Large Deviations for Stochastic Differential Equations on Associated with the Critical Sobolev Brownian Vector Fields
We obtain a large deviation principle for the stochastic differential equations on the sphere 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 associated with a critical Sobolev Brownian vector field which was constructed by Fang and Zhang .
Recall that Schilder theorem states that if is the real Brownian motion and is the space of real continuous functions defined on , null at 0, which endowed with the uniform norm, then for any open set and closed set , with
This result was then generalized by Freidlin and Wentzell in their famous paper  by considering the It equation They proved a large deviation principle for the above equation under usual Lipschitz conditions.
Recently, Ren and Zhang in  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 , 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 which is defined by the following SDE: where are eigenvector fields of Laplace operator on the sphere with respect to the metric . , , , and are the eigenspaces of eigenvalues and , respectively.
The authors in  consider the stochastic differential equations on Let , and there exists a real-valued Brownian motion such that therefore, the coefficients of SDEs which defined the Brownian motion on with respect to the metric are non-Lipschitz (see Lemma 4.2 or page 582–585  and Theorem 2.3 in ).
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 . We will adopt a different approach which is similar to those of Fang and Zhang  and Liang . We first work with the solution 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 is exponentially fast, which together with the special relation of rate functions guaranties that the large deviation estimate of can be transferred to , where 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 . 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.
Let be the Laplace operator on , acting on vector fields. The spectrum of is given by , where . Let be the eigenspace associated to and the eigenspace associated to . Their dimensions will be denoted by , . It is known (see ) that
Denote by for the orthonormal basis of and in ; that is, where is the Kronecker symbol and is the normalized Riemannian measure on , which is the unique one invariant by actions of . By Weyl theorem, the vector fields are smooth. For more detailed properties of the eigenvector fields, we refer the reader to Appendix A in .
Let and be the Sobolev space of vector fields on , which is the completion of smooth vector fields with respect to the norm
Then, is an orthonormal basis of . If we consider then
Let for be two family of independent standard Brownian motions defined on a probability space . Consider the series which converges in , uniformly with respect to in any compact subset of . According to (2.5), is a cylindrical Brownian motion in the Sobolev space . Moreover, takes values in the space for any . By Sobolev embedding theorem, in order to ensure that takes values in the space of vector fields, must be larger than . In this later case, the classical Kunita's framework  can be applied to integrate the vector field so that we obtain a flow of diffeomorphisms. For the case of small , the notion of statistical solutions was introduced in , and the phenomenon of phase transition appears. It was also shown in  that the statistical solutions give rise to a flow of maps if and the solution is not a flow of maps if The critical case was studied in . Instead of introducing as in (2.6), the authors in  consider first the stochastic differential equations on
3. Statement of the Result
We consider the abstract Wiener space associated with Wiener processes . is the Wiener measure and is the Cameron-Martin space associated with , where The purpose of this paper is to prove a large deviation principle for the family in the space and the collection of continuous functions from into with . To state the result, let us introduce the rate function. For any , let be the solution of And for any , let
We recall the definition of the good rate function.
Definition 3.1. A function mapping a metric space into is called a good rate function if for each , the level set is compact.
Our main result reads as follows.
Theorem 3.2. Let be the solution of (3.1) on , then satisfies a large deviation principle with a good rate function , ; that is,(i)for any closed subset , (ii)for any open set ,
4. Skeleton Equation and the Rate Function
Theorem 4.1. For any , (3.4) has a unique solution, denoted by .
Lemma 4.2. Let is defined respectively, by (2.14) and (2.13) in . Then, there exist some constants , such that for any ,
Proof of Theorem 4.1. Let be the solution of the following system:
Since are smooth, the solution of (4.3) exists.
For , consider the Riemannian distance defined by where denotes the inner product in . Let denote the Euclidean distance. We have the relation
Our aim is to show that converges to a solution of (3.4). By the chain rule, Let , then Let We have Using Proposition A.4 in  and Lemma 4.2, we see that Similarly, we have Therefore, Using the similar arguments as that in , the above inequality implies that there exist constants such that and are independent of . Hence, Thus, uniformly converges to some function in .
Next, we show that satisfies (3.4).
It suffices to show that for any , Set , , , and .
Fix , and by Proposition A.4 in  and Lemma 4.2, we have Thus, for any , there exists , for , By similar reasons, we also have On the other hand, because in , for any , one can find such that for , Therefore, for any , one can find such that for , Since is arbitrary, we obtain that The uniqueness is deduced from similar estimates.
Lemma 4.3. For any , the set is relatively compact in .
Proof. By the Ascoli-Arzela lemma, we need to show that is uniformly bounded and equicontinuous. The first fact is obvious, because for any . Next, we will show that is equicontinuous.
Let be an orthonormal basis of , and by Proposition A.4 in  and Lemma 4.2, we have where . Thus, which finishes the proof.
Lemma 4.4. The mapping is continuous from with respect to the topology on into .
Proof. Let with and assume that converges to in , then weakly in . By Lemma 4.2, is relatively compact. Let be a limit of any convergent subsequence of . We will finish the proof the lemma by showing that . Now, for simplicity, we drop the subindex k
It is sufficient to show that in .
Write with being given by Let , and by Proposition A.4 in  and Lemma 4.2, we haveLet Then, , because of Therefore, Combining above estimates, Hence, This implies which yields
Lemma 4.5. is a good rate function.
Proof. For any , The subset is a compact set in and is a continuous map for any . Therefore, is a compact set for any . So, is a good rate function.
5. The Proof of Theorem 3.2
Let be the solution to
We first have the following proposition.
Proposition 5.1. For any ,
Proof. Let . Using the similar estimates as that in  (see pages 582–585), there exists a real-valued Brownian motion such that
where , are defined as in Lemma 4.2.
Let , we have Introduce the function by Then, for any , as .
Define for , We have Without loss of generality, we may assume . Define . By Itö formula, we have Using Lemma 4.2, such that , Therefore, it follows from (5.9) that which implies that Since