛ(t))dt.
Introduce the function ψρ:(0,e-1)→R by
ψρ(ξ)=∫0ξdsslog(2π/s)+ρ.
Then, for any 0<ξ<1,
ψρ(ξ)↑ψ0(ξ)=∫0ξdsslog(2π/s)=+∞,
as ρ→0.

Define for λ>0,
Φρ,λ(ξ)=eλψρ(ξ).
We have
Φρ,λ′(ξ)(ξlog2πξ+ρ)=λΦρ,λ(ξ),Φρ,λ′′(ξ)=λ2Φρ,λ(ξ)1ξlog(2π/ξ)+ρ+λΦρ,λ(ξ)1-log(2π/ξ)(ξlog(2π/ξ)+ρ)2≤λ2Φρ,λ(ξ)1(ξlog(2π/ξ)+ρ)2,ifξ≤e-1.
Without loss of generality, we may assume δ<e-1. Define τn=inf{t≥0,θnɛ(t)>δ}. By Itö formula, we have
Φρ,λ(ξn(t∧τn))=1+∫0t∧τnΦρ,λ′(ξn(s))dξn(s)+12∫0t∧τnΦρ,λ′′(ξn(s))dξn(s)dξn(s)=1+2ɛ∫0t∧τnΦρ,λ′(ξn(s))(-θnɛ(s)σn(θnɛ(s)))dWn(s)+ɛ2∫0t∧τnΦρ,λ′(ξn(s))(σn2(θnɛ(s)))ds-2ɛ2∫0t∧τnΦρ,λ′(ξn(s))(θnɛ(s)Bn(θnɛ(s)))ds+2ɛ2∫0t∧τnΦρ,λ′′(ξn(s))(θnɛ(s)2σn2(θnɛ(s)))ds.
Using Lemma 4.2, ∃N such that n≥N,
λσn2(θnɛ(s))ξn(s)log(2π/ξn(s))+ρ≤λC(θnɛ(s)2log(2π/θnɛ(s))+2-n)(θnɛ(s))2log(2π/θnɛ(s)2)+ρ≤λC1,λθnɛ(s)(-Bn(θnɛ(s)))ξn(s)log(2π/ξn(s))+ρ≤λC(θnɛ(s)2log(2π/θnɛ(s))+2-n)(θnɛ(s))2log(2π/θnɛ(s)2)+ρ≤λC2,2λ2θnɛ(s)2σn2(θnɛ(s))(ξnlog(2π/ξn)+ρ)2≤2λ2θnɛ(s)2(θnɛ(s)2log(2π/θnɛ(s))+2-n)(θnɛ(s)2log(2π/θnɛ(s)2)+ρ)2≤2λ2C3.
Therefore, it follows from (5.9) that
E[Φρ,λ(ξn(t∧τn))]≤1+ɛ2C4(λ2+λ)E∫0tΦρ,λ(ξn(s∧τn))ds,
which implies that
E[Φρ,λ(ξn(t∧τn))]≤EC4(λ2+λ)ɛ2t.
Since
E[Φρ,λ(ξn(1∧τn))]≥E[Φρ,λ(ξn(1∧τn)),τn≤1]=eλψρ(δ2)P(τn≤1),
we have
P(sup0≤t≤1θnɛ(t)>δ)=P(τn≤1)≤e-λψρ(δ2)eC(λ2+λ)ɛ2.
Taking λ=1/ɛ2, we obtain that
limn→∞limε→0ɛ2logP(sup0≤t≤1θnɛ(t)>δ)≤-ψρ(δ2)+C⟶-∞.
Let ρ→0 to get (5.2). The proof is complete.

Define In(f)=inf{12‖hn‖H2,Sn,h(t)=f},
where hn=((hl,k1)1≤l≤2n,1≤k≤Dl,1,(hl,k2)1≤l≤2n,1≤k≤Dl,2),‖hn‖H2=∑l=12n{∑k=1Dl,1∫0T|ḣl,k1(t)|2dt+∑k=1Dl,2∫0T|ḣl,k2(t)|2dt}.
It is obvious that In(f)≥I(f).

Proof of Theorem <xref ref-type="statement" rid="thm3.2">3.2</xref>.

For any closed subset C⊂Cx([0,T],Sd) and δ>0,
P(xɛ∈C)≤P(‖xɛ-xn,ɛ‖≤δ,xɛ∈C)+P(‖xɛ-xn,ɛ‖≥δ,xɛ∈C)≤P(xn,ɛ∈Cδ)+P(‖xɛ-xn,ɛ‖≥δ),
where
Cδ={f∈Cx([0,T],Sd),infg∈C‖f-g‖≤δ}.
Therefore,
limsupɛ→0ɛ2logP(xɛ∈C)≤limsupɛ→0ɛ2logP(xn,ɛ∈Cδ)∨limsupɛ→0ɛ2logP(‖xɛ-xn,ɛ‖>δ)≤(-inff∈CδIn(f)∨limsupɛ→0ɛ2logP(‖xɛ-xn,ɛ‖>δ))≤(-inff∈CδI(f))∨limsupɛ→0ɛ2logP(‖xɛ-xn,ɛ‖>δ).
Let n→∞ to get
limsupɛ→0ɛ2logP(xɛ∈C)≤-inff∈CδI(f)⟶-inff∈CI(f)asδ⟶0,
which gives the upper bound of Theorem 3.2(i).

Let G⊂Cx([0,T],Sd) be an open subset. Take f∈G with I(f)<∞. Then, there exists h∈ℋ such that
f=Sh,I(f)=12‖h‖H.

Let fn=Sn,hn, hn be defined as (5.17). Then, fn⇒f as n→∞and also In(fn)≤(1/2)∥hn∥ℋ. Choose δ>0 such that Bf(2δ)={g∈Cx([0,T],Sd),∥f-g∥≤2δ}⊂G. Then, there exists N>0 such that for n>N,
‖fn-f‖<δ,Bfn(δ)⊂G.
Therefore,
P(xn,ɛ∈Bfn(δ))≤P(‖xn,ɛ-xɛ‖<δ,xn,ɛ∈Bfn(δ))+P(‖xn,ɛ-xɛ‖≥δ,xn,ɛ∈Bfn(δ))≤P(xɛ∈Bf(2δ))+P(‖xn,δ-xɛ‖≥δ)≤P(xɛ∈G)+P(‖xn,ɛ-xɛ‖≥δ).
Thus,
-12‖hn‖H2≤-In(fn)≤liminfɛ→0ɛ2logP(xn,ɛ∈Bfn(δ))≤liminfɛ→0ɛ2logP(xɛ∈G)∨limsupɛ→0ɛ2logP(‖xn,ɛ-xɛ‖≥δ).
Let n→∞ to obtain
liminfɛ→0ɛ2logP(xɛ∈G)≥-12‖h‖H=-I(f).
Because f is arbitrary,
liminfɛ→0ɛ2logP(xɛ∈G)≥-inff∈GI(f),
we complete the proof of Theorem 3.2.

Acknowledgments

The author thanks Professor Tusheng Zhang for very useful discussions and the referee and the editor for their suggestion which helped her to improve the paper in many ways. The research of the author is supported in part by NSFC no. 10971032 and NSFC no. 11026058.

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