SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS

The Annals of Probabiliy 2004, Vol. 32, No. 1A, 1 27 Insiue of Mahemaical Saisics, 2004 SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS BY DMITRY DOLGOPYAT, 1 VADIM KALOSHIN 2 AND LEONID KORALOV 3 Universiyof Maryland, California Insiue of Technology and Princeon Universiy We consider a sochasic flow driven by a finie-dimensional Brownian moion. We show ha almos every realizaion of such a flow exhibis srong saisical properies such as he exponenial convergence of an iniial measure o he equilibrium sae and he cenral limi heorem. The proof uses new esimaes of he mixing raes of he muli-poin moion. 1. Inroducion. The subjec of his paper is he sudy of he long-ime behavior of a passive subsance (or scalar) carried by a sochasic flow. Moivaion comes from applied problems in saisical urbulence and oceanography, Monin and Yaglom (1971), Yaglom (1987), Davis (1991), Isichenko (1992) and Carmona and Cerou (1999). The quesions we discuss here are also relaed o he physical basis of he Kolmogorov model for urbulence, Molchanov (1996). The physical mechanism of urbulence is sill no compleely undersood. I was suggesed in Ruelle and Takens (1971) ha he appearance of urbulence could be similar o he appearance of chaoic behavior in finie-dimensional deerminisic sysems. Compared o oher cases, he mechanism responsible for sochasiciy in deerminisic dynamical sysems wih nonzero Lyapunov exponens is relaively well undersood. I is caused by a sensiive dependence on iniial condiions, ha is, by exponenial divergence of nearby rajecories. I is believed ha a similar mechanism can be found in many oher siuaions, bu mahemaical resuls are scarce. Here we describe a seing where analysis similar o he deerminisic dynamical sysems wih nonzero Lyapunov exponens can be used. In he paper we shall consider a flow of diffeomorphisms on a compac manifold, generaed by soluions of sochasic differenial equaions driven by a finie-dimensional Brownian moion. We show ha he presence of nonzero exponens combined wih cerain nondegeneracy condiions (amouning roughly speaking o he assumpion ha he noise can move he orbi in any direcion) implies almos surely chaoic behavior in he following sense: Received Ocober 2001; revised Ocober 2002. 1 Suppored in par by he NSF and Sloan Foundaion. 2 Suppored in par by American Insiue of Mahemaics fellowship and Couran Insiue. 3 Suppored in par by an NSF posdocoral fellowship. AMS 2000 subjec classificaions. 37H15, 37A25, 37D25, 60F05, 60F10. Key words and phrases. Lyapunov exponens, sochasic flows, random diffeomorphisms, cenral limi heorems, passive scalar. 1

2 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV Exponenial, in ime, decay of correlaions beween he rajecories wih differen iniial daa. Equidisribuion of images of submanifolds. Cenral limi heorem, wih respec o he measure on a rich enough subse, which holds for almos every fixed realizaion of he underlying Brownian moion. In order o illusrae he las poin, le us consider a periodic flow on R n,and le ν be a Lebesgue probabiliy measure concenraed on an open subse. As a moivaing example one may hink of an oil spo on he surface of he ocean. The ulimae goal could be o remove he oil or a leas o preven i from ouching cerain places on he surface of ocean. Thus, we wish o predic he properies and he probabiliy laws governing he dynamics of he spo in ime. Le ν be he measure on R n induced from ν by ime map of he flow. We shall show ha almos surely ν is asympoically equivalen o a Gaussian measure wih variance of order. In oher words, for a sufficienly large posiive R for large ime, 99% of he oil spo is conained in he ball of radius R. Even hough we consider he random flows generaed by SDEs, very lile in our approach relies on he precise form of he noise, and in a fuure work we shall ry o generalize our resuls o oher random flows. Thus our work could be considered as a firs sep in exending deerminisic dynamical sysem picure o a more general seup. As a nex sep one may aemp o obain he same resuls for he so-called isoropic Brownian flows inroduced by Iô (1956) and Yaglom (1957). This is a class of flows for which he image of any simple poin is a Brownian moion and he dependence (he covariance ensor) beween wo differen poins is a funcion of disance beween hese poins. Relaed problems for his case have been sudied by Harris (1981), Baxendale (1986), Le Jan (1985), Carmona and Cerou (1999), Cranson, Scheuzow and Seinsalz (1999, 2000), Cranson and Scheuzow (2002), Lisei and Scheuzow (2001) and Scheuzow and Seinsalz (2002). The auhors have succesifully applied he resuls of his paper o he sudy of asympoic properies of passive scalar raspor problems in Dolgopya, Kaloshin and Karalov (2002, 2004). The precise saemens of our resuls are given in he nex secion. The proofs are carried ou in Secions 3 5. Secion 6 deals wih dissipaive flows. REMARK 1. One of he key ools in our work is he exponenial mixing esimae for he wo-poin moion (19) and almos sure equidisribuion resuls of Secions 4 and 6 which follow from (19). Afer having submied our paper we have learned ha hese mixing and equidisribuion resuls were obained earlier by Baxendale (1996) by slighly differen mehods. We are graeful o Professor Baxendale for elling us abou his ideas from Baxendale (1996), providing references o he work of Meyn and Tweedie (1993a) and he book of Arnold

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 3 (1998). This way we were able o dramaically simplify our proofs, especially in Secion 3. In addiion, Professor Baxendale proposed he simple proof of Lemma 5 and suggesed how o simplify proofs in Secion 5.2 by inroducing à (x) = A (x) f(x ). 2. Cenral limi heorems and an applicaion o periodic flows. 2.1. Measure-preserving nondegenerae sochasic flows of diffeomorphisms. Le M be a C smooh, conneced, compac Riemannian manifold wih a smooh Riemannian meric d. Consider on M a sochasic flow of diffeomorphisms d (1) dx = X k (x ) dθ k () + X 0 (x )d, where X 0,X 1,...,X d are C -vecor fields on M and θ()= (θ 1 (),...,θ d ()) is a sandard R d -valued Brownian moion. Since he differenials are in he sense of Sraonovich, he associaed generaor for he process is given by d (2) L = 1 2 Xk 2 + X 0. Le us impose addiional assumpions on he vecor fields X 0,X 1,...,X d.all ogeher we impose five assumpions, (A) (E). All hese assumpions, excep (A) (measure preservaion), are nondegeneracy assumpions and are saisfied for a generic se of vecor fields X 0,X 1,...,X d. Now we formulae hem precisely. (A) (Measure preservaion.) The sochasic flow {x : 0}, defined by (1), almos surely w.r.. he Wiener measure W of he Brownian moion θ preserves a smooh measure µ on M (his is equivalen o saying ha each X j preserves µ). (B) (Hypoellipiciy for x.) For all x M,wehave (3) Lie(X 1,...,X d )(x) = T x M, ha is, he linear space spanned by all possible Lie brackes made ou of X 1,...,X d coincides wih he whole angen space T x M a x. Denoe by (4) ={(x 1,x 2 ) M M : x 1 = x 2 } he diagonal in he produc M M. (C) (Hypoellipiciy for he wo-poin moion.) The generaor of he wo-poin moion {(x 1,x2 ) : >0} is nondegenerae away from he diagonal (M), hais, he Lie brackes made ou of ((X 1 (x 1 ), X 1 (x 2 )),..., (X d (x 1 ), X d (x 2 )) generae T x 1M T x 2M. To formulae he nex assumpion we need addiional noaions. For (, x) [0, ) M, ledx : T x0 M T x M be he linearizaion of x a ime. We

4 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV need an analog of hypoellipiciy condiion (B) for he process {(x,dx ) : >0}. Denoe by TX k he derivaive of he vecor field X k, hough as he map on TM, and by SM ={v TM: v =1}, he uni angen bundle on M. Ifwedenoeby X k (v) he projecion of TX k (v) ono T v SM, hen he sochasic flow (1) on M induces by a sochasic flow on he uni angen bundle SM is defined by he following equaion: (5) d x (v) = d X k ( x (v)) dθ k () + X 0 ( x (v)) d wih x 0 (v) = v, where v SM. Wih hese noaions we have condiion (D) (Hypoellipiciy for (x,dx ).) For all v SM,wehave Lie( X 1,..., X d )(v) = T v SM. The leading Lyapunov exponen λ 1 is defined by he following formula: 1 (6) λ 1 = lim log Dx (x). See Baxendale (1986), Carverhill (1985a, b) and Elworhy and Sroock (1986) for more informaion on Lyapunov exponens of sochasic flows. Our las assumpion is: (E) ( Posiive Lyapunov exponen.) λ 1 > 0. For measure-preserving sochasic flows wih condiions (D), Lyapunov exponens λ 1,...,λ dim M do exis by muliplicaive ergodic heorem for sochasic flows of diffeomorphisms [see Carverhill (1985b), Theorem 2.1]. Le us noe ha condiion (E) follows from condiions (A) (D) [we formulae i as a separae condiion because mos of our resuls are valid for dissipaive flows saisfying (B) (D), see Secion 6]. Indeed, under condiion (A) he sum of he Lyapunov exponens is equal o zero. On he oher hand, Theorem 6.8 of Baxendale (1989) saes ha under condiion (B) all of he Lyapunov exponens can be equal o zero only if, for almos every realizaion of he flow (1), one of he following wo condiions is saisfied: (a) here is a Riemannian meric ρ on M, invarian wih respec o he flow (1) or (b) here is a direcion field v(x) on M invarian wih respec o he flow (1). However, (a) conradics condiion (C). Indeed, (a) implies ha all he Lie brackes of {(X k (x 1 ), X k (x 2 ))} d are angen o he leaves of he foliaion {(x 1,x 2 ) M M : ρ (x 1,x 2 ) = Cons} and do no form he whole angen space. On he oher hand, (b) conradics condiion (D), since (b) implies ha all he Lie brackes are angen o he graph of v. This posiiviy of λ 1 is crucial for our approach.

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 5 2.2. CLTs for nondegenerae sochasic flows of diffeomorphisms. The firs CLT for such flows is CLT for addiive funcionals of he wo-poin moion. Denoe by {A (2) : >0} an addiive funcional of he wo-poin moion {(x 1,x2 ) : >0}. Suppose A(2) is governed by he Sraonovich sochasic differenial equaion (7) da (2) d (x0 1,x2 0 ) = α k (x 1,x2 ) dθ k() + a(x 1,x2 )d, where {α k } d and a are C -smooh funcions, such ha [ ] a(x 1,x 2 ) + 1 ( ) (8) 2 L(Xk,X k )α k (x 1,x 2 ) dm 2 (x 1,x 2 ) = 0, M M k where m 2 is he invarian measure of he wo-poin moion on M M \ [which exiss and is unique by Baxendale and Sroock (1988) under condiions (B) (E)] and where (L (Xk,X k )α k )(x 1,x 2 ) is he derivaive of α k along he vecor field (X k,x k ) on M M a he poin (x 1,x 2 ). This equaliy can be aained by subracing a consan from a(x 1,x 2 ). THEOREM 1. Le {A (2) : >0} be he addiive funcional of he wo-poin moion, defined by (7), and le he wo-poin moion {(x 1,x2 ) : >0} for x1 0 x2 0 be defined by he sochasic flow of diffeomorphisms (1), which in urn saisfies condiions (B) (E). Then as, we have ha A (2) / converges weakly o a normal random variable. Fix a posiive ineger n>2. The second CLT for sochasic flows, defined by (1) and saisfying assumpions (B) (E), is he CLT for addiive funcionals of he n-poin moion. In oher words, CLT for he wo-poin moion (Theorem 1) has a naural generalizaion o a CLT for he n-poin moion. Denoe by (n) (M) = { (x 1,...,x n ) M M : j i such ha x j = x i} he generalized diagonal in he produc M M (n imes). Replace (C) by he following condiion. (C n ) The generaor of he n-poin moion {(x 1,...,xn ) : >0} is nondegenerae away from he generalized diagonal (n) (M), meaning ha Lie brackes made ou of (X 1 (x 1 ),...,X 1 (x n )),..., (X d (x 1 ),...,X d (x n )) generae T x 1M T x nm. Similarly o he case of he wo-poin moion, denoe by {A (n) : >0} an addiive funcional of he n-poin moion {(x 1,...,xn ) : >0}. Suppose A(n) is governed by he Sraonovich sochasic differenial equaion (9) da (n) d (x0 1,...,xn 0 ) = α k (x 1,...,xn ) dθ k() + a(x 1,...,xn )d,

6 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV where {α k } d and a are C -smooh funcions, which saisfy [ a(x 1,...,x n ) M M (10) ] d ( ) L(Xk,...,X k )α k (x 1,...,x n ) dm n (x 1,...,x n ), + 1 2 where m n is he invarian measure for he n-poin moion and where we denoe by (L (Xk,...,X k )α k )(x 1,...,x n ) he derivaive of α k along he vecor field (X k,...,x k ) on M M (n imes) a he poin (x 1,...,x n ). As in he wo-poin case, his equaliy can be aained by subracing a consan from a(x 1,...,x n ).Forhe n-poin moion we have he following: THEOREM 2. Le {A (n) : >0} be he addiive funcional of he n-poin moion, defined by (9), and le he n-poin moion {(x 1,...,xn ) : >0} for pairwise disinc x0 i xj 0 be defined by he sochasic flow of diffeomorphisms (1), which in urn saisfies condiions (B), (C n ), (D), and (E). Then as we have ha A (n) / converges weakly o a normal random variable. The hird CLT for sochasic flows, defined by (1) and saisfying assumpions (A) (E), is he main resul of he paper. This CLT holds for a fixed realizaion of he underlying Brownian moion and is wih respec o he randomness in he iniial se, which is assumed o be a se of posiive Hausdorff dimension in M. Consider sochasic flow (1) and an addiive funcional {A : >0} of he onepoin moion saisfying d (11) da (x 0 ) = α k (x ) dθ k () + a(x )d wih C -smooh coefficiens. Define (12) â(x) = 1 2 d ( ) LXk α k (x) + a(x), where (L Xk α k )(x) is he derivaive of α k along he vecor field X k a poin x. Impose addiional assumpions on he coefficiens of (11). (F) (No drif or preservaion of he cener of mass.) (13) â(x)dµ(x) = 0, α k (x) dµ(x) = 0 for k = 1,...,d. M M This condiion can be aained by subracing appropriae consans from funcions α 1,...,α d,anda. ForA as above, when,wehavehaa / converges o a normal random variable wih zero mean and some variance D(A). Our nex

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 7 resul below shows how lile randomness in iniial condiion is needed for he CLT o hold. Le ν be a probabiliy measure on M, such ha for some posiive p i has a finie p-energy (14) dν(x)dν(y) I p (ν) = d p <. (x, y) In paricular, his means ha he Hausdorff dimension of he suppor of ν on M is posiive [see Maila (1995), Secion 8]. Le M θ be he measure on R defined on Borel ses R by { M θ ( ) = ν x M : Aθ (x) } (15). THEOREM 3. Le {x : >0} be a sochasic flow of diffeomorphisms (1), and le condiions (A) (F) be saisfied. Then for almos every realizaion of he Brownian moion {θ()} as he measure M θ converges weakly o he Gaussian measure wih zero mean and some variance D(A). 2.3. Applicaion o periodic flows. Consider he sochasic flow (1) on R N, wih he periodic vecor fields X k. Applicaion of Theorems 1 3 o he corresponding flow on he N-dimensional orus leads o he clear saemens on he behavior of he flow on R N. We formulae hose as Theorems 1 3 below. The usual CLT describes he disribuion of he displacemen of a single paricle wih respec o he measure of he underlying Brownian moion {θ()}. TheCLT formulaed below (Theorem 3 ), on he conrary, holds for almos every realizaion of he Brownian moion and is wih respec o he randomness in he iniial condiion. THEOREM 1. Le {X k } d k=0 be C periodic vecor fields in R N wih a common period, and le condiions (B) (E) be saisfied. Le x 1 and x 2 be he soluions of (1) wih differen iniial daa. Then for some value of he drif v, he vecor 1 (x 1 v,x 2 v) converges as o a Gaussian random vecor wih zero mean. THEOREM 2. Le {X k } d k=0 be C periodic vecor fields in R N wih a common period, and le he condiions (B), (C n ),(D),and (E) be saisfied. Le x 1,...,xn be he soluions of (1) wih pairwise differen iniial daa. Then for some value of he drif v, he vecor 1 (x 1 v,...,x n v) converges as o a Gaussian random vecor wih zero mean.

8 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV THEOREM 3. Le {X k } d k=0 be C periodic vecor fields in R N wih a common period, le ν be a probabiliy measure wih finie p-energy for some p>0 and wih compac suppor, and le condiions (A) (F) be saisfied. For he condiion (F) we ake α k = X k and a = X 0. Le x be he soluion of (1) wih he iniial measure ν. Then for almos every realizaion of he Brownian moion {θ()} he disribuion of x / induced by ν converges weakly as o a Gaussian random variable on R N wih zero mean and some variance D. PROOF OF THEOREMS 1 3. The funcions α k and a in he formulas (7), (9) and (11) could be considered o be vecor-valued, hus defining he vecor-valued addiive funcionals. Any linear combinaion of he componens of a vecor-valued addiive funcional is a scalar addiive funcional, for which Theorems 1 3 hold. If any linear combinaion of he componens of a vecor is a Gaussian random variable, hen he vecor iself is Gaussian. Therefore, Theorems 1 3 hold for vecor-valued addiive funcionals as well. I remains o rewrie equaion (1) in he inegral form and apply Theorems 1, 2 or 3 o he vecor-valued addiive funcional of he flow on he orus. The proofs of he CLT s occupy Secions 3 5. In he nex secion we prove CLT for addiive funcionals of he wo-poin and n-poin moion. The proof relies on resuls of Baxendale Sroock [Baxendale and Sroock (1988)] and he CLT for V -ergodic Markov processes of Meyn Tweedie [Meyn and Tweedie (1993b)]. In Secion 4 we prove ha a se of posiive Hausdorff dimension on M becomes uniformly disribued by he flow (1) in he limi as ime ends o infiniy. Secion 5 is devoed o he proof of CLT for measures (Theorem 3). In Secion 6 we exend he above CLT o he dissipaive case. 3. Proof of he CLT for wo-poin and muli-poin funcionals. We firs sae several lemmas, which will lead o he CLT for he wo-poin moion. In order o describe he wo-poin moion close o he diagonal M M we shall use he following resul of Baxendale Sroock [Baxendale and Sroock (1988)] showing, in paricular, ha a posiive Lyapunov exponen for (1) implies ha he exi ime from he r neighborhood of he diagonal has exponenial momens. LEMMA 1 [Baxendale and Sroock (1988), Theorem 3.19]. There are posiive consans α 0 and r 0, which depend on he vecor fields X 0,...,X d, such ha for any 0 <α<α 0 and 0 <r<r 0, here are p(α),k(α) wih he propery p(α) 0 as α 0, such ha for any pair of disinc poins x 1 and x 2 which are a mos r apar we have (16) K 1 d p (x 1,x 2 ) E z (e ατ ) Kd p (x 1,x 2 ), where τ is he sopping ime of x 1 s and x 2 s geing disance r apar, ha is, τ = inf{s >0:d(x 1 s,x2 s ) = r} and z = (x1 0,x2 0 ).

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 9 The nex resul of Meyn Tweedie will be used o demonsrae he exponenial ergodiciy of he wo-poin process on M M \. We sae i in slighly lesser generaliy han in Meyn and Tweedie (1993a). For a diffusion process z on a (possibly noncompac) manifold, we denoe he sochasic ransiion funcion by p (x, dy). For a measure µ and a posiive measurable funcion f we denoe µ f = sup µ(g). g f LEMMA 2 [Meyn and Tweedie (1993a), Theorem 6.1]. Le z be a posiive Harris recurren diffusion process on a manifold wih an invarian measure π. Assume ha for any compac se K he measures p 1 (x,dy),x K are uniformly bounded from below by a nonrivial posiive measure. Suppose ha here exiss a coninuous posiive funcion V(z), such ha: (a) For any N>0 here is a compac se K, such ha V(z) N for z/ K. (b) For some c>0, d<, (17) LV(z) cv (z) + d, where L is he generaor of he diffusion process. Then here exis θ>0 and C<, such ha (18) p (z, ) π V CV (z)e θ. Le us now apply Lemma 2 o he case of he wo-poin moion z on M M \. Recall ha by (C) he wo-poin process has he unique invarian measure m 2 on M M \. Inside he r-neighborhood of he diagonal he funcion V(z)= E z (e ατ ) saisfies he relaion LV(z)= αv (z), and, by Lemma 1 i saisfies (16). Exending V(z)from he r/2 neighborhood of he diagonal o he res of M M \ as a smooh posiive funcion, we find ha Lemma 2 applies o he process z. Thus, we obain he following: COROLLARY 3. Le B C (M M) be a funcion wih zero mean wih respec o he invarian measure. For any poin z M M \, le ρ z,b () = E z (B(z )), where z is he wo-poin moion wih z 0 = z = (x, y). Then for sufficienly small posiive θ, here are posiive C(θ) and p(θ), wih he propery ha p(θ) 0 when θ 0, such ha (19) ρ z,b () Ce θ d p (x, y).

10 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV PROOF OF THEOREM 1. Consider he Markov chain (z n,ξ n ) on M M \, where ξ n = A n (2) A (2) n 1.LeP n(z, ) denoe he sochasic ransiion funcion for his chain (noe ha i only depends on he firs componen of he original poin). This chain is posiive Harris recurren, and is invarian measure is given by (f ) = P 1 (z, f ) dπ(z). Le W(z,ξ)= V(z)+ ξ 2,andlef be a funcion on (M M \ ) R, such ha f W. This choice of W is due o he fac ha we need he inequaliy ξ 2 W for he resuls of Meyn and Tweedie below o be applicable. Then Noe ha E z ( f(zn,ξ n ) ) = E z ( P1 (z n 1,f) ). P 1 (f ) P 1 (V + ξ 2 ) C 1 V + C 2 C 3 V. Therefore, from Lemma 2 i follows ha he following esimae holds: (20) P n W DWe θn. Now by he general heory of V -ergodic Markov chains [Meyn and Tweedie (1993b), Theorem 17.0.1]A n / n is asympoically normal. To ge Theorem 1 wrie A = A [] + (A A [] ) and noice ha he second erm converges o 0 in probabiliy. PROOF OF THEOREM 2. The proof is he same as for Theorem 2, excep ha we need a new proof of he exisence of he funcion V n (x 1,x 2,...,x n ) saisfying (17). Le L n be he generaor of he n-poin moion, and le V n (x 1,x 2,...,x n ) = i,j V(x i,x j ). Then (L n V n )(x 1,x 2,...,x n ) = (LV )(x i,x j ), i,j so he required inequaliy follows from (17) for he wo-poin moion. 4. Equidisribuion. In his secion we prove ha images of smooh submanifolds become uniformly disribued over M as. More generally, we prove ha if a measure ν has finie p-energy, defined in (14), for some p>0, hen he image of his measure under he dynamics of sochasic flow (1), saisfying condiions (A) (E), becomes uniformly disribued on M.

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 11 LEMMA 4. Le ν be a measure on M which has finie p-energy for some p>0. Le b C (M) saisfy b(x)dµ(x) = 0. Then here exis posiive γ independen of ν and b, and C independen of ν such ha for any posiive 0 { } (21) P sup b(x )dν(x) >CI p(ν) 1/2 e γ 0 Ce γ 0. 0 PROOF. Wihou loss of generaliy, we may assume ha ν(m) = 1 (oherwise we muliply ν by a consan). By he exponenial mixing of he wo-poin processes (Corollary 3) we have ( 2 ( E b(x )dν(x)) = E (x,y) b(x )b(y ) ) dν(x)dν(y) (22) Therefore, (23) M M dν(x)dν(y) C 1 b 2 e θ d p (x, y) C 2 I p (ν)e θ. E b(x )dν(x) C 3I p (ν) 1/2 e θ/2. This shows ha he expecaion of he inegral decays exponenially fas. Now we shall use sandard argumens based on he Borel Canelli lemma o show ha b(x )dν(x)iself decays o zero exponenially fas. Fix small posiive α and κ, o be specified laer. Le τ n,j = n + je κn, 0 j e κn. By he Chebyshev inequaliy, (23) implies { P b ( } ) x τn,j dν(x) >I p (ν) 1/2 e αn C 3 e (α θ/2)n. Taking he sum over j, { P max b ( } ) (24) x τn,j dν(x) >I p (ν) 1/2 e αn C 3 e (κ+α θ/2)n. 0 j<e κn Nex we consider he oscillaion of b(x )dν(x)on he inerval [τ n,j,τ n,j+1 ].By Iô s formula, due o (1), b(x )dν(x) b ( ) x τn,j dν(x) (25) = d τ n,j β k (x s )dν(x)dθ k (s) + τ n,j β(x s )dν(x)ds, where β k, k = 0,...,d, are some bounded funcions on M. Each of he inegrals τn,j βk (x s )dν(x)dθ k (s) can be obained from a Brownian moion by a random ime change. This ime change has a bounded derivaive, since β k s are bounded.

12 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV The absolue value of he las erm on he righ-hand side of (25) is no greaer han e κn/4 /(2(d + 1)) for sufficienly large n. Therefore, for suiable C 4,C 5,C 6 > 0, which are independen of ν since ν(m) = 1, { P sup b ( ) x 1 dν(x) b ( } ) x 2 dν(x) e κn/4 (26) τ n,j 1 2 τ n,j+1 { d P sup w( 1 ) w( 2 ) e κn/4 0 1 2 C 4 e κn d + 1 ( (e κn/4 /(d + 1)) 2 ( ) κn C 5 exp 2C 4 e κn ( κn = C 5 exp( C 6 e κn/2 ) exp 2 ). ) exp Combining his wih (24), we obain { } P sup b(x )dν(x) >I p(ν) 1/2 (e αn + e κn/2 ) n n+1 C 3 e (κ+α θ/2)n + C 5 e 3κn/2 exp( C 6 e κn/2 ). This implies (21) if we ake α = κ = γ = θ/10 in he las inequaliy and ake he sum over all n such ha n 0 1. 5. CLT for measures. 5.1. Energy esimae. We prove he lemma which is needed in order o conrol he growh of he he p-energy of a measure. Le ν ( ) = ν(x ).Alsoweshall wrie ν(f) = fdν. (27) LEMMA 5. If p is small enough, hen for some C 1,C 2,θ >0, PROOF. E ( I p (ν ) ) C 1 e θ I p (ν) + C 2. We apply Lemma 2 o he wo-poin process o obain E (x,y) d p (x,y ) d p (x, y) dµ(x) dµ(y) C 1 d p (x, y)e θ. [This formula follows from (18) once we recall ha V(x,y) is esimaed by d p from above and below.] From (27) i follows ha E (x,y) d p (x,y ) C 1 e θ d p (x, y) + C 2. 2 } Inegraing wih respec o dν(x)dν(y), we obain he lemma.

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 13 5.2. Momen esimaes and he proof of he main resul. This secion is devoed o he proof of he main resul of he paper: CLT for he passive racer (Theorem 3). Recall ha we sar wih a nonrandom measure ν of finie p-energy I p (ν) < for some p>0, a sochasic flow of diffeomorphisms (1), and an addiive funcional {A : >0} of he one-poin moion, defined by he sochasic differenial equaion (11). The assumpion in (F) ha â(x) has mean zero implies he exisence of a smooh funcion f(x)such ha Lf (x) =â(x). Define he new addiive funcional à (x) = A (x) f(x ).Sincef is bounded, hen Ã/ has he same asympoic properies as A /. Now by Iô s formula, dã(x) = d ( αk (x ) ( L Xk f ) (x ) ) dθ k (). Noice ha since each X k preserves he measure µ we have (L Xk f)dµ= 0. Hence, he new funcion α k (L Xk f)dµ saisfies (F). Thus, wihou loss of generaliy, we can assume ha â 0 and, in paricular, ha A is a maringale. Le χ(,ξ) = ν(exp{ iξ A }) be he characerisic funcion of he funcional A (x). Below we shall see ha χ(,ξ) is equiconinuous in ξ when ξ K, N on a se or realizaions of randomness of measure 1 δ, whereδ>0anda compac se K are arbirary. We shall furher see ha ν( A A [] ) (28) lim = 0 almos surely. Finally, we shall see ha here exiss D(A), such ha for any ξ fixed ( lim χ(n,ξ)= exp D(A)ξ2 ) (29) almos surely. n 2 Moreover, we shall prove: LEMMA 6. Wih he noaions above for any posiive ρ,n, here is C>0 such ha for any >0, we have { ( P exp D(A)ξ2 ) ( + ρ χ(,ξ) exp D(A)ξ2 )} ρ 1 C N. 2 2 Combining (29) wih (28) above, and wih he equiconinuiy of χ, we obain Theorem 3. Thus i remains o verify he equiconinuiy of χ and formulas (28) and (29). In he nex hree lemmas we esimae he growh rae of A and of is momens. LEMMA 7. Le δ,n 0 and N be arbirary posiive numbers. There exiss C = C(δ,N 0,N), such ha for any >0, and any signed measure ν, which saisfies ν (M) 1, I p ( ν ) N 0, we have P{ ν(a ) > δ } C N.

14 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV PROOF. Wrie he equaion for ν(a ) in Iô s form, (as shown above we can assume ha â 0) d ν(a ) = α k (x s )dν(x)dθ k (s). By Lemma 4, { (30) P sup δ/3 s 0 α k (x s )dν(x) >e γδ/3 N 0/2 } Ce γδ/3. The sum d δ/3 0 [ α k (x s )dν(x)] dθ k (s) is esimaed using he facs ha α k are bounded and ha he sochasic inegrals can be viewed as Brownian moions wih a random ime change. The same inegrals over he inerval [ δ/3,] are similarly esimaed using (30). The proof of (28) is similar o he proof of Lemma 7. Rewrie (28) as ν((a A [] ) 2 ) lim = 0 almos surely. Now one can wrie he expression for ν((a A [] ) 2 ) in Iô s form, and hen use he fac ha he sochasic inegral can be viewed as a ime-changed Brownian moion. Alernaively, (28) follows from a more general resul in Lisei and Scheuzow (2001). LEMMA 8. iniial poin x, (31) There exiss a consan C>0, such ha for any k 0 and any { } A P x >k C exp ( k2 C ). PROOF. Recall ha A = d 0 α k (x s )dθ k (s). For each of he sochasic inegrals, recall ha 0 α k (x s )dθ k (s) can be viewed as a ime-changed Brownian moion, wih he derivaive of he ime change bounded. Therefore, { 0 } α k (x s )dθ k (s) P x >k (32) { } sups c W s P >k C exp ( k2 ) C for some C>0. Therefore he esimae (31) holds, wih possibly a differen consan C.

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 15 LEMMA 9. For any posiive δ and any N N, here exiss a consan C>0, such ha for any measure ν wih ν (M) 1 and any n N, (33) P { ν ( A n )>n! (1/2+δ)n} C N δn/2. PROOF. Wihou loss of generaliy we may assume ha ν is a probabiliy measure. By Jensen s inequaliy, ( A n dν) l A nl dν for l N. I is herefore sufficien o esimae he probabiliy P{ν( A nl )>(n!) l (1/2+δ)nl }.By he Chebyshev inequaliy, P { ν( A nl )>(n!) l (1/2+δ)nl} Ex A nl dν(x) (34) (n!) l (1/2+δ)nl. Take l> N δ + 1 2. Then he righ-hand side of (34) is no greaer han sup x E x A nl / ((n!) l 1/2nl ) N δn/2.thisislesshanc N δn/2 by Lemma 8. Pu n =[ 1/3 ], τ = /n and for each 0 <s< denoe he incremen of he funcional A (x) from ime s o ime by (35) s, (x) = A (x) A s (x). We spli he ime inerval [0,] ino n equal pars and decompose n 1 (36) A (x) = jτ,(j+1)τ (x). j=0 The idea is o prove ha his is a sum of weakly dependen random variables and ha he CLT holds for almos every realizaion of he Brownian moion. We need an esimae on he correlaion beween he inpus from differen ime inervals. For any posiive τ,s and l wih τ l, we denoe ν( l τ,l l,l+s ) = l τ,l (y) l,l+s (y) dν(y). LEMMA 10. Le some posiive c 1,c 2,γ 1 and γ 2 be fixed, and consider s and τ, which saisfy c 1 l γ 1 s,τ c 2 l γ 2. For any posiive δ, N and N 0 here exiss a consan C such ha for l C, and any measure ν, which saisfies ν (M) 1, I p ( ν ) l N 0, we have (37) P{ ν( l τ,l l,l+s ) >l δ } l N. PROOF. Wihou loss of generaliy we may assume ha ν is a probabiliy measure. We sar by decomposing each of he segmens [l τ,l], [l,l + s] ino wo: [l τ,l]= 1 2 =[l τ,l ln 2 l] [l ln 2 l,l], [l,l + s]= 3 4 =[l,l + ln 2 l] [l + ln 2 l,l + s].

16 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV We denoe a,b = a,b χ { a,b (b a) 2 }. By Lemma 8 here is a consan C, such ha (E x 2 a,b )1/2 C(1 + b a) and ( Ex ( a,b a,b ) 2) 1/2 Ce (b a)/c. Therefore for wo segmens, [a,b] and [c,d], such ha b c, E x ( a,b c,d a,b c,d ) = ( ) ( E x ( a,b a,b ) c,d + Ex a,b ( c,d c,d ) ) C(1 + b a + d c ) ( e (b a)/c + e (d c)/c). Afer using Chebyshev s inequaliy, we obain ha for any posiive k, P{ ν( a,b c,d ) ν( a,b c,d ) >k} (38) C(1 + b a + d c )(e (b a)/c + e (d c)/c ). k In he same way one obains (39) P{ ν( a,b c,d ) ν( a,b c,d ) >k} C(1 + b a + d c )(e (b a)/c + e (d c)/c ). k The conribuion from ν( 2 3 ) is esimaed using esimae (38) wih k = l δ :for any posiive δ and N for sufficienly large l, P{ ν( 2 3 ) >l δ } l N. The conribuion from each of he oher hree producs is esimaed using he fac ha he segmens are separaed by a disance of order ln 2 l. Le us, for example, prove ha P{ ν( 1 3 ) >l δ } l N.Byakingk = l δ in (39), we obain P{ ν( 1 3 ) ν( 1 3 ) >l δ } l N. In order o esimae ν( 1 3 ) we apply he change of measure (40) ν( 1 3 ) = l τ,l ln 2 l l,l+ln 2 l dν(x) = ln 2 l,2ln 2 l (x) d ˆν(x), where ˆν(A) = χ {xl ln 2 l A} l τ,l ln 2 l (x) dν(x).

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 17 The measure ˆν has a densiy wih respec o ν l ln 2 l, which is bounded by l2 since l τ,l ln 2 l is bounded. Therefore, by Lemma 5, for any posiive N, here exiss M such ha The righ-hand side of (40) is wrien as d (41) ln 2 l,2ln 2 l (x) d ˆν(x) = P{I p (ˆν) > l M } l N. 2ln 2 l ln 2 l α k (x s )dˆν(x)dθ k (s). We now proceed as in he proof of Lemma 7. The conribuion from ν( 1 4 ) and ν( 2 4 ) is esimaed in exacly he same way. Our nex saemen concerns he asympoic behavior of he second momen of he funcional A. LEMMA 11. iniial poin x: (42) The following limi exiss and he convergence is uniform in he D(A) = lim n E x (A 2 n ). n PROOF. E x (A 2 n lim ) 1 n = lim n n n n 0 E x ( d using he uniform ergodiciy of he one-poin moion. ) ( ) d α k (x ) 2 d = α k (x ) 2 dµ(y) The nex lemma provides a linear bound (in probabiliy) on he growh of ν(a 2 ). Noe ha such a bound implies he equiconinuiy of χ(,ξ) in he sense discussed above. LEMMA 12. For any posiive N,N 0 and ρ>0, here is C>0, such ha for any measure ν which saisfies ν (M) 1, I p ( ν ) N 0, we have P{ ν(a 2 ) ν(m)d(a) ρ} C N. PROOF. Le us prove ha P{ν(A 2 ) ν(m)d(a) ρ} C N. The esimae from below can be proved similarly. Consider he even Q ha I p (ν jτ ) N 1 for all j<n.bylemma5wehavep{q } 1 N,ifa sufficienly large N 1 is seleced. Fix any δ wih 0 <δ<1/20. Le R be he even

18 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV ha ν( 2 jτ,(j+1)τ ) τ 1+δ for all j<n.bylemma9wehavep{r } 1 N. Le β j = ν( 2 (j 1)τ,jτ )χ {Q R } and B j = j β k. We shall prove ha { ( )} Bj j(ν(m)d(a)+ ρ)τ (43) E exp (1 ρ 1/6 /2) j. 5/6 The proof will proceed by inducion on j. Firs we show ha { ( )} βj (ν(m)d(a) + ρ)τ (44) E exp (1 ρ 1/6 /2). 5/6 Indeed, using he Taylor expansion, { ( )} βj (ν(m)d(a) + ρ)τ E exp (45) 5/6 = 1 + E β j (ν(m)d(a) + ρ)τ 5/6 ( ) βj (ν(m)d(a) + ρ)τ k/ + E k=2 5/6 (k!). Since by definiion β j ν( 2 jτ,(j+1)τ ),wehaveeβ j Eν( 2 jτ,(j+1)τ ).From Lemma 11 i easily follows ha Eν( 2 jτ,(j+1)τ ) (ν(m)d(a) + ρ/4)τ for large. The expecaion of he infinie sum is less han ρ 1/6 /4, since β j τ 1+δ. This proves (44). Assume ha (43) holds for some j. Then, { ( )} Bj+1 (ν(m)d(a) + ρ)(j + 1)τ E exp (46) 5/6 { ( Bj (ν(m)d(a) + ρ)jτ = E exp ( E exp 5/6 ) ( βj+1 (ν(m)d(a) + ρ)τ 5/6 ) Fjτj )}. Due o (44) and since I p (ν jτ ) N 1 on Q by he Markov propery, he condiional expecaion on he righ-hand side of (46) is no greaer han 1 ρ 1/6 /2. Therefore, { ( )} Bj+1 (ν(m)d(a) + ρ)(j + 1)τ E exp (47) (1 ρ 1/6 /2)E 5/6 { ( Bj (ν(m)d(a) + ρ)jτ exp This proves (43). I follows from (43) wih j = n ha 5/6 P { B n ν(m)d(a) ρ } C N. )}.

Recall ha SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 19 { P B n n 1 j=0 ν ( 2 jτ,(j+1)τ ) } C N by Lemmas 5 and 9. Finally, direc applicaion of Lemma 11 o pair producs of s a differen ime segmens gives ha for any posiive δ, wehave n 1 P{ ν(a2 ) ν ( 2 ) } (48) jτ,(j+1)τ > 1/3+δ C N j=0 for sufficienly large. This complees he proof of he lemma. LEMMA 13. Le ρ,n 0, and N be arbirary posiive numbers. There exiss a consan C>0, such ha for any >0, and any signed measure ν, which saisfies ν (M) 1, I p ( ν ) N 0 we have ( { }) ) iξ P{ ν exp 0,τ ν(m) (1 D(A)ξ2 } ρ 1/3 C N. 2 1/3 PROOF OF LEMMA 13. Consider he Taylor expansion of he funcion exp( iξ 0,τ (x)), ( { }) iξ ν exp 0,τ (49) = ν(m) + iξ ν ( ) ξ 2 0,τ 2 ν( 2 ) ( ) iξ k 0,τ + ν ( k ) 0,τ. k=3 By Lemma 7 for any δ>0, we have { iξ P ν ( 0,τ (x) ) } (50) > 1/2+δ <C N. By Lemma 12 almos cerainly, we have ν ( 2 0,τ (x) ) (51) ν(m)d(a)τ ρτ. To esimae he ail we apply Lemma 9. This proves he lemma. PROOF OF THEOREM 3. show ha (52) P{ ν ( { }) iξ exp 0,(j+1)τ 2 1/3 I remains o demonsrae ha (29) holds. Firs we (1 D(A)ξ2 ) ( { }) } iξ ν exp 0,jτ ρ 1/3 C N.

20 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV Wrie (53) ( { }) iξ ν exp 0,(j+1)τ ( { } { }) iξ iξ = ν exp 0,jτ exp jτ,(j+1)τ ( { iξ ( ) }) =ˆν exp 0,τ T jτ, where ˆν is a random measure, defined by { iξ ˆν(A) = exp 0,jτ }χ {xjτ A} dν(x) and T s is ime shif by s. Noe ha by Lemma 5 for some N 0, P{I p ( ˆν )> N 0 } C N. Thus he righ-hand side of (53) can be esimaed wih he help of Lemma 13: ( { }) ) iξ P{ ˆν exp 0,τ ˆν(M) (1 D(A)ξ2 } ρ 1/3 C N. 2 1/3 This is exacly he same as (52). Applying (52) recursively for j = n 1,...,1 we obain ha for any posiive N and ρ here is C>0 such ha {( P 1 D(A)ξ2 ) + ρ n { }) ( iξ 2 1/3 ν( exp 0,jτ 1 D(A)ξ2 ) ρ n } 2 1/3 1 C N. This implies Lemma 6 and formula (29), which also complees he proof of he heorem. 6. The dissipaive case. In his secion we exend our CLT for measures (Theorem 3), proved in he previous secion for measure-preserving sochasic flows [see condiion (A)], o he dissipaive case. In oher words, we consider sochasic flows defined by he sochasic differenial equaion (1) saisfying condiions (B) (E). Noice ha wihou measure-preservaion assumpion i is no longer rue ha generically he larges Lyapunov exponen is posiive. However, he case when all he Lyapunov exponens are negaive is well undersood [Le Jan (1986b)], so we shall concenrae on he case wih a leas one posiive exponen. The main resul of his secion is CLT for measures (Theorem 5). Le m be he invarian measure of he one-poin process, which is unique by hypoellipiciy assumpion (B). Le m 2 be he invarian measure of he wo-poin process which is suppored away from he diagonal. Such a measure exiss and is unique for he processes wih posiive larges exponen by he resuls of Baxendale

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 21 and Sroock (1988). Moreover, we have exponenial convergence o m 2. Now he normalizaion condiion becomes more suble since he Lebesgue measure is no longer invarian under each realizaion of he sochasic flow. I is convenien o define saisical equilibrium measure by aking he Lebesgue measure and ieraing i from. Noe ha his involves defining sochasic flow backward, as well as forward in ime. The simples way o do so is o ake counably many independen copies of sochasic flows defined on ime inerval [0,1]. Denoe by he canonical space of he wo-sided d-dimensional Brownian moion wih he Wiener measure P. We denoe by φ s, he diffeomorphism obained by evolving our sochasic flow beween imes s and. THEOREM 4. Wih he noaions above here is a family of probabiliy measures {µ : >0} such ha: (a) For any measure ν of finie p-energy, lim n φ n, (ν) = µ almos surely. (b) The process µ is Markovian and push forward φ by he ime sochasic flow (1) saisfies φ (µ 0) = µ. (c) For any coninuous funcion b for any measure ν of posiive p-energy, ν(b(x )) µ (b) 0 as + almos surely (see Lemma 17 for a more precise saemen). REMARK 2. The measure µ has been exensively sudied and par (b) is well known, see Arnold (1998). Dimensional characerisics of µ have been sudied in several papers [Ledrappier and Young (1988a, b) and Le Jan (1985, 1986a, b)]. The quesions which we discuss here are differen from he ones sudied in he book and hese papers and we will no use any of heir resuls. Also, do no confuse µ s wih he Riemannian measure µ we have on he Riemannian manifold M. Le {A : >0} be an addiive funcional of he one-poin moion given by (11) and saisfying (54) â(x)dm(x) = 0, α k (x) dm(x) = 0 fork = 1,...,d, M M where â is given by (12). Denoe by ā() = µ (a), ᾱ k () = µ (α k ) for k = 1,...,d averages wih respec o µ and define wo addiive funcionals d dc = ᾱ k () dθ k () +ā()d, (55) db (x) = d ( αk (x ) ᾱ k () ) dθ k () + ( a(x ) ā ) d.

22 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV Funcional B (x) for each becomes a random variable on M wih he produc measure P ν. Denoe C = C, B = B. by THEOREM 5. (a) Le M θ, be he measure on R defined on Borel ses R (56) M θ, ( ) = ν { x M : B θ, }. Then here is a consan D (A) such ha almos surely M θ, converges weakly o a Gaussian measure wih zero mean and variance D (A). (b) C is asympoically Gaussian wih zero mean and some variance D (A). (c) B and C are asympoically independen. We can reformulae Theorem 5 as follows. COROLLARY 14. (a) Almos surely for large he measure M θ defined by (15) is asympoically Gaussian wih a random drif C and deerminisic variance D (A). (b) As + he disribuion of he drif C is asympoically Gaussian wih zero mean and he variance D (A). PROOF OF THEOREM 4. Given a measure ν of finie p-energy denoe µ (n) (ν) = φ n, ν. LEMMA 15. There is a consan ρ<1 such ha for each coninuous funcion b on M, each >0 and any pair of probabiliy measures ν 1 and ν 2, each of finie p energy, almos surely here exiss a consan C = C(θ) such ha (57) µ (n) (ν 1 )(b) µ (n) (ν 2 )(b) Cρ n. Moreover, almos surely he limi lim n µ (n) (ν i ) exiss and is no dependen on i. (58) REMARK 3. This proves par (a) of Theorem 4. PROOF OF LEMMA 15. E ( µ (n) Noice ha (ν 1 )(b) µ (n) (ν 2 )(b) ) = E(b(x )) dν 1 (x n ) E(b(x )) dν 2 (x n ) Consρ n 1,

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 23 since boh erms are exponenially close o m(b) by he exponenial mixing of one-poin process. Likewise, E ([ µ (n) (ν 1 )(b) µ (n) (ν 2 )(b) ] 2 ) = E b(x )b(y )d(ν 1 ν 1 )(x n,y n ) (59) + E b(x )b(y )d(ν 2 ν 2 )(x n,y n ) 2E b(x )b(y )d(ν 1 ν 2 )(x n,y n ) Consρ n 2, since he firs wo erms are exponenially close o m 2 (b b) and he las erm is exponenially close o 2m 2 (b b). Thus, he Borel Canelli lemma implies he firs par of he lemma. This, in urn, implies independence of he limi lim n µ (n) (ν i ) from i. To prove exisence of his limi noice ha almos surely here is a random consan C = C(θ) such ha (60) µ (n+1) (ν)(b) µ (n) (ν)(b) Cρ n. The proof of (60) is similar o he proof of (57) and can be lef o he reader. By he consrucion we have par (b) of Theorem 4. LEMMA 16. For all b 1 C (M), b 2 C (M M), E ( µ (b 1 ) ) = m(b 1 ), E ( µ µ (b 2 ) ) = m 2 (b 2 ). For small p we have E(I p (µ )) < and is independen of by saionariy. PROOF. The firs par is obained by aking expecaion in he limis (φ n, m)(b ( 1) µ (b 1 ), (φ n, φ n, ) (m m) ) (b 2 ) (µ µ )(b 2 ). The second follows from he firs by Lemma 5. LEMMA 17. Le ν be a measure on M which has finie p-energy for some p>0. Le b C (M). Then here exis posiive γ independen of ν and b, and C independen of ν, such ha for any posiive 0, { } (61) P sup b(x )dν(x) µ (b) >CI p(ν) 1/2 e γ 0 Ce γ 0. 0 REMARK 4. This is par (c) of Theorem 4.

24 D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV PROOF OF LEMMA 17. Following he argumen of he proof of Lemma 15, we ge, for any wo measures ν 1 and ν 2 of finie p-energy, P{ ν 1 (b(x )) ν 2 (b(x )) r} Cons[I p(ν 1 )I p (ν 2 )] 1/2 e γ 1. r Taking ν 2 = µ 0,wege P{ ν(b(x )) µ (b) r} Cons[I p(ν 1 )] 1/2 e γ 1. r The res of he proof is similar o he proof of Lemma 4. PROOF OF THEOREM 5. The proof of par (a) is he same as he proof of Theorem 3. To prove par (b) observe ha similarly o discussion a he beginning of Secion 5.2 we can wihou he loss of generaliy assume ha A is a maringale. Thus, da = d α k (x )dθ k (). Then d C = ᾱ k (s) dθ k (s) is a maringale. So o prove (b) i suffices o show ha for each k, 1 ᾱk 2 (s) ds c k in probabiliy where 0 0 (62) c k = m 2 (α k α k ). Now ( 1 ) E ᾱk 2 (s) ds c k 0 by Lemmas 15 and 16. So i is enough o show ha ( 1 2 2 E [ᾱk 2 (s) c k] ds) 0. 0 This follows from he esimae E (( ᾱk 2 (s )(ᾱ2 )) 1) c k k (s 2 ) c k Conse θ s 1 s 2, which can be proven by he argumens of Lemma 2. Par (c) follows from he fac ha C depends only on he noise (and no on he iniial condiions) and B is asympoically independen of he noise by par (a) of his heorem.

SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS 25 REMARK 5. In he conservaive case µ m, so ā and ᾱ k are nonrandom and Theorem 5 reduces o Theorem 3. Conversely, suppose ha D (A) = 0forall addiive funcionals A. Le A = a(x s )dθ 1 (s), 0 where m(a) = 0. Then C = µ s (a) dθ(s). 0 By (62), 0 = D (A) = m 2 (a a). This implies ha for an arbirary funcion [no assuming ha m(a) = 0] we have m 2 (a a) = m(a) 2. By polarizaion, for any pair of coninuous funcions a,b on M,wehave a(x)b(y)dm 2 (x, y) = m(a)m(b). Hence, m 2 = m m. By Kunia (1990) his implies ha each φ s, preserves m. Hence, we can characerize he conservaive case by he condiion ha he drif in Corollary 14 is nonrandom. Acknowledgmens. This paper was sared during he conference Nonlinear Analysis, 2000. D. Dolgopya is graeful o IMA for he ravel gran o aend his conference. During he work on his paper we enjoyed he hospialiy of IMPA, Rio de Janeiro and of Universidad Auonoma de Madrid. We are especially graeful o J. Palis, M. Viana, A. Cordoba and D. Cordoba for providing excellen working condiions in Rio de Janeiro and Madrid. The auhors would like o hank P. Friz and S. R. S. Varadhan for useful commens. We are also graeful o P. Baxendale and he anonymous referee for consrucive criicism on he original version of his paper. REFERENCES ARNOLD, L. (1998). Random Dynamical Sysems. Springer, Berlin. BAXENDALE, P. (1986). The Lyapunov specrum of a sochasic flow of diffeomorphisms. Lyapunov Exponens. Lecure Noes in Mah. 1186 322 337. Springer, Berlin. BAXENDALE, P. (1989). Lyapunov exponens and relaive enropy for a sochasic flow of diffeomorphisms. Probab. Theory Relaed Fields 81 521 554. BAXENDALE, P. (1996). Lecure a AMS meeing a Pasadena. BAXENDALE,P.andHARRIS, T. (1986). Isoropic sochasic flows. Ann. Probab. 14 1155 1179. BAXENDALE, P. and STROOCK, D. W. (1988). Large deviaions and sochasic flows of diffeomorphisms. Probab. Theory Relaed Fields 80 169 215. BIRKHOFF, G. (1967). Laice Theory. Amer. Mah. Soc., Providence, RI.

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