Schilder theorem for the Brownian motion on the diffeomorphism group of the circle

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1 Journal of Functional Analysis ) Schilder theorem for the Brownian motion on the diffeomorphism group of the circle Jiangang Ren a,b,, Xicheng Zhang a a Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 4374, P.R. China b School of Mathematics and Computational Science, Zhongshan University, Guangzhou, Guangdong 51275, P.R. China Received 26 May 24; accepted 19 August 24 Communicated by Paul Malliavin Available online 15 December 24 Abstract We prove a large deviation principle for flows associated to stochastic differential equations with non-lipschitz coefficients. As an application we establish a Schilder Theorem for the Brownian motion on the group of diffeomorphisms of the circle. 24 Elsevier Inc. All rights reserved. MSC: Primary: 6H1, 6F1; Secondary: 6C2, 34F5 Keywords: Stochastic flow; Large deviation; Non-Lipschitz; Brownian motion; Homeomorphism 1. Introduction This paper is a continuation of our previous work [17]. Our motivation is to prove a Schilder theorem on the asymptotic behavior for the Brownian motion on the group of homeomorphisms of the circle which is constructed by Malliavin [16]. Corresponding author. Department of Mathematics, Huazhong University of Science & Technology, Wuhan, Hubei 4374, P.R. China. addresses: ren@mathematik.uni-bielefeld.de J. Ren), xczhang@hust.edu.cn X. Zhang) /$ - see front matter 24 Elsevier Inc. All rights reserved. doi:1.116/j.jfa

2 18 J. Ren, X. Zhang / Journal of Functional Analysis ) Recall that the original Schilder theorem [18] states that if B is the real Brownian motion and C [, 1] the space of real continuous functions defined on [,1], null at, and endowed with the uniform convergence norm, then for any Borel set A C [, 1], lim sup ε ε 2 log P {εb A} ΛĀ), lim inf ε ε 2 log P {εb A} ΛA ), where Ā and A denote the closure and interior of A respectively, and ΛA) := inf γ A λγ) with λγ) = { 12 1 γs) 2 ds, γ absolutely continuous,, otherwise. 1) This result was then generalized by Freidlin and Wentzell in their famous paper [1] by considering the Itô equation { dx ε t) = εσx ε t)) dw t + bx ε t)) dt t 1, X ε ) = x. 2) They proved, under broad conditions mainly the Lipschitzness and the boundedness of the coefficients), that for every Borel set A of C x [, 1], R d ), ΛA ) lim inf ε ε 2 log PX ε x A) lim sup ε ε 2 log PX ε x A) ΛĀ), where C x [, 1], R d ) is the space of continuous functions defined on [, 1] and valued in R d, null at. Here also, the closure and the interior are taken in the topology of uniform convergence. Changing the angle we can let x run over R d and regard the solution as a random variable taking values in C[, 1],DR d )) where DR d ) is the space of C functions on R d endowed with the topology of compact uniform convergence of derivatives of all orders, provided that the coefficients are smooth and of linear growth see [12,14,15]). Thus one can also consider the large deviation principle LDP in abbreviation) in this context, namely the asymptotic estimates of probabilities PX ε A) where A BC[, 1],DR d ))). See e.g. [2,11] among others for studies along this line. Now let us return to the Brownian motion on the group of homeomorphisms of the circle. It is defined by a SDE as in 2) below. The main features of this equation are i) The coefficients are not Lipschitz, not to mention differentiable; ii) Infinitely many driving Brownian motions are involved. To deal with such an equation we will first work in a more general context by considering flows generated by SDE in R d with non-lipschitz coefficients and driven

3 J. Ren, X. Zhang / Journal of Functional Analysis ) by infinitely many Brownian motions. An advisable choice of tools to treat this situation would be the weak convergence approach to the theory of large deviations which is systematically developed in [8] and which proves to be highly efficient for equations with irregular coefficients see [5]). This will be done in Section 3. In the last Section, Section 4, we will prove our main result, Theorem 4.5 for the Brownian motion on the group of homeomorphisms of the circle. Here the point is that Theorem 4.5 is not a simple consequence of Theorem 3.2 since we will consider the natural topology in the space of homeomorphisms which is stronger than that of uniform convergence in the space of continuous functions. This will need a deeper analysis of the flows. Throughout the paper, C with or without indexes will denote different constants depending on the indexes) whose values are not important. Now, let us begin by introducing notions and notations. 2. Preliminaries Fixing T,R >, we denote by C T the space of all continuous maps from [,T] R d to R d and C R T the space of all continuous maps from [,T] D R to R d, where D R := {x R d di=1, x R}, x := xi 2 being the usual Euclidean norm in R d. Then C R T is a Banach space with the uniform norm f C R T := sup t,x) [,T ] D R ft,x) and C T is a Polish space with the seminorm: f CT := R=1 ) 2 R f C R 1. T Let l 2 denote the usual Hilbert space of R-valued sequences with the inner product, l 2. The norm in l 2 is denoted by l 2. Let H denote the Hilbert space of absolutely continuous functions from [,T] to l 2 with square integrable derivatives, i.e., H := { T } h :[,T] l 2 ; ḣs) 2 l 2 ds <. Let W T denote the space of all real continuous functions ω defined on [,T] with ω) =, endowed with the uniform convergence topology, and μ T the Wiener measure on W T, BW T )). Denote by WT, BW T ), μ T ), which will be our underlying probability space, the product space of denumerably many copies of W T, BW T )).

4 11 J. Ren, X. Zhang / Journal of Functional Analysis ) Then W T is a Polish space with the metric: dω, ω ) = 2 j ω j ω j 1). j=1 Define the projection W j of W T onto the jth coordinate: W j ω) = ω j W T ω W T ). Then W := {W j ; j = 1, 2, } is an infinite sequence of independent standard Brownian motions on the probability space WT, BW T ), μ T ).Fort T, let F t be the complete σ-algebra generated by {Ws),s t}. Let A denote the class of l 2 -valued F t predictable processes h, satisfying T ) E ḣs) 2 l 2 ds < +. We recall the following variational representation formula for the infinite-dimensional cylindrical) Brownian motion W [6, Corollary 3.7] for the finite-dimensional one see [4]). Theorem 2.1. Let F be a bounded Borel measurable function on WT log Ee FW) ) = inf E FW + h) + 1 ) h A 2 h 2 H.. Then we have Here and hereafter the expectation E is taken with respect to the probability measure μ T. We recall now the definition of Laplace principle. Let {Z ε,ε > } be a family of measurable mappings from W T to a Polish space E. Definition 2.2. A function I mapping E to [, ] is called a rate function if for each a<, the level set {f E : If ) a} is compact. Definition 2.3. Let I be a rate function on E. We say that {Z ε,ε > } satisfies the Laplace principle on E with rate function I if for all real bounded continuous functions g on E: [ ]) lim ε log E exp gzε ) = inf {gf ) + If )}. ε ε f E

5 J. Ren, X. Zhang / Journal of Functional Analysis ) For N>, let B N denote the ball in H with radius N. Then by Kolmogorov and Fomin [13, Theorem III.1]: Lemma 2.4. B N is metrizable as a compact Polish space under the weak topology in H. B N will be always endowed with this topology throughout the paper. We will need the following concave function: ρ η x) := { x log x 1, x η, η log η 1 + log η 1 1)x η), x > η, where < η < 1/e. Then we have see [3] [17, Example 2.2]). Lemma 2.5. Let gs), qs) be two strictly positive functions on R + satisfying g) <η and gt) g) + qs)ρ η gs)) ds, t. Then gt) g)) exp{ qs)ds}. 3) Here we list some simple properties of ρ η that will be used in the sequel. Their proofs are elementary and so omitted. Lemma ) ρ η is decreasing in η, i.e. ρ η1 ρ η2 if 1 > η 1 > η 2. 2) For any p>1, we have x p ρ η x) ρ η x 1+p )/1 + p). 3. A general result 3.1. Statement of the result Consider the following stochastic differential equation on W T, BW T ), μ T ): { dxt = σx t ) dw t + bx t )dt, t [,T], X = x R d, 4)

6 112 J. Ren, X. Zhang / Journal of Functional Analysis ) where σ : R d R d l 2 and b : R d R d are measurable functions, W := W 1,W 2, ) where W i, i = 1, 2, are defined in the preceding section. We make the following assumptions on the coefficients: A1) d σ i x) σ i y) 2 l 2 C x y ρ η x y ). i=1 A2) bx) by) x y γ x y ). where γ is a continuous positive function on R +, bounded in [1, ), such that lim x γx) log x =. First we prove Theorem 3.1. Let h H. Under A1) and A2), the following equation: has a unique solution. Xt h x) = x + s σx h x)), ḣ s l 2 ds + bxs h x)) ds. 5) Proof. Let X n be the solution of the following recursive system: X n t = x + σxs n n ), ḣ s l 2 ds + bxs n n )ds, where s n =[s2 n ]/2 n. It is not hard to find that there is a constant C such that for any t [,T] Set X n t X n t n C2 n. 6) φ n t) = X n+1 t X n t.

7 Nowby6) wehave J. Ren, X. Zhang / Journal of Functional Analysis ) φ n t) σxs n+1 n+1 ) σxs n t n ), ḣ s l 2 ds + bxs n+1 n+1 ) bx n s n )ds Cρ η 2 n t ) + σxs n+1 ) σxs n ), ḣ t s l 2 ds + Cρ η 2 n ) + C φ n s)ρ η φ n s)) ḣ s l 2 ds + Cρ η 2 n ) + ρ η φ n s)) C1 + ḣ s l 2)ds, bxs n+1 ) bxs n )ds φ n s)γφ h n s)) ds where < η < η is small enough. By Lemma 2.5, we obtain φ n t) [Cρ η 2 n )] exp{ C 1+ ḣ u l 2 )du} [Cρ η 2 n )] exp{ CT 1+ h H)} C2 αn for n sufficiently large and some α >. Hence X n t) converges uniformly on [,T] and taking the limit we obtain the existence. The uniqueness is deduced from a similar estimate. For h H, we will set Sh)t, x) := Xt hx) C T, where Xt h x) is the unique solution of 5). Consider the small perturbation to 4) { dx ε t = ε σxt ε) dw t + bxt ε )dt, t [,T], X ε = x Rd. The unique solution is denoted by Xt ε x). Under assumptions A1) and A2), we proved in [17,21] that the mapping t, x) Xt εx) is in C T a.s.. Letting μ ε denote the law of X ε in C T, we can now state our main result in this section. Theorem 3.2. For any A BC T ), we have inf If ) lim inf ε log μ f A ε A) lim sup ε log μ ε A) inf If ), ε ε f Ā

8 114 J. Ren, X. Zhang / Journal of Functional Analysis ) where A denotes the interior of A, Ā denotes the closure of A, and If ) is the rate function defined by If ) := 1 2 Here we use the usual convention infø) =+. inf {h H: Sh)=f } h 2 H, f C T. 7) The rest of this section is devoted to the proof of this theorem Rate function If ) First we have to show that If ) is indeed a rate function. Lemma 3.3. For any N>, the set {Sh); h H N} is relatively compact in C T. Proof. By the Ascoli Arzelà lemma, we only need to prove the following two facts for any R N: i) sup h H N Sh) C R < +. T ii) {t, x) Sh)t, x); t, x) [,T] D R, h H N} is equi-continuous. Set φ h t) := X h t x) Xh t y). By Eq. 5) wehave φ h t) x y + σxs h x)) σxh s y)), ḣ s l 2 ds + bxs h x)) bxh s y))) ds x y + C φ h s)ρ η φ h s)) ḣ s l 2 ds + φ h s)γφ h s)) ds x y + ρ η φ h s)) C1 + ḣ s l 2)ds, where < η < η is small enough. By Lemma 2.5, we obtain for x y η. sup φ h t) sup x y exp{ C1+ ḣ u l 2 )du} t T, h H N t T, h H N x y exp{ CT 1+N)} 8)

9 J. Ren, X. Zhang / Journal of Functional Analysis ) By the linear growth of σ and b, it is clear that there is a constant C = CR,T,N) such that for all s, t [,T] sup Xt h x) Xh s x) C s t 1/2. 9) x D R, h H N Both i) and ii) follow from 8) and 9). Then we have: Lemma 3.4. The mapping h Sh) is continuous from B N to C T. Proof. Let h n h weakly in B N. Set g n t, x) := Then for every t, x) [,T] D R σx h s x)), ḣn s ḣ s l 2 ds. lim g nt, x) =. n By the same method as in the above lemma, we know that {g n,n N} is relatively compact in C R T, and so Put By Eq. 5) wehave φ n t) g n C R T + + sup φ n t) := s,x) [,t] D R g n C R T + sup s,x) [,t] D R s lim g n n C R =. 1) T sup Xs hn s,x) [,t] D R s x) Xh s x). σx hn u x)) σxh u x)), ḣn u l 2 du bx hn u x)) bxh u x))) du C φ n s)ρ η φ n s)) ḣ n s l 2 ds + φ n s)γφ n s)) ds g n C R + ρ T η φ n s)) C1 + ḣ n s l 2)ds, where < η < η is small enough.

10 116 J. Ren, X. Zhang / Journal of Functional Analysis ) By Lemma 2.5, we obtain X hn X h C R T ] T exp{ C = φ n T ) [ g n 1+ ḣn u l 2 )du} C RT [ g n C R T ] exp{ CT 1+N)}, which together with 1) finishes the proof. Using the above two lemmas we now prove Lemma 3.5. i) For any f C T, if If ) <, then there is an h H such that 2If ) = h 2 H. ii) If ) is a rate function on C T. Proof. i) By the definition of If ), there is a sequence h n H such that h n 2 H 2If ) and Sh n ) = f. Put N := sup n h n H, then there is a subsequence h nk and h such that h nk h in B N.So h 2 H lim inf k h nk 2 H = 2If ). By Lemma 3.4 we have Sh ) = f, hence 2If ) = h 2 H. ii) For any a<, obviously A := {f : If ) a} {Sh); h 2 H 2a}. By Lemma 3.3, we only need to prove that A is closed in C T. Let A f n f in C T.Byi) we can choose h n B 2a such that Sh n ) = f n. By the compactness of B 2a, there is a subsequence h nk and h B 2a such that h nk h in B 2a. By Lemma 3.4 we have f nk = Sh nk ) Sh), hence f = Sh) and A is closed Tightness of the laws of X ε,hε For any h A, consider the following SDE: { dx h t = σx h t ) dw t + σx h t ), ḣ t l 2 dt + bx h t )dt, X h = x Rd. 11) Then we have: Theorem 3.6. Under Assumptions A1) and A2), Eq. 11) has a unique strong solution which will be denoted by X h t x). Proof. The proof is similar to [7,19,2], so we omit the details. We have the following representation formula for X.

11 J. Ren, X. Zhang / Journal of Functional Analysis ) Theorem 3.7. Let X be the unique strong solution to 4). Then for any bounded Borel function f : C T R, the following representation holds: log Ee fx) ) = inf E fx h ) + 1 ) h A 2 h 2 H. Proof. Since X is a strong solution to SDE4), there exists a BW T )/BC T )-measurable function such that Φ : W T C T 12) X t x) = ΦW )t, x), for all t, x) [,T] R d a.s. On the other hand, ΦW + h)t, x) satisfies the following equation: ΦW + h)t, x) = x + = x + σφw + h)s, x)) dw + h) s + σφw + h)s, x)) dw s + + bφw + h)s, x)) ds. Then by the uniqueness of solution to 11) wehave bφw + h)s, x)) ds σφw + h)s, x)), ḣ s l 2 ds X h = ΦW + h). Therefore by Theorem 2.1 log Ee fx) ) = log Ee f ΦW ) ) = inf E f ΦW + h) + 1 ) h A 2 h H = inf E fx h ) + 1 ) h A 2 h H. To proceed we need the following simple lemma.

12 118 J. Ren, X. Zhang / Journal of Functional Analysis ) Lemma 3.8. Let f be a smooth function from R + to R + satisfying f x) 1; fx)= x, x < 1 ; fx)= 1, x 2. 4 Let Z t be a continuous R d -valued semimartingale, ps) 2 an absolutely continuous adapted process. Then we have f Z t ) pt) = f Z ) p) + p s)f Z s ) ps) log f Z s )ds + ps)f Z s ) ps) 1 f Z s ) Z s / Z s,dz s R d ps)ps) 1)f Z s ) ps) 2 f Z s ) 2 Zi s Zj s Z s 2 d Zi,Z j s ps)f Z s ) ps) 1 f Z s ) Zi s Zj s Z s 2 d Zi,Z j s [ ] ps)f Z s ) ps) 1 f 1i=j Z s ) Z s Zi s Zj s Z s 3 d Z i,z j s log and α / = 1 α=1 for α 1 by convention). Proof. Let gx,y) := f y ) x, then gx,y) is twice continuously differentiable on [2, ) R and x gx,y) = f y ) x log f y ), yi gx,y) = xf y ) x 1 f y )y i / y, yi yj gx,y) = xx 1)f y ) x 2 f y ) 2 y j y i / y 2 +xf y ) x 1 f y )y j y i / y 2 +xf y ) x 1 f y ) [ 1 i=j / y y i y j / y 3]. Applying Itô s formula to gpt), Z t ) yields the formula. For N>, define A N := {h A : hω) H N, a.s.}.

13 J. Ren, X. Zhang / Journal of Functional Analysis ) Given a family {h ε,ε >} in A N, let Xt ε,hε x) be the unique solution to { dx ε,hε t = εσx ε,hε t X ε,hε = x R d. ) dw t + σxt ε,hε ), ḣ ε t l 2 dt + bxε,hε)dt The following lemma is the core estimate to derive the large deviation principle for stochastic flow by the method of weak convergence. Lemma 3.9. Let f be as in Lemma 3.8. There are three positive numbers p, β and C independent of ε such that E f X ε,hε t )) p x) Xt ε,hε y) C x y d+1+β for all ε [,ε ] and x y < η, provided ε = ε T, N) and η = η T, N) are sufficiently small. Proof. In the following, for the simplicity of notation we write X t x) = Xt ε,hε x). Put Since h ε A N we have pt) := d t + ḣ ε s l 2 ds. t d + 1 pt) d T + TN =: A T,N, t [,T]. By A2), we can choose δ 1 = δ 1 T, N) small enough such that and Put and bx) by) { 1 2A T,N x y log x y 1, < x y δ 1, C T,N x y, x y > δ 1 13) C A T,N log x log x, <x<δ1. 14) Z t := X t x) X t y) G ij s := σ i X s x)) σ i X s y)), σ j X s x)) σ j X s y)) l 2.

14 12 J. Ren, X. Zhang / Journal of Functional Analysis ) Then by A1) G ij s Z s ρ η Z s ). By Lemma 3.8 we have f Z t ) pt) = f Z ) p) + Mt ε + where 6 i=1 ξ i s) ds, M ε t := ε i ps)f Z s ) ps) 1 f Z s ) Zi s Z s σi X s x)) σ i X s y))) dw s, ξ 1 s) := 1 + ḣ ε s l 2)f Z s ) ps) log f Z s ), ξ 2 s) := i ξ 3 s) := i ξ 4 s) := ε 2 ξ 5 s) := ε 2 ξ 6 s) := ε 2 ps)f Z s ) ps) 1 f Z s ) Zi s Z s [bi X s x)) b i X s y))], ps)f Z s ) ps) 1 f Z s ) Zi s Z s σi X s x)) σ i X s y)), ḣ ε s l 2, i,j i,j ps)ps) 1)f Z s ) ps) 2 f Z s ) 2 Zi s Zj s Z s 2 Gij s, ps)f Z s ) ps) 1 f Z s ) Zi s Zj s Z s 2 Gij s, ps)f Z s ) ps) 1 f Z s ) i,j [ ] 1i=j Z s Zi s Zj s Z s 3 G ij s. If Z s < 1 4, then ξ 5 s) =. If Z s < δ 1, then by 13) wehave ξ 2 s) 1 2 fz s) ps) log Z s.

15 J. Ren, X. Zhang / Journal of Functional Analysis ) If Z s < δ 1 η, then by 14) wehave If Z s < η and ε<ε 1 := ξ 3 s) C A T,N f Z s ) ps) log Z s ḣ ε s l 2 fz s ) ps) log Z s ḣ ε s l 2. 1 C A T,N A T,N 1)2d 2, then ξ 4 s) + ξ 6 s) ε 2 C A T,N A T,N 1)2d 2 fz s ) ps) log Z s 1 2 fz s) ps) log Z s. So if Z s < δ 2 := 4 1 δ 1 η and ε<ε 1, then ξ 1 s) + ξ 2 s) + ξ 3 s) + ξ 4 s) + ξ 5 s) + ξ 6 s). If Z s δ 2, then it is easy to see that there exists a constant C T,N such that 6 ξ i s) C T,N 1 + ḣ s l 2)f Z s ) ps). i=1 Hence for ε<ε 1, we always have f Z t ) pt) f Z ) d+1 + Mt ε + C T,N + ḣ 1 ε s l 2)f Z s ) ps) ds. Now by Hölder s inequality, Lemma 2.6 and Jensen s inequality, we have E f Z t ) 2pt)) ) 3f Z ) 2d+1) + 3εA 2 T,N C E f Z s ) 2ps) 1 f Z s ) 2 ρ η Z s ) ds +3CT,N 2 E + ḣ 1 ε s l 2)2 ds 3f Z ) 2d+1) + 3εA 2 T,N C ρ η1 E +3CT,N N) E f Z s ) 2ps)) ds ) f Z s ) 2ps) ds f Z s ) 2ps))) /2ps)) ds 3f Z ) 2d+1) + εc N,T ρ η E f Z s ) 2ps))) ds, provided that η = η T, N)< η 1 < η) is sufficiently small.

16 122 J. Ren, X. Zhang / Journal of Functional Analysis ) Since fx) 1, we obtain by Lemma 2.5 ) E f Z t ) 2d+1+T + TN) E f Z t ) 2pt)) for x y < η. We choose ε = ε T,N)<ε 1 so small that C x y 2d+1) exp 3εT A2 T,N C /4), 15) 2d + 1) exp 3ε TA 2 T,N C /4) >d+ 1, then from 15) we get the desired estimate. The proof of the following lemma is standard, so we omit it. Lemma 3.1. There are three positive numbers p, β and C independent of ε such that E X ε,hε t for all ε [,ε ] and s, t [,T],x D R. We need also the following: ) p x) Xs ε,hε x) C s t d+1+β Lemma i) For every ε [,ε ], X ε,hε C T. ii) The laws of {h ε,x ε,hε, W ), ε [,ε ]} in B N C T WT is tight. iii) There exists a probability space Ω, F,P) and a sequence still indexed by ε for simplicity) { h ε, X ε, h ε, W ε )} and h, X h, W) defined on this probability space and taking values in B N C T WT such that a) h ε, X ε, h ε, W ε ) has the same law as h ε,x ε,hε,w) for each ε. b) h ε, X ε, h ε, W ε ) h, X h, W) in B N C T WT P.-a.s. as ε. c) h, X h ) solves the following ordinary differential equation: Xt h x) = x + s σx h x)), ḣ s l 2 ds + bxs h x)) ds 16) for any t [,T] and x R d. Proof. i) Let f be a function as in Lemma 3.9. Since fx+ y) fx)+ fy), for all x,y,

17 J. Ren, X. Zhang / Journal of Functional Analysis ) we can define a new complete metric on R d disx, y) := f x y ). Then by Lemmas 3.9 and 3.1, there are three positive constants p, β and C such that E disx ε,hε t ) p x), Xs ε,hε y)) C s t + x y ) d+1+β 17) for all ε [,ε ], s, t [,T] and x,y D R, x y < δ. By Kolmogorov s criterion, we know that X ε,hε C R T for all R N, and therefore in C T. ii) The tightness of the laws of {h ε,x ε,hε,w)} in B N C T WT follows from 17) and sup h ε H N a.s.. iii) The existences of { h ε, X ε, h ε, W ε )} and h, X h, W) and a), b) follow from ii) by Skorohod representation theorem. For c), noticing that W ε has the same law as W for each ε, we know that W ε is a Brownian motion on Ω, F,P). Let Φ be defined by 12). By a) and X ε,hε = ΦW + h ε ) μ T a.s., we have X ε, h ε = Φ W ε + h ε ) P a.s.. satisfies the following equa- Now by virtue of the uniqueness of strong solution, X ε, h ε tion: X ε, h ε t = x + ε σ X ε, h ε s ) d W s ε + σ X ε, h ε s ), h ε s l 2 ds + b X ε, h ε s )ds. By extracting a subsequence if necessary and then taking the limit as ε, we obtain that h, X h ) satisfies Eq. 16) Laplace principle We now prove a Laplace principle for X ε in C T by the approach of Budhiraja Dupuis [6], which will in turn give Theorem 3.2 via the equivalence of the Laplace principle and the large deviation principle [8]. Theorem {X ε,ε >} satisfies the Laplace principle in C T with the rate function If ).

18 124 J. Ren, X. Zhang / Journal of Functional Analysis ) Proof. Let g be a real bounded and continuous function on C T. Lower bound: By Theorem 3.7, we have [ ]) ε log E exp gxε ) = inf ε E gx ε, εh ) + ε ) h A 2 h 2 H = inf h A E Fix δ >. For every ε> there is a h ε A such that gx ε,h ) + 1 ) 2 h 2 H. 18) inf E gx ε,h ) + 1 ) h A 2 h H E gx ε,hε ) + 1 ) 2 hε 2 H δ. Since g is bounded, we have 1 ) 2 sup E h ε 2 H 2 g + δ. ε> Define τ ε N := inf {t [,T]: } ḣ ε s) 2 l 2 ds N and h ε N t) := hε t τ ε N ). Then h ε N t) A N and μ T ω : hε N ω) hε ω) H = ) = μ T ω : τε N ω) <T) 22 g + δ). N Therefore, E gx ε,hε ) + 1 ) 2 hε 2 H δ E gx ε,hε N ) + 1 ) 2 hε N 2 H 2 g 2 g + δ) δ. N By Lemma 3.11 we have lim inf ε E gx ε,hε N ) + 1 ) 2 hε N 2 H

19 J. Ren, X. Zhang / Journal of Functional Analysis ) = lim inf ε E P g X ε, h ε N ) h ε N 2 H E P gsh)) h 2 H inf gf ) + 1 ) {f,h) C T H: f =Sh)} 2 h 2 H inf f C T gf ) + If )). ) ) Therefore, lim inf ε ε log E [ ]) exp gxε ) ε inf gf ) + If )) 2 g 2 g + δ) δ. f C T N Finally, letting N and δ yields the lower bound. Upper bound: Fix δ >. Since g is bounded, there is an f C T Then choose h H such that gf ) + If ) inf f C T gf ) + If )) + δ. 1 2 h 2 H = If ) and f = Sh ). By 18) and Lemma 3.11 we have [ lim sup ε log E exp gxε ) ε ε = lim sup ε lim sup ε inf E h A E ]) gx ε,h ) h 2 H gx ε,h ) + 1 ) 2 h 2 H ) such that = gsh )) h 2 H gf ) + If ) inf f C T gf ) + If )) + δ. Letting δ, we obtain the upper bound.

20 126 J. Ren, X. Zhang / Journal of Functional Analysis ) Brownian motion on the group of homeomorphisms of the circle Let S 1 be the unit circle. The distance de iθ,e iθ ) between two points e iθ and e iθ on the circle is given by the absolute value of the angle between the vectors associated to e iθ, e iθ. Thus, for θ < 2π and θ < 2π, wehavede iθ,e iθ ) = inf θ θ, 2π θ θ ) and de iθ,e iθ ) π. We denote θ θ =de iθ,e iθ ). Since S 1 [, 2π) R/2π Z), a function f : S 1 S 1 can be regarded as a function f on R in a natural way with the property fθ + 2π) = fθ) + 2π, and vice versa. Moreover, f is Lipschitz, Hölder) continuous with respect to the usual Euclidean metric if and only if so is f with respect to. Below we will not distinguish between f and f. Define { H T := g :[,T] S 1 S 1 is continuous, t [,T], } g t is a homeomorphism on S 1, and g 1 ) is also bicontinuous. Define a metric on H T by d H g 1,g 2 ) := sup sup g 1 t, θ) g 2 t, θ) t [,T ] θ S 1 + sup sup 1 t, θ) g 1 2 t, θ). 19) t [,T ] θ S 1 g 1 Then Proposition 4.1. H T is a Polish space under d H. Proof. The completeness is obvious. The separability follows from the density of the set of continuous, piecewise linear, and strictly increasing functions which change their directions only at rational points at rational rates. Let α k := 1 2 hk + c 12 k3 k)), where h, c are two positive constants for the meaning of h and c see [16]). Define σ 2k 1 x) := coskx) α k, σ 2k x) := sinkx) α k.

21 Then we have see [1]) J. Ren, X. Zhang / Journal of Functional Analysis ) for some C > and η >. Consider the following SDE: dθ t = σx) σy) 2 l 2 C x y ρ η x y ) σ k θ t )dw k t), θ) = θ R; t [,T]. 2) k=1 The bi-continuous solution of Eq. 2) with the initial value θ is denoted by {θ t θ )} t. Since the coefficients are periodic functions with period 2π, wehave θ t θ + 2π) = θ t θ ) + 2π. Therefore by [1,17] t, θ ) θ t θ ) H T. For h H, let H h Sh)t, θ ) := θ h t θ ) H T denote the unique solution to the following equation Theorem 3.1): θ h t θ ) = θ + σθ h s θ )), ḣ s l 2 ds. 21) We consider the small perturbation of 2): dθ ε t = ε σ k θ ε t )dwk t), θ ε ) = θ R; t [,T]. k=1 It is easy to see that θ ε : ω W T θε,wω)) HT is Borel measurable. Let ν ε denote the law of θ ε in H T. Our main result in this section is: Theorem 4.2. For any A BH T ), we have inf If ) lim inf ε log ν ε A) lim sup ε log ν ε A) inf If ), f A ε ε f Ā

22 128 J. Ren, X. Zhang / Journal of Functional Analysis ) where the closure and the interior are taken in the topology induced by the metric 19), and If ) is the rate function defined by If ) := 1 2 inf {h H: Sh)=f } h 2 H, f HT. 22) Let us first prove that If ) is indeed a rate function. Lemma 4.3. The map S is continuous from B N to H T and I is a rate function on H T. Proof. Let h n h weakly in B N. Let θ hn t θ ) := Sh n )t, θ ) and θ t θ ) := Sh)t, θ ). We only need to prove that and sup t [,T ] sup θ S 1 θ hn t θ ) θ t θ ) 23) sup t [,T ] sup θ hn, 1 t θ ) θ 1 t θ ). 24) θ S 1 Eq. 23) follows from Lemma 3.4. Let us look at 24). Denote by ˆθ t,h the solution of the following equation: Then ˆθ t,h s θ ) = θ s σˆθ t,h u θ )), ḣ t u l 2 du,for all s [,t]. 25) [θ h t θ )] 1 = Similar to Lemma 3.4, we can prove that ˆθ t,h t θ ). lim n t,hn t,h ˆθ s θ ) ˆθ s θ ) = for every s [,t] and θ S 1. In particular, lim n [θhn t θ )] 1 [θ h t θ )] 1 lim n [θhn t θ )] 1 [θ h t θ )] 1 =.

23 J. Ren, X. Zhang / Journal of Functional Analysis ) Now if we can prove that {[θ hn t θ )] 1,n N} is relatively compact in H T, then the result will follow. From Eq. 25), we easily deduce θ hn, 1 t θ ) θ hn, 1 t θ t,hn t,hn ) ˆθ t θ ) ˆθ t θ ) C θ θ α, θ hn, 1 s θ ) θ ˆθ s,hn θ ) s ˆθ s,hn θ ) C s for some α, 1). On the other hand, since θ hn t θ hn s θ )) = θ hn s+t θ ) we have θ hn, 1 s+t θ ) = θ hn, 1 s θ hn, 1 t θ )). So θ hn, 1 s+t θ ) θ hn, 1 t θ ) θ hn, 1 s θ hn, 1 t θ )) θ hn, 1 t θ ) + θ hn, 1 t θ ) θ hn, 1 t θ ) C s + θ θ α ). Finally, Ascoli Arzelá lemma yields the relative compactness. Given a family {h ε,ε >} in A N. Let θ ε,hε t θ ) be the unique solution to θ ε,hε t θ ) = θ + ε σθ ε,hε s θ )) dw s + σθ ε,hε s θ )), ḣ ε s l 2 ds. Let [θ ε,hε t θ )] 1 denote the inverse function of θ ε,hε t θ ) for any fixed t [,T]. Then we have Lemma 4.4. There are three positive constants p, β,c independent of ε such that E [θ ε,hε t θ )] 1 [θ ε,hε s θ )] 1 p C t s 2+β + θ θ 2+β ), 26) provided ε [,ε ], where ε = ε N, T ) is small enough. In particular, the conclusions of Lemma 3.11 still hold if we replace C T by H T and X ε,hε by θ ε,hε. Proof. In the following proof, we shall omit the superscript of θ ε,hε t θ ), i.e. we write θ t θ ) for θ ε,hε t θ ). We have the following:

24 13 J. Ren, X. Zhang / Journal of Functional Analysis ) Claim. Let {ˆθ t s θ ), s [,t]} solve the following equation: ˆθ t s θ ) = θ + s ε σˆθ t s u θ )) dŵ u + σˆθ t u θ )), ḣ ε t u l 2 du, s [,t]27) where Ŵ s := W t s W t is a Brownian motion with respect to the backward filtration ˆF s t := σ{ŵ u,u [,s]}. Then for all t [,T] Proof. Put Then a simple calculation gives that θ 1 t θ ) = ˆθ t t θ ) a.s.. σ n θ) := σ 1 θ),, σ 2 nθ),, ). σ n θ) σθ) l 2 C2 n. For every n, let {θ n s θ ), s [,t]} and {ˆθ n s θ ), s [,t]} be respectively the solutions to and θ n s θ ) = θ + s ε = θ + ε s ˆθ n s θ ) = θ + s ε = θ + ε s s σ n θ n u θ )) dw u + σ n θ n u θ )) dw u + σ n ˆθ n s u θ )) dŵ u + σ n ˆθ n u θ )) dŵ u + where the stochastic contractions disappear since 2 n k=1 s s σ n θ n u θ )), ḣ ε u l 2 du σ n θ n u θ )), ḣ ε u l 2 du σ n ˆθ n u θ )), ḣ ε t u l 2 du σ n ˆθ n u θ )), ḣ ε t u l 2 du, 2n 1 σ k x)σ k x) = σ2k 1 x)σ 2k 1 x) + σ 2kx)σ 2k x)) k=1 2 n 1 k coskx) sinkx) = α 2 k k=1 =. + ) k sinkx) coskx) α 2 k

25 J. Ren, X. Zhang / Journal of Functional Analysis ) Since the coefficients are smooth with bounded derivatives, by the classic theory of stochastic flows we have a.s. θ n t s θ ) = ˆθ n s θn t θ )), ˆθ n t s θ ) = θ n s ˆθ n t θ ))for all s [,t]. 28) In order to passing to the limit, we need the following uniform estimates: there are four positive constants p, C,α, β independent of ε such that E θ n s θ ) θ s θ ) p C2 βn, E ˆθ n s θ ) ˆθ s θ ) p C2 βn, E θ n s θ ) θ n s θ ) p C θ θ 3+α, E ˆθ n s θ ) ˆθ n s θ ) p C θ θ 3+α, E θ n s θ ) θ n s θ ) p C s s 3+α, E ˆθ n s θ ) ˆθ n s θ ) p C s s 3+α for ε [,ε ], where ε = ε N, T ) is small enough. The first four ones can be obtained by using the same argument as in Lemma 3.9. The last two ones are easy. Now taking the limits in 28) yields θ t s θ ) = ˆθ t s θ tθ )), ˆθ t t s θ ) = θ s ˆθ t t θ ))for all s [,t]. Letting s = t gives ˆθ t t θ tθ )) = θ, θ t ˆθ t t θ )) = θ which means that θ 1 t θ ) = ˆθ t t θ ), and the proof of the claim is complete. Using the same argument as in the proof of Lemma 3.9, we can prove that for p big enough E ˆθ t s θ ) ˆθ t s θ ) p C θ θ 5+β. In particular, we have E θ 1 t θ ) θ 1 t θ ) p C θ θ 5+β. 29)

26 132 J. Ren, X. Zhang / Journal of Functional Analysis ) It is direct from Eq. 27) that E ˆθ t s θ ) ˆθ t s θ ) p C s s p/2 s, s [,t]. 3) So Let w +,t u := w u+t w t, then θ s θ t θ, w), w +,t ) = θ t+s θ,w). θ 1 t θ 1 s Thus by the independence of θ 1 s θ,w +,t ), w) = θ 1 t+s θ,w). θ,w +,t ) and θ 1 E θ 1 t+s θ,w) θ 1 t θ,w) p = E θ 1 t θ 1 s t θ,w),29) and 3) we obtain θ,w +,t ), w) θ 1 θ,w) p CE θ 1 s θ,w +,t ) θ 5+β C s 5+β)/2. t Eq. 26) is thus proved. Applying the same argument as in the proof of Theorem 3.12 we obtain the following Laplace principle, which will yield Theorem 4.2. Theorem 4.5. {θ ε,ε >} satisfies the Laplace principle in H T If ). with the rate function Acknowledgments It is our great pleasure to thank deeply Professor Paul Malliavin for his encouragement and for telling us the existence of [19]. Financial support by Project 973 and NSF grant No of China is also gratefully acknowledged. References [1] H. Airault, J. Ren, Modulus of continuity of the canonic Brownian motion on the group of diffeomorphisms of the circle, J. Funct. Anal ) 22) [2] G. Ben Arous, F. Castell, Flow decomposition and large deviations, J. Funct. Anal. 14 1) 1996) [3] I. Bihari, A generalization of a lemma of Belmman and its application to uniqueness problem of differential equations, Acta. Math. Acad. Sci. Hungar )

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