Large Deviations for Stochastic Differential Equations on Sd Associated with the Critical Sobolev Brownian Vector Fields

The purpose of our paper is to prove a large deviation principle on the asymptotic behavior of the stochastic differential equations on the sphere S 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 ,


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 S d 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 C 0 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 ⊂ C 0 0, 1 and closed set F ⊂ C 0 0, 1 , lim inf

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This result was then generalized by Freidlin and Wentzell in their famous paper 2 by considering the It ö equation 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 S d which is defined by the following SDE: where A i ,k are eigenvector fields of Laplace operator Δ on the sphere S d with respect to the metric H d 2 /2 .D ,1 dim G , D ,2 dim D , G , and D are the eigenspaces of eigenvalues −c ,d and −c ,δ , respectively.
The authors in 1 consider the stochastic differential equations on S d dx n t 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 x n,ε 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 x n,ε → x ε is exponentially fast, which together with the special relation of rate functions guaranties that the large deviation estimate of x n,ε can be transferred to x ε , where x ε 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 S d .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.
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Framework
Let Δ be the Laplace operator on S d , acting on vector fields.The spectrum of Δ is given by spectrum Let G be the eigenspace associated to c ,d and D the eigenspace associated to c ,δ .Their dimensions will be denoted by as −→ ∞.

2.1
Denote by {A i ,k ; k 1, . . ., D ,i , ≥ 1} for i 1, 2 the orthonormal basis of G and D in L 2 ; that is, where δ ij is the Kronecker symbol and dx is the normalized Riemannian measure on S d , which is the unique one invariant by actions of g ∈ SO d 1 .By Weyl theorem, the vector fields {A i ,k } are smooth.For more detailed properties of the eigenvector fields, we refer the reader to Appendix A in 1 .
Let s > 0 and H s S d be the Sobolev space of vector fields on S d , which is the completion of smooth vector fields with respect to the norm 2 be two family of independent standard Brownian motions defined on a probability space Ω, F, P .Consider the series In this later case, the classical Kunita's framework 7 can be applied to integrate the vector field W t 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 W t t≥0 as in 2.6 , the authors in 1 consider first the stochastic differential equations on S d dx n t Using the specific properties of eigenvector fields, it was proved that x n t x converges uniformly in t, x ∈ 0, T × S d to a solution of the sde 2.8 below.We quote the following result from 1 .
admits a unique strong solution x t x t≥0 , which gives rise to a flow of homeomorphisms.
In the case of the circle S 1 , this property of flows of homeomorphisms was discovered in 8 then studied in 9, 10 .

Statement of the Result
Consider the small perturbation of 2.8 Equation 3.1 has a unique strong solution x ε t x t≥0 according to Theorem A, denoted by x ε t .
We consider the abstract Wiener space Ω, H, F, P associated with Wiener processes . P is the Wiener measure and is the Cameron-Martin space associated with W, where The purpose of this paper is to prove a large deviation principle for the family {x ε , ε > 0} in the space C x 0, T , S d and the collection of continuous functions f from 0, T into S d with f 0 x.To state the result, let us introduce the rate function.For any h ∈ H, let {S h t , t ∈ 0, T } be the solution of

3.4
And for any f ∈ C x 0, T , S d , let 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 x ε t be the solution of 3.1 on C x 0, T , S d , then {x ε t , ε > 0} satisfies a large deviation principle with a good rate function In order to prove Theorem 4.1, we now introduce the following estimates which is Theorem 2.3 in 1 .

4.1
U n , V n 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,

4.2
Proof of Theorem 4.1.Let S n,h be the solution of the following system:

4.3
Since A i ,k are smooth, the solution of 4.
Our aim is to show that S n,h converges to a solution of 3.4 .By the chain rule,

4.6
Let θ n t d S n,h t , S n 1,h t , then Let S n,h s , A 2 ,k S n 1,h s ḣ2 ,k s ds.

4.8
We have 4.9

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Using Proposition A.4 in 1 and Lemma 4.2, we see that

4.10
Similarly, we have 2 n a b ds.

4.12
Using the similar arguments as that in 1 , the above inequality implies that there exist constants C 1 , C 2 such that

4.17
By similar reasons, we also have

4.18
On the other hand, because S n,h ⇒ S h in C x 0, T , S d , for any ε > 0, one can find N 1 > 0 such that for n > N 1 ,

4.19
Therefore, for any ε > 0, one can find N 2 > 0 such that for n > N 2 ,

4.20
Since ε is arbitrary, we obtain that

4.21
The uniqueness is deduced from similar estimates.
where θ t d S h t , u i .Thus, which finishes the proof.
Lemma 4.4.The mapping h → S h is continuous from {h : h H ≤ N} with respect to the topology on Ω into C x 0, T , S d .

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Proof.Let h n ∈ H with h n H ≤ N and assume that h n converges to h in Ω, then h n → h weakly in H.By Lemma 4.2, {S h n , n ≥ 1} is relatively compact.Let g ∈ C x 0, T , S d be a limit of any convergent subsequence of {S h n , n ≥ 1}.We will finish the proof the lemma by showing that g S h .Now, for simplicity, we drop the subindex k

4.24
It is sufficient to show that S h n ⇒ S h in C x 0, T , S d .Write S h n t − S h t I 3 − I 4 with I 3 , I 4 being given by

4.25
Let θ t d S h t , S h n t , and by Proposition A.4 in 1 and Lemma 4.2, we have

4.31
This implies Proof.For any a > 0, The subset { h H ≤ √ 2a} is a compact set in Ω and h → S h 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.

The Proof of Theorem 3.2
Let x n,ε t be the solution to

5.7
We have

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Using Lemma 4.2, ∃N such that n ≥ N,

5.10
Therefore, it follows from 5.9 that we have

5.18
It is obvious that

5.19
Proof of Theorem 3.2.For any closed subset C ⊂ C x 0, T , S d and δ > 0, Let f n S n,h n , h n be defined as 5.17 .Then, f n ⇒ f as n → ∞ and also I n f n ≤ 1/2 h n H . Choose δ > 0 such that B f 2δ {g ∈ C x 0, T , S d , f − g ≤ 2δ} ⊂ G.Then, there exists N > 0 such that for n > N, f n − f < δ, B f n δ ⊂ G.

Theorem 4 . 1 .
For any h ∈ H, 3.4 has a unique solution, denoted by S h t .
3 exists.For x, y ∈ S d , consider the Riemannian distance d x, y defined by cos d x, y x, y , 4.4 where •, • denotes the inner product in R d 1 .Let | • | denote the Euclidean distance.We have the relation

ε 2
log P x ε ∈ G ≥ −inf f∈G I f , 5.29we complete the proof of Theorem 3.2.
takes values in the space H s S d for any 0 < s < α/2.By Sobolev embedding theorem, in order to ensure that W t takes values in the space of C 2 vector fields, α must be larger than d 2.
Lemma 4.3.For any N > 0, the set {S h : h H ≤ N} is relatively compact in C x 0, T , S d .Proof.By the Ascoli-Arzela lemma, we need to show that {S h : h H ≤ N} is uniformly bounded and equicontinuous.The first fact is obvious, because S h 1 for any h ∈ H. Next, we will show that {S h : h H ≤ N} is equicontinuous.Let {u i , i 1, . . ., d 1} be an orthonormal basis of R d 1 , and by Proposition A.4 in 1 and Lemma 4.2, we have .15 Let ρ → 0 to get 5.2 .The proof is complete.