Central limit theorem of Brownian motions in pinched negative curvature
aa r X i v : . [ m a t h . DG ] A ug CENTRAL LIMIT THEOREM OF BROWNIAN MOTIONSIN PINCHED NEGATIVE CURVATURE
JAELIN KIMAbstract. We prove the central limit theorem of random variables induced by distances to Brownianpaths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinchednegative curvature. We further provide some ergodic properties of Brownian motions and an applicationof the central limit theorem to the dynamics of geodesic flows in pinched negative curvature. Introduction
Let f M be a simply connected complete Riemannian manifold of dimension d ≥ with pinchednegative curvature; its sectional curvature is uniformly bounded between two negatives. We furtherassume that f M admits a finite-volume quotient M and the first derivative of the sectional curvature isuniformly bounded.The Brownian motion ( e ω t ) t ∈ R + on f M starting from x is transient as f M is negatively curved.Therefore, the distance d( x, e ω t ) goes to infinity as t → ∞ with probability 1 and its asymptotic growthis linear ([14]): there is ℓ > such that ℓ = lim t →∞ t d( x, e ω t ) . Due to the pinched negative curvature, the Green function G( x, y ) on f M tends to zero as d( x, y ) → ∞ .Hence G( x, e ω t ) → as t → ∞ and it decays exponentially fast with probability 1 ([19]): there exists h > such that h = lim t →∞ − t log G( x, e ω t ) . Our main result is the central limit theorem of random processes Y ℓt ( e ω ) = d( x, e ω t ) − tℓ and Y ht ( e ω ) = log G( x, e ω t )+ th . It was proved for co-compact negatively curved manifolds by F. Ledrappierin [25]. Theorem 1.
The distributions of σ b √ t Y ℓt and σ k √ t Y ht are asymptotically normal for some positiveconstants σ b , σ k . More precisely, for every x ∈ f M , P x (cid:20) Y ℓt σ b √ t ≤ r (cid:21) , P x (cid:20) Y ht σ k √ t ≤ r (cid:21) → √ π Z r −∞ exp (cid:18) − s (cid:19) ds, as t → ∞ , where P x is the probability measures on the space C ( R + , f M ) of continuous sample paths which definesthe Brownian motion on f M starting from x . Mathematics Subject Classification.
Key words and phrases.
Foliated Brownian motion, central limit theorem, geodesic flow in negative curvature.
F. Ledrappier introduced a double process to provide a lower bound for the expectation of theGromov product at Brownian points in [25]. The lower bound implies the contraction property of thefoliated Brownian motion, which plays an important role in the proof of the central limit theorem.However, since the double process argument is not valid in the absence of compactness, we insteadprovide an argument using the C -convergence of the normalized distance functions to the Busemannfunction in pinched negatively curved manifolds. Although the resulting lower bound is less sharpthan the lower bound by the double process argument, it is sufficient for the proof of the contractionproperty.As in [25], we use the contraction property of the foliated Brownian motion (Theorem 3) onHölder spaces to solve the leafwise heat equation on the unit tangent bundle for the foliated Laplacian.We construct Martingales from the solutions of the heat equation with the initial conditions of theBusemann function and the logarithm of the Martin kernel of the Brownian motion. We prove that theyare asymptotically normal and have the same distributions with the random variables of our interest.As a consequence of the central limit theorem, we provide a characterization for the asymptoticharmonicity of f M with an assumption for thermodynamic formalism. f M is said to be asymptoticallyharmonic if the mean curvature of the horospheres of f M is constant. If f M is asymptotically harmonicthen the Liouville measure on the unit tangent bundle of M has maximal entropy for the geodesicflow. The characterization reveals an interplay between the stochastic properties, the geometry and thedynamics of the geodesic flow of f M . The Martin kernel of the Brownian motion gives rise to a Höldercontinuous function F BM on T M , which helps us understand the asymptotic behavior of Brownianpaths and correlation with geodesics. An equilibrium state of F BM is a geodesic flow-invariant Borelprobability measure on T M which maximizes the pressure of F BM . For compact manifolds, everyHölder continuous function admits a unique equilibrium states ([11]) while the existence is not alwaysguaranteed for finite-volume manifolds. Theorem 2. If F BM admits an equilibrium state, then σ ≥ h. The equality holds if and only if f M is asymptotically harmonic. In Section 2, we introduce the heat kernel and the Brownian motion on f M . We also recall pre-liminaries of the geometry, the ergodic theory and thermodynamic formalisms for geodesic flows ofmanifolds with pinched negative curvature. We prove Theorem 1 in Section 3 while Section 4 isdevoted to the proof of the contraction property (Theorem 3). Section 4 also contains a diagonalestimate of the heat kernel and the proof of exponential ergodicity of the Brownian motion on M . InSection 5, we prove ergodic properties of the Brownian motions which generalize the results in [22].We conclude the section with the proof of Theorem 2. Acknowledgement . It is a pleasure to thank François Ledrappier and Seonhee Lim for sharingtheir insights and helpful comments. The work is supported by Samsung Science and TechnologyFoundation under Project Number SSTF-BA1601-03. Preliminaries
Let ( M , g ) be a complete finite-volume Riemannian manifold of dimension d ≥ . We say that M has pinched negative curvature if − b ≤ sec M ≤ − a LT IN PINCHED NEGATIVE CURVATURE 3 for some positive numbers b > a > .We assume that M has pinched negative curvature and |∇ sec M | ≤ c for some c > . Let f M → M be the universal cover with the group of deck transformation Γ acting isometrically on f M .We also denote the lift of the metric on M to f M by g . Let d be the Riemannian distance of f M and vol := vol f M the Riemannian volume on f M .2.1. Geometry of pinched negative curvature.
Since f M has pinched negative curvature, the metricspace ( f M , d) is a CAT(0)-space. Hence we consider its boundary at infinity ∂ f M , also called the visual boundary . Fix x ∈ f M . A sequence ( z n ) in f M converges to a point ξ in ∂ f M if and only if z n → ∞ and the sequence of normalized distance functions f n ( y ) = b( y, x, z n ) := d( y, z n ) − d( x, z n ) converges uniformly on compact sets in C ( f M ) . We denote the limit function by b( y, x, ξ ) , which wecall the Busemann function based at ξ . The convergence of z n to ξ is independent of the choice of x .An important remark is that f n converges to the Busemann function C -uniformly on compact sets: Proposition 2.1. ( [3] ) Let f n ( y ) = d( y, z n ) − d( x, z n ) and z n → ξ ∈ ∂ f M . Then ∇ f n → ∇ b( · , x, ξ ) , ∇ v ∇ f n → ∇ v ∇ b( · , x, ξ ) , uniformly on compact sets. ∇ b( · , x, ξ ) means the covariance derivative of y b( y, x, ξ ) . Let ∆ = div ∇ be the Laplace-Beltrami operator on ( f M , g ) . If { e , . . . , e d } is an orthonormalframe on an open set U , for each C -function f on U ,(1) ∆ f = d X j =1 h e j , ∇ e j ∇ f i g on U . Applying Proposition 2.1 to each summand of (1), we obtain the following result. Proposition 2.2.
Let f n ( y ) = d( y, z n ) − d( x, z n ) and z n → ξ ∈ ∂ f M . Then ∆ f n converges to ∆b( · , x, ξ ) uniformly on compact sets. The visual boundary ∂ f M is equipped with a distance. For ξ, η ∈ ∂ f M with z n , w n ∈ f M whichconverge to ξ, η respectively, we define the Gromov product of ξ and η at x ∈ f M by ( ξ | η ) x := lim n →∞ d( x, z n ) + d( x, w n ) − d( z n , w n ) . Then for τ > small enough, d x,τ ∞ ( ξ, η ) := exp[ − τ ( ξ | η ) x ] is a distance function on the visualboundary ∂ f M (see [4]).Let π : T f M → f M be the tangent bundle of f M . We endow T f M with a Riemannian metric g T called the Sasaki metric , induced by the Riemannian structure g of f M and its Levi-Civita connection ∇ . We consider the unit tangent bundle T f M = { v ∈ T f M : k v k = h v , v i g = 1 } of f M , whichis a submanifold of T f M and also a sphere bundle of f M . We denote the geodesic flow on T f M by g t : T f M → T f M . We also denote by g t the geodesic flow on the unit tangent bundle T M of M . JAELIN KIM
We introduce the stable foliation f W s and the strong unstable foliation f W su of T f M which willplay an important role in the following sections. Their leaves are defined by f W s (v) = n w ∈ T f M : lim t →∞ d( γ v ( t + s ) , γ w ( t )) = 0 , ∃ s o , f W su (v) = n w ∈ T f M : lim t →∞ d( γ v ( t ) , γ w ( t )) = 0 o , Where γ v is the geodesic generated by v . Note that f W su consists of unit normal bundles of level setsof Busemann functions and leaves are transversal to the stable foliation with angle uniformly boundedaway from zero (Lemma 7.4. in [27]).The stable distribution e E s of T f M is a rank d -subbudle of the tangent bundle T T f M → T f M of T f M whose fibers are tangent spaces of stable leaves: e E s v := T v f W s (v) . Since f W s (v) is diffeomorphicto f M via π : T f M → f M for v ∈ T f M , we endow stable leaves of f W s with a metric g s inducedfrom the metric g on f M : for v ∈ T x f M , define g s on e E s v = T v f W s (v) from g on T x f M .For each point x ∈ f M and point at infinity ξ ∈ ∂ f M , there is a unique unit vector v in T f M suchthat γ v ( t ) converges to ξ at t → ∞ . Conversely, for every geodesic γ , γ ( t ) converges to a point ξ in ∂ f M . We denote the limit point ξ of γ v ( t ) by v + . This gives a useful identification of T f M with f M × ∂ f M . With such identification, we have that for v = ( x, ξ ) , f W s (v) = f M × { ξ } . Moreover, ∇ y b( y, x, ξ ) = ( y, ξ ) .Let X : T f M → e E s be a section of the stable distribution which is leafwise C , i.e., the restriction X | f W s ( x,ξ ) is C on f W s ( x, ξ ) for each ( x, ξ ) ∈ T f M . We identify X | f W s ( x,ξ ) with a C -vector field X ξ on f M for each ξ . We define the g s -divergence div s by div s X ( x, ξ ) = div X ξ ( x ) . Let u ∈ C ( T f M ) be a leafwise C -function; u | f W s (v) is C on f W s (v) . Thus for each ξ ∈ ∂ f M , u ξ ( x ) := u ( x, ξ ) is C on f M . We define the foliated Laplacian ∆ s by ∆ s u = div s ∇ u, where ∇ u ( x, ξ ) := ∇ u ξ ( x ) .2.2. Brownian motions.
The heat kernel ℘ : (0 , ∞ ) × f M × f M → (0 , ∞ ) is the fundamental solutionof the heat equation: ∂ t ℘ ( t, x, y ) = ∆ y ℘ ( t, x, y ) , lim t ↓ ℘ ( t, x, y ) = δ x ( y ) . The limit in the last equation means that for each f ∈ C b ( f M ) , lim t ↓ Z f M ℘ ( t, x, y ) f ( y ) d vol f M ( y ) = f ( x ) . Since the curvature of f M is negatively pinched, ∆ is (weakly) coercive, i.e., the Green function of ∆G( x, y ) := Z ∞ ℘ ( t, x, y ) dt is finite for x = y ∈ f M . LT IN PINCHED NEGATIVE CURVATURE 5
For κ < , if ℘ H d ( κ ) ( t, x, y ) is the heat kernel on the d -dimensional hyperbolic space H d ( κ ) ofconstant curvature κ , ℘ H d ( κ ) ( t, x, y ) depends only on t and d H d ( κ ) ( x, y ) . The following comparisontheorem of the heat kernel is also due to the pinched negative curvature. Proposition 2.3. (Heat kernel comparison theorem, [18] ) ℘ H d ( − b ) ( t, d( x, y )) ≤ ℘ ( t, x, y ) ≤ ℘ H d ( − a ) ( t, d( x, y )) . Note that ℘ ( t, x, y ) determines a unique family of probability measures on the space e Ω = C ( R + , f M ) of sample paths. For each x ∈ f M , we define the probability measure P x on the cylinder sets in e Ω by P x [ e ω t i ∈ A i , t < · · · < t k ] = Z A k · · · Z A ℘ ( t , x, y ) ℘ ( t − t , y , y ) × · · · × ℘ ( t k − t k − , y k − , y k ) d vol( y ) · · · d vol( y k ) . By Kolmogorov extension theorem, P x extends to a unique probability measure on e Ω . For s ≥ ,we denote the projection map e ω e ω s by π s : e Ω → f M . Let F t = F t ( f M ) := σ { π s } ≤ s ≤ t be thesmallest σ -algebra for which the projections π s are measurable. The canonical process e Z t ( e ω ) := e ω t of the filtered space ( e Ω , { F t } ≤ t ≤∞ ) forms a Markov process with respect to P x , which is called the Brownian motion on f M with initial distribution δ x , for each x ∈ f M .Let Ω = C ( R + , M ) . For each x ∈ M and its lift e x ∈ f M , we also denote the push-forwardmeasure of P e x by P x . Then the canonical process Z t of (Ω , ( F t ( M )) ≤ t ≤∞ , ( P x ) x ∈M ) is a Markovprocess, which we call the Brownian motion on M . This process is the projected process of theBrownian motion on f M . The stationary measure of the Brownian motion is the probability measurewhich defines the Brownian motion with initial distribution m : P m = R M P x d m( x ) where m is thenormalized Riemannian volume on M . The shift dynamical system on the path space (Ω , S t , P m ) isergodic since M is connected, where S t ω s = ω t + s for ω ∈ Ω .Let r ( ω, t ) = d( e ω , e ω t ) where e ω is a lift of ω . Then since r is a sub-additive cocycle, that is, r ( ω, t + s ) ≤ r ( ω, t ) + r ( S t ω, s ) for every s, t > , there exists a positive constant ℓ , which is called the linear drift of the Brownian motion, such that for every x ∈ f M and for a.s. ω ∈ Ω ℓ = lim t →∞ t r ( ω, t ) = lim t →∞ t d( x, e ω t ) due to the subadditive ergodic theorem ([20]).For a fixed x ∈ f M , the exponential map at x induces a polar coordinate on f M \ { x } : (0 , ∞ ) × T x f M → f M \ { x } ( r, v) exp x r v . Note that T x f M inherits the Riemannian metric g S of the unit sphere S d − from ( f M , g ) and write g as g = dr + λ x ( r, v) g S , for some smooth function λ x on f M \ { x } = (0 , ∞ ) × T x f M .For e ω ∈ e Ω , we write r ( e ω, t ) = d( e ω , e ω t ) and let θ ( e ω, t ) be the unit vector in T e ω f M with exp e ω [ r ( e ω, t ) θ ( e ω, t )] = e ω t . Proposition 2.4. ( [31] , [29] ) For every x ∈ f M and P x -a.e. e ω , the limit lim t →∞ θ ( e ω, t ) exists. JAELIN KIM
Since r ( e ω, t ) → ∞ as t → ∞ for P x -a.e. e ω , the limit e ω ∞ := lim t →∞ e ω t exists for P x -a.e. e ω . Inaddition, the Brownian path roughly follows the geodesic γ θ ( e ω, ∞ ) ([22]):(2) lim t →∞ t d ( e ω t , exp x [ r ( e ω, t ) θ ( e ω, ∞ )]) = 0 . We can replace r ( e ω, t ) by ℓt . We denote the asymptotic distribution of Brownian paths starting from x by ν x , i.e., ν x ( U ) := P x [ e ω : e ω ∞ ∈ U ] , for U ⊂ ∂ f M . Since the family ( P x ) is Γ -equivariant, ( ν x ) x ∈ f M is also Γ -equivariant: γ ∗ ν x = ν γx for each γ ∈ Γ .Moreover, ( ν x ) x ∈ f M is absolutely continuous and we denote the Radon-Nikodym derivative, called the Martin kernel , by k( x, y, ξ ) := dν y dν x ( ξ ) . The Martin kernel is also characterized by the limiting behavior of the Green function.
Proposition 2.5. ( [2] ) For each sequence ( z n ) in f M with z n → ξ ∈ ∂ f M , k( x, y, ξ ) = lim n →∞ G( y, z n )G( x, z n ) . We introduce another invariant of the Brownian motion called the stochastic entropy of the Brownianmotion denoted by h . The stochastic entropy was first introduced by V. Kaimanovich in [19] for co-compact manifolds with negative curvature. The stochastic entropy determines whether the Poissonboundary is trivial or not. The argument in [23] easily extends to manifolds with finite volume. Proposition 2.6.
For each x ∈ f M , P x -a.e. e ω , the following limits exist and coincide: h = lim t →∞ − t log ℘ ( t, x, e ω t )= lim t →∞ − t log G( x, e ω t ) . There is another characterization of the stochastic entropy analogous to the definition of the topo-logical entropy as the exponential growth of dynamically separated sets (see [19], [23]).
Proposition 2.7.
For x ∈ f M , T > and < δ < , h = lim T →∞ T log N ( x, T, δ ) , where N ( x, T, δ ) := inf { Card( E ) : P x [ d ( e ω T , E ) ≤ ≥ δ } .Proof. Fix ε > . Let C T,x := { e ω = x, ℘ ( T, e ω , e ω T ) ≤ e − T ( h − ε ) } , D T,x := { e ω : d( e ω t , γ θ ( e ω, ∞ ) ( ℓT )) ≤ εT, ℘ ( T, x, γ θ ( e ω, ∞ ) ( ℓT )) ≥ e − T ( h + ε ) } . LT IN PINCHED NEGATIVE CURVATURE 7
Choose a sufficiently large T such that − δ ≤ P x ( C T,x ) = P x [ e ω T ∈ π T C T,x ] . We denote by E x theexpectation with respect to P x . For each finite set E such that P x [ d ( e ω T , E ) ≤ ≥ δ , δ ≤ E x [ d ( e ω T , E ) ≤ P x [ { d ( e ω T , E ) ≤ } ∩ C T,x ] + P x [ { d ( e ω T , E ) ≤ } \ C T,x ] ≤ e − T ( h − ε ) X y ∈ E vol B ( y,
1) + 1 − (1 − δ ≤ Ce − T ( h − ε ) Card( E ) + δ where C = sup z vol B ( z, . Thus, δ C e T ( h − ε ) ≤ Card( E ) and we have h ≤ lim T →∞ T log N ( x, T, δ ) . For the converse inequality, Let E be a minimal set satisfying d ( e ω T , E ) ≤ for every e ω ∈ D T,x and F ⊂ { γ θ ( e ω, ∞ ) ( ℓT ) : e ω ∈ D T,x } a maximal -separated set. Note that Card( E ) ≥ N ( x, T, P x ( D T,x )) and Card( F ) ≤ C ′ e T ( h + ε ) . For each f ∈ F , N ( f ) := { e ∈ E : ∃ e ω ∈ D T,x s.t. d( f, γ θ ( e ω, ∞ ) ( ℓT )) ≤ , d( e ω T , e ) ≤ } . Then
Card N ( f ) ≤ e C ′′ εT . Therefore, we have N ( x, T, P x ( D T,x )) ≤ Card( E ) ≤ e C ′′ εT Card( F ) ≤ C ′ e T [ h +(2+ C ′′ ) ε ] . Given δ , for each T large enough, N ( x, T, δ ) ≤ N ( x, T, D T.,x ) . (cid:3) The stochastic entropy is related to the spectral information of f M , the bottom of the spectrum λ := inf Spec(∆ f M ) of the Laplacian on f M . It was proved in Proposition 3 of [24] for co-compactmanifolds. The proof is valid for pinched negative curvature and even the co-finiteness is not required. Proposition 2.8. λ ≤ h. Proof.
Since ℘ ( t, x, y ) is a solution of the heat equation, ℘ ( t, x, y ) log ℘ ( t, x, y ) = Z t ∂∂s ( ℘ ( s, x, y ) log ℘ ( s, x, y )) ds = Z t (1 + log ℘ ( s, x, y )) ∂∂s ℘ ( s, x, y ) ds = Z t (1 + log ℘ ( s, x, y ))∆ y ℘ ( s, x, y ) ds. JAELIN KIM
By applying this equation, h = lim t →∞ − t Z f M ℘ ( t, x, y ) log ℘ ( t, x, y ) d vol( y )= lim t →∞ t Z t Z f M h∇ log ℘ ( s, x, y ) , ∇ ℘ ( s, x, y ) i g d vol( y ) ds = lim t →∞ t Z t Z f M (cid:13)(cid:13)(cid:13) ∇ p ℘ ( s, x, y ) (cid:13)(cid:13)(cid:13) d vol( y ) ds ≥ t Z t λ ds = 4 λ . The inequality is due to Rayleigh’s theorem (see [7]). (cid:3)
Thermodynamic formalisms in pinched negative curvature.
We provide some general theoryof thermodynamic formalisms for geodesic flows in pinched negative curvature. Notions and detailedarguments can be found in [27]. A function F on T M is called a potential on T M if it is boundedand Hölder continuous. For a g t -invariant Borel probability measure µ , if h µ is the measure-theoreticentropy of the dynamical system ( T M , g , µ ) , we denote the pressure of F for µ by P ( F, µ ) : P ( F, µ ) = h µ + Z T M F dµ.
If we wirte P F := sup P ( F, µ ) where the supremum is taken among all g t -invariant Borel probabilitymeasure µ , An equilibrium state µ F for F is a g t -invariant Borel probability measure which attainsthe above supremum: P F = P ( F, µ F ) . Given a potential F on T M , we denote the lift to T f M by e F . We define a line integral of apotential by Z yx e F := Z d( x,y )0 e F ( g t v yx ) dt, where v yx ∈ T x f M is the unit vector at x pointing y : γ v yx (d( x, y )) = y . A Patterson-Sullivan density for F of dimension δ is a family ( µ x ) x ∈ f M of finite Borel measures absolutely continuous to each otheron ∂ f M satisfying γ ∗ µ x = µ γx ,dµ y ( ξ ) = exp ( C F − δ ( x, y, ξ )) dµ x ( ξ ) , for each x, y ∈ f M , γ ∈ Γ where C F ( x, y, ξ ) := lim z → ξ Z zy e F − Z zx e F .
We denote by µ T x the spherical measure at x , the push-forward measure of µ x via the inverse ofhomeomorphism T x f M → ∂ f M for each x ∈ f M .Let v ∈ T M with a lift e v to a vector in T f M . Define the Bowen ball around v by B (v , T, T ′ , r ) := { w ∈ T M : sup t ∈ [ − T ′ ,T ] d( γ e v ( t ) , γ e w ( t )) < r, ∃ a lift e w ∈ T f M} , One can construct a Gibbs measure from a Patterson-Sullivan density. That is, if a Patterson-Sullivandensity ( µ x ) for F of dimension P F is given, there is a g t -invariant Borel measure e µ on T f M which LT IN PINCHED NEGATIVE CURVATURE 9 is Γ -invariant and whose induced measure µ on T M has a Gibbs property (see Section 3.8 of [27]):For each compact set K ∈ T f M , there exist r > and c K,r > such that for every T, T ′ ≥ and forevery v , c − K,r exp Z T − T ′ (cid:0) F ( g t v) − P F (cid:1) dt ≤ µ ( B (v , T, T ′ , r ) ≤ c K,r exp Z T − T ′ (cid:0) F ( g t v) − P F (cid:1) dt. We call e µ the Gibbs measure of F and ( µ x ) . The Gibbs measure determines whether an equilibriumstate for F exists or not. Proposition 2.9. ( [27] ) F is Hölder continous with P F < ∞ . (1) there is a Patterson-Sullivan density ( µ x ) for F of dimension P F unique up to multiplicativeconstants. (2) If the corresponding Gibbs measure e µ F induces a finite measure µ F on T M then µ F is theunique equilibrium state for F and µ F is ergodic. Otherwise, there is no equilibrium state for F . V. Pit and B. Schapira found a necessary and sufficient condition for the finiteness of Gibbs measurein [30]. One can find the same statement also in [27].
Proposition 2.10.
A Hölder continuous potential F admits an equilibrium state if and only if for everymaximal parabolic subgroup Π of Γ , the following series converges: X γ ∈ Π d( x, γx ) exp Z γxx e F .
We have an ergodic theorem for the geodesic flow with respect to spherical measures. We alsoderive a Gibbs property for spherical measures (see [22]).
Proposition 2.11.
If a bounded Hölder continuous potential F admits an equilibrium state µ then forevery φ ∈ C b ( T M ) , x ∈ M and for µ T x -a.e. v in T M , t Z t φ ( g s v) ds → Z T M φ dµ as t → ∞ , (3) lim t →∞ − t log µ T x ( B (v , t, , ε )) = h µ for some ε > . (4) Proof.
Since µ is ergodic, the set G of the vectors for which the convergence (3) holds is a union ofstable leaves with µ ( G ) = 1 . Thus G ∩ T x M is also a µ T x M -full set. From the P F -Gibbs propertyof µ , for every v ∈ G ∩ T x M , lim t →∞ − t log µ ( B (v , t, , ε )) = P F − lim t →∞ t Z t F ( g s v) ds = P F − Z F dµ = h µ . Since the Bowen ball consists of local stable manifolds, the limit also holds for µ T x . (cid:3) There are two important potentials. The first is the zero potential, whose equilibrium state is themeasure of maximal entropy, also called the
Bowen-Margulis measure if it admits an equilibrium state.The measure class of the Patterson-Sullivan density for the zero potential is called the visibility class .The other is the geometric potential F su induces from the Γ -invariant function g F su (v) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 log det T v g t | E su (v) on f M , where T v g t : T v T f M → T g t v T f M is the tangent map of the flow map g t at v and E su (v) = T v W su (v) is the strong unstable distribution. Due to the pinched negative curvature and the uniformbound on the first derivatives of the sectional curvature, the angles between the stable leaves and thestrong unstable leaves have positive lower bound and the foliations are Hölder continuous. Thus F su isHölder continuous and the Liouville measure on T M is the equilibrium state for F su . The existencewith an assumption on the pressure of F su is proved in Chapter 7 of [27] and [32] proves that theassumption is true in our case. The measure class determined by the Patterson-Sullivan density iscalled the Lebesgue class .
3. Central limit theorem of Brownian motions
Foliated Brownian motions.
We shall introduce a Markov process on T M called the foliatedBrownian motion for the stable foliation of T M . The foliated Brownian motion was first introducedin the way to develop the ergodic theory of foliations (See [5], [12]).Fix a fundamental domain M ⊂ f M of Γ . Identify M , T M with M , M × ∂ f M , respectively.Note that f W s ( x, ξ ) = f M × { ξ } is projected onto W s ( x, ξ ) := { ( y, γ − ξ ) ∈ T M : y ∈ M , γ ∈ Γ } . The stable foliation W s = {W s (v) : v ∈ T M} of T M is the collection of the projected stableleaves. Similarly, we define the stable distribution E s of T M . The stable leaves of W s inherit theRiemannian metric from g s on the leaves of f W s which is also denoted by g s . We denote the inheriteddifferentials by div s and ∆ s . Definition 1.
Let P ( T M ) be the space of probability measures on T M . We define a transitionkernel P : (0 , ∞ ) × T M → P ( T M ) by d P [ t, v](w) = X γ ∈ Γ ℘ ( t, x, γy ) dδ γ − ξ ( η ) d vol | M ( y ) , for v = ( x, ξ ) , w = ( y, η ) ∈ T M . The transition kernel defines a unique family { P ( x,ξ ) } ( x,ξ ) ∈T M of Borel probability measures on the space T Ω := C ( R + , T M ) of sample paths on T M . Thecanonical filtration is the collection of the smallest σ -algebras F t = F t ( T M ) := σ { π s : 0 ≤ s ≤ t } for which the projections π s ( ω ) = ω s on T Ω are measurable. The canonical process Z t ( ω ) = ω t ofthe filtered space ( T Ω , { F t } ≤ t ≤∞ ) is a Markov process with respect to P ( x,ξ ) , which is called thefoliated Brownian motion for W s with initial distribution δ ( x,ξ ) , for each ( x, ξ ) ∈ T M . We define the Markov operator Q t : C b ( T M ) → C b ( T M ) on the space of bounded continuousfunctions on T M by(5) Q t f (v) := Z T M f d P [ t, v] = X γ ∈ Γ Z M f ( y, γ − ξ ) ℘ ( t, x, γy ) d vol( y ) . Note that the foliated Brownian motion for W s is the projected process of a Markov process, calledthe foliated Brownian motion for f W s , with the transition kernel(6) d e P [ t, v](w) = ℘ ( t, x, y ) dδ ξ ( η ) d vol( y ) . Let e Q t be the Markov operator on T f M . For any f ∈ C b ( T M ) and for each ( x, ξ ) ∈ M × ∂ f M , Q t f ( x, ξ ) = Z f M e f ( y, ξ ) ℘ ( t, x, y ) d vol( y ) = e Q t e f ( x, ξ ) , LT IN PINCHED NEGATIVE CURVATURE 11 where e f is the Γ -invariant lift of f to T f M . Note that the infinitesimal generator of the Markovoperator is the foliated Laplacian: ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Q t f = ∆ s f. L. Garnett proved in [12] that the Markov operator e Q admits an invariant measure m e Q on T f M ofthe form d m e Q ( x, ξ ) = dν x ( ξ ) d e m( x ) = k( x , x, ξ ) d e m( x ) dν x ( ξ ) , where e m = M ) vol and ν x is the harmonic measure. We have an induced probability measure m Q := m e Q | M × ∂ f M on T M . By Γ -equivariance of ν x , Z T M Q t f d m Q = 1vol( M ) Z M Z ∂ f M X γ Z M e f ( y, γ − ξ ) ℘ ( t, x, γy ) d vol( y ) dν x ( ξ ) d vol( x )= 1vol( M ) Z M X γ Z M Z ∂ f M e f ( y, ξ ) ℘ ( t, γ − x, y ) dν γ − x ( ξ ) d vol( x ) ! d vol( y ) . Since we know dν γ − x ( ξ ) = k( y, γ − x, ξ ) dν y ( ξ ) , the integrand in the right-handed side is: X γ Z M Z ∂ f M e f ( y, ξ ) ℘ ( t, γ − x, y ) dν γ − x ( ξ ) d vol( x )= Z ∂ f M e f ( y, ξ ) X γ Z M ℘ ( t, γ − x, y )k( y, γ − x, ξ ) d vol( x ) dν y ( ξ )= Z ∂ f M e f ( y, ξ ) Z f M ℘ ( t, x, y )k( y, x, ξ ) d vol( x ) dν y ( ξ )= Z ∂ f M e f ( x, ξ ) dν y ( ξ ) . We used the harmonicity of the Martin kernel in the last equality: Z f M ℘ ( t, x, y )k( y, x, ξ ) d vol( x ) = k( y, y, ξ ) = 1 . Therefore, we have the Q t -invariance of m Q . The stationary measure of the foliated Brownian motionis P m Q = R T M P ( x,ξ ) d m Q ( x, ξ ) and is ergodic for the shift map on T Ω .We have an integral expression of the linear drift and the stochastic entropy. Propsosition 2.9 and2.16 in [26] prove the same descriptions for the Brownian motion on co-compact negatively curvedmanifolds. The identities for co-finite manifolds follow in the same way. Proposition 3.1. ℓ ≤ h . Moreover, ℓ = Z M Z ∂ f M ∆ y b( y, x, ξ ) dν y ( ξ ) d e m( y )= Z M Z ∂ f M h−∇ y b( y, x, ξ ) , ∇ y log k( x, y, ξ ) i g dν y ( ξ ) d e m( y ) , and h = Z M Z ∂ f M |∇ y log k( x, y, ξ ) | dν y ( ξ ) d e m( y ) . Proof.
We only verify the second equality. The other equalities follow immediately from the sameargument in [26]. ℓ = Z M Z ∂ f M ∆ y b( y, x, ξ ) dν y ( ξ ) d e m( y )= Z ∂ f M Z M ∆ y b( y, x, ξ )k( x, y, ξ ) d e m( y ) dν x ( ξ )= Z ∂ f M Z M h−∇ y b( y, x, ξ ) , ∇ y k( x, y, ξ ) i g d e m( y ) dν x ( ξ )= Z M Z ∂ f M h−∇ y b( y, x, ξ ) , ∇ y log k( x, y, ξ ) i dν y ( ξ ) d e m( y ) . (cid:3) Leafwise heat equation.
We prove the contraction property on Hölder spaces of the foliatedBrownian motion. Let τ > . We define a τ -Hölder norm of f in the space C b ( T M ) of boundedcontinuous functions by k f k L τ = k f k ∞ + sup x ∈M sup ξ,η ∈ ∂ f M | e f ( x, ξ ) − e f ( x, η ) | d x,τ ∞ ( ξ, η ) , and we denote the corresponding Hölder space by L τ = { f ∈ C b ( T M ) : k f k L τ < ∞} . The following statement corresponds to the uniqueness of a Q t -invariant measure for compact nega-tively curved manifolds (see [25]). In [16], it was shown that the uniqueness for the (∆ s + Y ) -diffusionon compact negatively curved manifolds holds for a stably closed vector field Y on T M with positivepressure. Proposition 3.2.
Every Q t -invariant measure η on T M and for each f ∈ L τ , Z f dη = Z f d m Q . Proof. If η is a Q t -invariant measure on T M , its Γ -invariant lift e η to T f M is disintegrated into d e η ( x, ξ ) = d e η x ( ξ ) d e m( x ) over the fibration T f M = f M× ∂ f M ([12]). We denote by E x the expectationwith respect to P x . From the Q t -invariance, we have Z T M f dη = Z M Z ∂ f M Q t f ( x, ξ ) d e η x ( ξ ) d e m( x )= Z M Z ∂ f M Z f M ℘ ( t, x, y ) e f ( y, ξ ) d vol( y ) d e η x ( ξ ) d e m( x )= Z M E x (cid:20)Z ∂ f M e f ( e ω t , ξ ) d e η x ( ξ ) (cid:21) d e m( x ) . Note that given ε > , x ∈ M and f ∈ L τ , there is θ > such that for every y ∈ f M and ξ, η ∈ ∂ f M with ∠ y ( ξ, η ) < θ | e f ( y, ξ ) − e f ( y, η ) | < ε k η x k . LT IN PINCHED NEGATIVE CURVATURE 13 If y is distant enough from x , then due to the uniform visibility of f M , e η x (cid:8) ξ : ∠ y ( ξ, (v xy ) + ) > θ (cid:9) < ε | f | ∞ . Hence we have (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ f M e f ( y, ξ ) d e η x ( ξ ) − e f (v xy ) (cid:12)(cid:12)(cid:12)(cid:12) < ε. Since for any x ∈ M lim t →∞ E x (cid:20)Z ∂ f M e f ( e ω t , ξ ) dη x ( ξ ) (cid:21) = lim t →∞ E x h e f (v x e ω t ) i , and γ (cid:0) v xy (cid:1) = v γxγy , we have Z f dη = lim t →∞ Z M E x h e f (v x e ω t ) i d e m( x )= lim t →∞ Z M Z M X γ ∈ Γ ℘ ( t, x, γy ) e f (v xγy ) d vol( y ) d e m( x )= lim t →∞ Z M Z M X γ ∈ Γ ℘ ( t, y, γ − x ) e f (v γ − xy ) d e m( y ) d vol( x )= Z M E y h e f ( y, e ω ∞ ) i d e m( y )= Z M Z ∂ f M e f ( y, ξ ) dν y ( ξ ) d e m( y ) . Therefore, R f dη = R f d m Q . (cid:3) We denote by N the integration operator on C b ( T M ) : N ( f ) := Z T M f d m Q . The Markov operator Q t converges to N on L τ . Furthermore the following theorem shows the rate ofconvergence is exponentially fast. We postpone the proof until Section 4. Theorem 3. Q t : L τ → L τ defines a one-parameter semigroup of continuous operators for smallenough τ > . Furthermore, there is C = C ( τ ) > such that for every t > , kQ t − N k L τ ≤ e − Ct . Given f ∈ L τ , if R f d m Q = 0 , then the L τ -limit of R T Q t f dt exists by the contraction property.The limit u := lim T →∞ R T Q t f dt is a weak solution of the leafwise heat equation ∆ s u = − f , thus astrong solution in L τ . Since a leafwise harmonic u is Q t -invariant, the uniqueness also follows fromthe contraction property (See [25] for the detail). Therefore we obtain the following corollary. Corollary 1.
For small enough τ > and every f ∈ L τ with R f d m Q = 0 , there exists a solution u ∈ L τ to the leafwise heat equation ∆ s u = − f which is unique up to additive constants. In addition, u is C along the stable leaves. Let α : T M → E s ∗ be a continuous section of the dual bundle E s ∗ of the stable distribution E s of T M and e α : T f M → e E s be the lift of α . The section α is called a leafwise closed 1-form of class C if e α | f W s (v) is a closed 1-form on f W s (v) of class C for any v ∈ T f M . For each ( x, ξ ) ∈ T f M ,since f W s ( x, ξ ) = f M × { ξ } is diffeomorphic to f M , there is a 1-form e α ξ on f M which agrees withthe pull-back of e α | f W s ( x,ξ ) . Furthermore, if α is a leafwise closed 1-form of class C , then there exists A ξ ∈ C ( f M ) such that dA ξ = e α ξ . Hence if α is a leafwise closed 1-form of class C , we define foreach foliated Brownian path ω ∈ T Ω starting from ( x, ξ ) ∈ T M , Z ω t ω α := A ξ ( e ω t ) − A ξ ( e ω ) for every t ≥ , where e ω is a Brownian path on T f M such that ( e ω t , ξ ) ∈ T f M is a lift of ω t .We denote by δ s the leafwise codifferential g s -dual to − div s , that is, δ s α = − div s α where α : T M → E s is the continuous section g s -dual to α . Since δ s e α ( x, ξ ) = − div s e α ( x, ξ ) = − div ∇ A ξ ( x ) = − ∆ A ξ ( x ) , by Itô’s formula (see Chapter 3 of [18]),(7) X t ( ω ) = Z ω t ω α + Z t δ s α ( ω r ) dr = A ξ ( e ω t ) − A ξ ( e ω ) − Z t ∆ A ξ ( e ω r ) dr is a martingale on ( T Ω , { F t ( T M ) } ≤ t ≤∞ , P m Q ) having the quadratic variation d h X , X i t ( ω ) = (∆( A ξ ) − A ξ ∆ A ξ )( e ω t ) dt = 2 k α ( ω t ) k dt. If β is a leafwise closed 1-form of class C such that δ s β is Hölder continuous on T M , applyingCorollary 1, there is u ∈ L τ such that ∆ s u = δ s β − R δ s βd m Q . Hence, due to the equation (7) for α = β + du , we have a martingale(8) X t = Z ω t ω ( β + du ) + Z t δ s ( β + du )( ω r ) dr = Z ω t ω β + u ( ω t ) − u ( ω ) + t Z δ s βd m Q with the quadratic variation h X , X i t ( ω ) = 2 R t k α + ∇ u k ( ω t ) ds .3.3. Proof of Theorem 1.
For ( x, ξ ) ∈ T f M , let B ( x, ξ ) := b( x, x , ξ ) , K ( x, ξ ) := log k( x , x, ξ ) .Note that ∆ s B ( x, ξ ) = ∆ x b( x, x , ξ ) is Hölder continuous due to uniform bounds of the first derivatives of curvature. On the other hand, ∆ s K ( x, ξ ) = k∇ x log k( x , x, ξ ) k is Hölder contunous due to [2], [15]. By Corollary 1 for f = ∆ s B , ∆ s K there exist u b , u k ∈ L τ forwhich we obtain square-integrable martingales B t ( ω ) = b( e ω t , e ω , ξ ) − tℓ + u b ( ω t ) − u b ( ω ) , K t ( ω ) = log k( e ω , e ω t , ξ ) + th + u k ( ω t ) − u k ( ω ) , for ω ∈ T Ω with a lift ( e ω, ξ ) ∈ T f M , by the Itô formula (8) for β ( x,ξ ) = dB ξx , dK ξx , respectively.Their quadratic variations are h B , B i t ( ω ) = 2 Z t k∇ B + ∇ u b k ( ω s ) ds, h K , K i t ( ω ) = 2 Z t k∇ K + ∇ u k k ( ω s ) ds. (9) LT IN PINCHED NEGATIVE CURVATURE 15
We denote by E ( x,ξ ) the expectation with respect to P ( x,ξ ) . From the equalities (9) of quadraticvariations, E ( x,ξ ) (cid:20) t h B , B i t ( ω ) (cid:21) = 2 t Z t Q s k∇ B + ∇ u b k ( x, ξ ) ds, E ( x,ξ ) (cid:20) t h K , K i t ( ω ) (cid:21) = 2 t Z t Q s k∇ K + ∇ u k k ( x, ξ ) ds. Due to the ergodicity of m Q , for m Q -a.e. ( x, ξ ) , lim t →∞ E ( x,ξ ) (cid:20) t h B , B i t ( ω ) (cid:21) = 2 Z k∇ B + ∇ u b k d m Q , (10) lim t →∞ E ( x,ξ ) (cid:20) t h K , K i t ( ω ) (cid:21) = 2 Z k∇ K + ∇ u k k d m Q . (11)Using Markov property, we have E ( x,ξ ) (cid:20) t + 1 h M , M i t +1 (cid:21) = E ( x,ξ ) (cid:20) tt + 1 E ω (cid:20) t h M , M i t (cid:21)(cid:21) = tt + 1 E ( x,ξ ) (cid:20) t Z t Q r F ( ω ) dr (cid:21) . for M = B or K and F = 2 k∇ B + ∇ u b k or k∇ K + ∇ u k k , respectively. Given x ∈ f M , for ν x -a.e. ξ and P ( x,ξ ) -a.e. ω , lim t →∞ t Z t Q r F ( ω ) dr = Z F d m Q . Hence for each x , there is ξ for which we have the limits (10) and (11). We denote the square root of thelimits by σ b and σ k , respectively. Note that both of σ b , σ k are positive since B and K are unboundedwhile u b and u k are bounded. We have σ b , σ k < ∞ since both of k∇ B + ∇ u b k or k∇ K + ∇ u k k are bounded. Thus for every x , there is ξ such that the distributions of B t σ b √ t and K t σ k √ t under P ( x,ξ ) converge to N (0 , as t → ∞ due to the following lemma : Lemma 3.1. ( [17] ) Let ( M t ) ≤ t ≤∞ be a continuous, centered, square-integrable martingale on afiltered probability space with stationary increments. If M = 0 and there is σ > such that lim t →∞ E [ | t h M, M i t − σ | ] = 0 , then the distribution of σ √ t M t is asymptotically normal. Let W ℓt ( ω ) := d( e ω , e ω t ) − tℓ . Since the distribution of W ℓt under P ( x,ξ ) and the distribution of Y ℓt under P x coincide, it is enough to show that W ℓt and B t have the same P ( x,ξ ) -distribution. For P ( x,ξ ) -a.e. ω and a lift ( e ω, ξ ) , since B ( ω t ) − B ( ω ) − d( e ω , e ω t ) = b( e ω t , e ω , ξ ) − d( e ω , e ω t ) → − ξ | e ω ∞ ) e ω and | ( ξ | e ω ∞ ) e ω | < ∞ , lim t →∞ σ b √ t [ B ( ω t ) − B ( ω ) − d( e ω , e ω t )] = 0 . Hence the distribution of σ b √ t W ℓt under P ( x,ξ ) also converges to the normal distribution since W ℓt ( ω ) = [d( e ω , e ω t ) − B ( ω t ) + B ( ω )] − [ u b ( ω t ) − u b ( ω )] + B t ( ω ) , and σ b √ t | u b ( ω t ) − u b ( ω ) | ≤ σ b √ t k u b k ∞ → , as t → ∞ . Let W ht ( ω ) := log G( e ω , e ω t ) + th . Since the P ( x,ξ ) -distribution of W ht and the P x -distributionof Y ht are the same, to verify that σ k √ t W ht is asymptotically normal, it is sufficient to show that for P ( x,ξ ) -a.e. ω with a lift ( e ω, ξ ) to T f M ,(12) lim sup t →∞ | log G( e ω , e ω t ) − K ( ω t ) + K ( ω ) | < ∞ . Note that for P ( x,ξ ) -a.e. ω with a lift ( e ω, ξ ) , K ( ω t ) − K ( ω ) = log k( e ω , e ω t , ξ ) and e ω ∞ = ξ . Wedenote by z t the closest point to e ω on the geodesic ray [ e ω t , ξ ) generated by ( e ω t , ξ ) , z t converges toa point z ∞ ∈ f M on the geodesic ( e ω ∞ , ξ ) joining two boundary points e ω ∞ and ξ . We have that forevery y on [ e ω t , ξ ) , | log G( e ω , e ω t ) − log k( e ω , e ω t , ξ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) log G( e ω , e ω t )G( z t , e ω t ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log G( z t , e ω t ) (cid:16) G( y, e ω t )G( y,z t ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:16) G( y, e ω t )G( y,z t ) (cid:17) k( e ω , e ω t , ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Applying the Harnack inequality to the first term in the right handed side, since { d( e ω , z t ) } t ≥ isbounded, it follows that (cid:12)(cid:12)(cid:12) log G( e ω , e ω t )G( z t , e ω t ) (cid:12)(cid:12)(cid:12) ≤ C for some constant C = C ( e ω ) > dependent of e ω butnot t . And by the Ancona inequality ([1]), the second term in the right handed side is also boundedby C ( e ω ) . Letting y tend to ξ , we see that the last term converges to (cid:12)(cid:12)(cid:12) log k( z t , e ω t ,ξ )k( e ω , e ω t ,ξ ) (cid:12)(cid:12)(cid:12) which is alsobounded by C ( e ω ) due to the Harnack inequality. Therefore we have (12) and this completes the proofof Theorem 1.
4. Proof of Theorem 3
In this section, we prove the contraction property on Hölder spaces of the foliated Brownian motion.For the Hölder semi-norm, we prove a lower bound of the expectation of the Busemann functions atBrownian points which depends only on the dimension and the curvature bounds and linearly on time T . The lower bound follows from the fact that the Laplacian of the Busemann function has the samelower bound with the Laplacian of the distance function due to the Rauch comparison theorem. Wealso show the Doeblin property of the Brownian motion for the estimate of the uniform norm. Proposition 4.1.
For sufficiently small τ , there exists C > such that for each t > , sup x ∈M sup ξ,η ∈ ∂ f M |Q t f ( x, ξ ) − Q t f ( x, η ) | d x,τ ∞ ( ξ, η ) ≤ k f k L τ e − C t . Proof.
Since we have that |Q t f ( x, ξ ) − Q t f ( x, η ) | d x,τ ∞ ( ξ, η ) < Z f M ℘ ( t, x, y ) (cid:12)(cid:12)(cid:12) e f ( y, ξ ) − e f ( y, η ) (cid:12)(cid:12)(cid:12) d x,τ ∞ ( ξ, η ) d vol f M ( y ) < k f k L τ Z f M ℘ ( t, x, y ) d y,τ ∞ ( ξ, η ) d x,τ ∞ ( ξ, η ) d vol f M ( y )= k f k L τ E x (cid:20) d y,τ ∞ ( ξ, η ) d x,τ ∞ ( ξ, η ) (cid:21) , it is sufficient to find C > such that sup x,ξ,η E x " d e ω t ,τ ∞ ( ξ, η ) d x,τ ∞ ( ξ, η ) < e − C t . LT IN PINCHED NEGATIVE CURVATURE 17
Due to the Markov property of the Brownian motion, sup x,ξ,η E x " d e ω t + s ,τ ∞ ( ξ, η ) d x,τ ∞ ( ξ, η ) = sup x,ξ,η E x " d e ω s ,τ ∞ ( ξ, η ) d x,τ ∞ ( ξ, η ) E x " d e ω t + s ,τ ∞ ( ξ, η ) d e ω s ,τ ∞ ( ξ, η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F s ( f M ) ≤ sup x,ξ,η E x " d e ω t ,τ ∞ ( ξ, η ) d x,τ ∞ ( ξ, η ) sup x,ξ,η E x " d e ω s ,τ ∞ ( ξ, η ) d x,τ ∞ ( ξ, η ) . Let us write g ( e ω t ) := ( ξ | η ) e ω t − ( ξ | η ) x . Applying the Taylor theorem to the function R exp( − τ R ) and substituting g ( e ω t ) for R , we have d e ω t ,τ ∞ ( ξ, η ) d x,τ ∞ ( ξ, η ) ≤ − τ g ( e ω t ) + τ d( x, e ω t ) e τ d( x, e ω t ) . By Proposition 2.3, for some constant C ′ > ,(13) sup x E x h d( x, e ω t ) e τ d( x, e ω t ) i < C ′ . Therefore, with (13) and Lemma 4.1 below, we have sup ≤ t ≤ T sup x,ξ,η E x " d e ω t ,τ ∞ ( ξ, η ) d x,τ ∞ ( ξ, η ) ≤ − τ ( d − a + τ C ′ . Fix T ≥ and sufficiently small τ such that − τ ( d − a + τ C ′ < . For such small τ , put C = (1 − a ( d − τ + C ′ τ ) T and the inequality follows. (cid:3) Lemma 4.1.
For every T ≥ , inf x ∈M inf ξ = η E x [( ξ | η ) e ω T − ( ξ | η ) x ] ≥ ( d − aT. Proof of Lemma 4.1.
Due to the equation ( ξ | η ) x − ( ξ | η ) y = 12 b( x, y, ξ ) + 12 b( x, y, η ) , it suffices to show that E x [b( e ω T , x, η )] ≥ ( d − aT. Choose z n ∈ f M such that z n → ξ as n → ∞ and write f n ( y ) := b( y, x, z n ) = d( y, z n ) − d( x, z n ) . By the Rauch’s comparison theorem (see [28], for instance), ∆ f n ( y ) = ∆ y d( y, z n ) ≥ ( d − sn ′− a (d( y, z n )) sn − a (d( y, z n ))= a ( d −
1) coth ( a d( y, z n )) (14)where sn − a ( t ) = a sinh( at ) .Let f ( y ) = b( y, x, ξ ) . Then, since ∆ is the generator of Q , E x [b( e ω T , x, ξ )] = Q T f ( x ) = Z T Q t ∆ f ( x ) dt = Z T E x [∆b( e ω t , x, ξ )] dt. Due to (14) and Proposition 2.2, E x [b( e ω T , x, ξ )] ≥ ( d − aT for every x ∈ f M and every ξ ∈ ∂ f M . (cid:3) Write P ( t, x, y ) = P γ ∈ Γ ℘ ( t, x, γy ) for x, y ∈ M . We have lim t →∞ P ( t, x, y ) = M ) (see [6]).In particular, P ( t, x, x ) decreases as t → ∞ . We also have that (cid:18)Z M (cid:12)(cid:12)(cid:12)(cid:12) P ( t, x, y ) − M ) (cid:12)(cid:12)(cid:12)(cid:12) d vol( y ) (cid:19) ≤ vol( M ) Z M (cid:12)(cid:12)(cid:12)(cid:12) P ( t, x, y ) − M ) (cid:12)(cid:12)(cid:12)(cid:12) d vol( y ) (15) = vol( M ) (cid:18) P (2 t, x, x ) − M ) (cid:19) . (16)Hence the integral on the left-handed side decreases to zero as t goes to infinity. Indeed, it decaysexponentially fast (see [9]). The following lemma shows that it has uniform exponential decay rate. Lemma 4.2.
There exists a constant C = C ( d, b ) > such that for each x ∈ M , Z M (cid:12)(cid:12)(cid:12)(cid:12) P ( t, x, y ) − M ) (cid:12)(cid:12)(cid:12)(cid:12) d vol( y ) ≤ Ce − λ t , where λ = inf { λ > λ ∈ Spec(∆ M ) } . Remark 1.
Since the bottom of the ( L -)esssential spectrum λ ess := inf Spec ess (∆ M ) of the Lapla-cian is positive ( [8] ) and Spec (∆ M ) ∩ [0 , λ ess ) is discrete ( [9] ), the smallest nonzero the spectrum λ is also positive.Proof. If we consider P t f ( x ) := R ( P ( t, x, y ) − vol( M ) − ) f ( y ) d vol( y ) as an operator acting on thespace L ( M ) of square-integrable functions with zero integral, ∆ | L ( M ) is the generator of P t withthe bottom of the spectrum λ . Therefore the operator norm satisfies(17) kP t k ≤ e − λ t for every t > (see the proof of Proposition V.1.2 in [10]).For every x ∈ M , if we denote f t ( y ) = P ( t, x, y ) − M ) , then f t + t ( y ) = P t f t ( y ) . It followsfrom (15) and (17) that Z M (cid:12)(cid:12)(cid:12)(cid:12) P ( t + t , x, y ) − M ) (cid:12)(cid:12)(cid:12)(cid:12) d vol( y ) ≤ (cid:18) vol( M ) Z M | f t + t ( y ) | d vol( y ) (cid:19) / ≤ kP t k k f t k ≤ e − λ t | f t ( x ) | / = e − λ t (cid:12)(cid:12)(cid:12)(cid:12) P (2 t , x, x ) − M ) (cid:12)(cid:12)(cid:12)(cid:12) / . Thus it suffices to prove that the diagonal supremum sup x ∈M P (2 t , x, x ) of the heat kernel on M isfinite for some t > .Recall that we identify the fundamental domain M with M and P ( t, x, y ) = P γ ∈ Γ ℘ ( t, x, γy ) .In order to estimate the diagonal supremum sup x ∈M P ( t, x, x ) of the heat kernel on M , we shall usethe Gaussian upper bound of the heat kernel on f M ([13]): there is a constant C = C ( d, b ) s.t.(18) ℘ ( t, x, y ) ≤ C (cid:18) x, y ) t (cid:19) d exp (cid:18) − d( x, y ) t − λ t (cid:19) . LT IN PINCHED NEGATIVE CURVATURE 19
Fix x ∈ M . For a cuspidal point ξ ∈ Π( M ) := ∂ f M ∩ M , we denote the horoball of level n based at ξ by H ( n, ξ ) := { y ∈ M : n ≤ b( x , y, ξ ) } . If γ is in the stabilizer Γ ξ of ξ and we choosea point x n in H ( n, ξ ) \ H ( n + 1 , ξ ) , then x and γx are in the horosphere of the same level based at ξ . This implies that(19) e − b ( n +1) d( x , γx ) ≤ d( x n , γx n ) ≤ e − an d( x , γx ) . Applying (19) to the Gaussian bound (18),(20) ℘ ( t, x n , γx n ) ≤ Ce − λt d( x , γx ) d +2 exp (cid:18) d( x , γx ) te b ( d − n +1) − a ( d + 2) n (cid:19) . Note that d( x n , γx n ) is increasing as n → ∞ unless γ ∈ Γ ξ . We want to show that given δ > , thereis t > such that for every n large enough,the right-hand side of (20) ≤ e − δ d( x ,γx ) . To simplify the notation, we put f n,ξ ( R ) := R d +2 exp (cid:18) − R te b ( d − n +1) + δR (cid:19) . Since its derivative is f ′ n,ξ ( R ) = R d +1 (cid:18) d + 2 − R t e − b ( d +2)( n +1) + δR (cid:19) exp (cid:18) − R t e − b ( d +2)( n +1) + δR (cid:19) , the positive nonzero extreme point of f n,ξ is R n := tδe b ( n +1) + p t δ e b ( n +1) + 2 t ( d + 2) e b ( n +1) .Thus f n,ξ has the maximum on R + at R n : f n,ξ ( R ) ≤ f n,ξ ( R n )= R d +2 n exp (cid:18) − R n t + δR n − a ( d + 2) n (cid:19) ≤ (cid:16) tδe b ( n +1) (cid:17) d +2 exp − tδ e b ( n +1) tδ e b ( n +1) − a ( d + 2) n ! = h (3 tδ ) e b ( n +1) − an i d +2 e − δ t exp (cid:18) − δ t e b ( n +1) − (cid:19) . Therefore, there is N ξ ( δ ) such that f n,ξ ( R ) ≤ C − t − − d e λt , hence ℘ ( t, x n , γx n ) ≤ e − δ d( x ,γx ) whenever n > N ξ ( δ ) . We conclude that X γ ∈ Γ ξ ℘ ( t, x n , γx n ) ≤ X γ ∈ Γ ξ e − δ d( x ,γx ) = Q Γ ξ ,x ( δ ) , where Q G,x ( δ ) := P g ∈ G e − δ d( x,gx ) denotes the Poincaré series of a discrete group G of isometrieson f M . We denote the abscissa of convergence of Q G,x , which is called the critical exponent of G , by δ G .Put N ξ := N ξ (cid:0) δ Γ ξ + 1 (cid:1) and let N := max ξ ∈ Π( M ) N ξ , M N := M \ [ ξ ∈ Π( M ) H ( N ξ , ξ ) . Note that M N is a pre-compact domain. Take x ∈ M N . Since d( x, γx ) ≥ d( x , γx ) when x ∈ ∪ ξ ∈ Π( M ) H ( N ξ , ξ ) P ( t, x, x ) ≤ C X γ ∈ Γ (cid:18) d( x, γx ) t (cid:19) d exp (cid:18) − d( x, γx ) t − λt (cid:19) ≤ C X γ / ∈∪ ξ ∈ Π( M Γ ξ (cid:18) d( x , γx ) t (cid:19) d exp (cid:18) − d( x , γx ) t − λt (cid:19) + C X γ ∈∪ ξ ∈ Π( M Γ ξ Q Γ ξ ,x ( δ Γ ξ + 1) ≤ CQ Γ ,x ( δ Γ + 1) < ∞ . Therefore, sup x ∈M P ( t, x, x ) ≤ max { sup x ∈M N P ( t, x, x ) , C ′ Q Γ ,x ( δ + 1) } . (cid:3) We are ready to verify the exponential decay of uniform norm and complete the proof of Theorem3. It is enough to show that the exponential decay of the supremum norm since we have already provedthe exponential decay of Hölder norm in Proposition 4.1.
Proposition 4.2.
There exists a constant C > such that for every f ∈ L , t > kQ t f − N f k ∞ ≤ k f k τ e − C t . Proof.
Denote F t ( x ) := R Q t f ( x, ξ ) dν x ( ξ ) . (cid:12)(cid:12)(cid:12)(cid:12) Q t f ( x, ξ ) − Z f d m Q (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Q t f ( x, ξ ) − Z Q t f d m Q (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Q t f ( x, ξ ) − Q t F t ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Q t F t ( x ) − Z Q t f d m Q (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Q t (cid:16) Q t f ( x, ξ ) − F t ( x ) (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Q t F t ( x ) − Z Q t f d m Q (cid:12)(cid:12)(cid:12)(cid:12) . By Lemma 4.2, the last term of the last inequality decays exponentially: (cid:12)(cid:12)(cid:12)(cid:12) Q t F t ( x ) − Z Q t f d m Q (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z M P ( t/ , x, y ) F t ( y ) d vol( y ) − Z M F t ( y ) d e m ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k F t k ∞ Z M (cid:12)(cid:12)(cid:12)(cid:12) P ( t/ , x, y ) − M ) (cid:12)(cid:12)(cid:12)(cid:12) d vol( y ) ≤ k f k τ e − λ t . For the first term, it follows from Proposition 4.1 that (cid:12)(cid:12)(cid:12) Q t (cid:16) Q t f ( x, ξ ) − F t ( x, ξ ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ sup y ∈M (cid:12)(cid:12)(cid:12) Q t f ( y, ξ ) − F t ( y, ξ ) (cid:12)(cid:12)(cid:12) ≤ sup y ∈M Z (cid:12)(cid:12)(cid:12) Q t f ( y, ξ ) − Q t f ( y, η ) (cid:12)(cid:12)(cid:12) dν y ( η ) ≤ k f k τ e − C t . (cid:3) LT IN PINCHED NEGATIVE CURVATURE 21
5. Ergodic properties of Brownian motions
In this section, we discuss the thermodynamic formalisms for the harmonic potential, which arisesfrom the Brownian motion and an equidistribution theorem of Brownian paths. Using such ergodicproperties of the Brownian motion, we also provide a characterization of the asymptotic harmonicityas an application of the central limit theorem to the ergodic theory of the geodesic flow on M .5.1. Harmonic potentials.
We introduce another natural potential F BM on T M induced from theBrownian motion, which we call the harmonic potential . Define a function ] F BM on T f M by ] F BM (v) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 log k( γ v (0) , γ v ( t ) , v + ) where v + denote the end point at infinity lim t →∞ γ v ( t ) of the geodesic γ v generated by v and k( x, y, ξ ) is the Martin kernel of the Brownian motion on f M . Since ] F BM is a Γ -invariant Hölder continuousfunction on T f M ([15]), it induces a Hölder potential, which is denoted by F BM , on T M .Note that F BM has the harmonic measure ( ν x ) x ∈ f M as a Patterson-Sullivan density of dimension .Since the harmonic measure does not have atom, the topological pressure of F BM vanishes; P F BM = 0 .We denote by e ν the Gibbs measure on T f M of F BM and ( ν x ) . Proposition 2.9 for F BM demonstratesthat F BM admits an equilibrium state ν on T M for F BM if and only if e ν (cid:0) T M (cid:1) is finite and ν agrees with the induced measure on T M by e ν . From Proposition 2.10 it follows that F BM admits anequilibrium state if and only if for every parabolic subgroup Π of Γ , X γ ∈ Π d( x, γx )k( x, γx, (v γxx ) + ) < ∞ , where v yx ∈ T x f M such that g d( x,y ) v yx ∈ T y f M . We shall provide dynamical aspects of Brownianmotions using the ergodic theory of ν .Recall that given x ∈ f M we identify ( r, v) ∈ (0 , ∞ ) × T x f M with exp x ( r v) ∈ f M \ { x } and g = dr + λ x ( r, v) g S . Now we denote the density of volume at z = ( r, v) with respect to the polarcoordinate at x by A x ( z ) : d vol( z ) = A x ( z ) drd vol S (v) . Note that A x ( z ) = λ d − x ( r, v) . Recall that we denote by θ ( e ω, t ) the unit vector in T x f M such that e ω t = ( r, θ )( e ω, t ) := ( r ( e ω, t ) , θ ( e ω, t )) . Then the following proposition means that Brownian paths areequidistributed with respect to ν . The proof follows the argument for compact manifolds ([22]). Proposition 5.1.
Assume that F BM admits an equilibrium state ν . For every x ∈ f M , for each boundedcontinuous function φ ∈ C b ( T M ) and for P x -a.e. e ω , Z φ dν = lim t →∞ ℓt Z r ( e ω,t )0 e φ ( g s θ ( e ω, t )) ds. Proof.
For v , w ∈ T f M , let d t (v , w) be the distance on the geodesic sphere S ( x, t ) between g t v and g t w . Then d t (v , w) ≤ d s (v , w) sinh( at )sinh( as ) for every < t < s due to the curvature upper bound sec f M ≤ − a < . Since the Sasaki distance isHölder equivalent to the distance d (v , w) := sup ≤ t ≤ d ( γ v ( t ) , γ w ( t )) , (cid:12)(cid:12)(cid:12)(cid:12)Z t e φ ◦ g s (v) ds − Z t e φ ◦ g s (w) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( a, φ )d( γ v ( t ) , γ w ( t )) . Hence the proposition follows from of Proposition 2.11 and the limit (2): for P x -a.e. e ω , lim t →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓt Z r ( e ω,t )0 e φ ( g s θ ( e ω, t )) ds − Z φdν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim t →∞ ℓt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z r ( e ω,t )0 e φ ( g s θ ( e ω, t ) ds − Z ℓt e φ ( g s θ ( e ω, ∞ )) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + lim t →∞ (cid:12)(cid:12)(cid:12)(cid:12) ℓt Z ℓt e φ ( g s θ ( e ω, ∞ )) ds − Z φdν (cid:12)(cid:12)(cid:12)(cid:12) = lim t →∞ ℓt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z r ( e ω,t )0 e φ ( g s θ ( e ω, t )) ds − Z r ( e ω,t )0 e φ ( g s θ ( e ω, ∞ )) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 0 ≤ lim t →∞ C ( a, φ ) ℓt d( e ω t , ( r, θ )( e ω, t )) = 0 . We used (3) of Proposition 2.11 in the equation and (2) in the last inequality. (cid:3)
The equidistribution of Brownian paths provides another stochastic invariant, which helps under-standing the relation between the harmonic measure class and the Lebesgue measure class. The proofin [22] extends to the finite-volume case.
Theorem 4.
For each x ∈ f M and for P x, -a.e. e ω , if F BM admits an equilibrium state ν , the followinglimit exists: Υ = lim t →∞ t log A ( x, e ω t ) . Moreover,
Υ = ℓ Z F su dν ≥ h, and the equality holds if and only if the harmonic measure class and the Lebesgue class agree.Proof. Let T v g t be the tangent map of the flow map g t at v = ( x, ξ ) ∈ T f M . Since the angle betweenstable distribution E s (v) and T v g t ( T v T x f M ) , where T v T x f M is the tangent space of the sphere T x f M ,is bounded away from zero uniformly on v and t > , lim t →∞ t log A ( π v , π g t v) = lim t →∞ t log det T g t | T v T x f M = lim t →∞ t log det T g t | E uu (v) = lim t →∞ − t Z t F su ( g s v) ds. Therefore, by Proposition 5.1, for P x -a.e. e ω , lim t →∞ t log A ( x, e ω t ) = lim t →∞ − t Z r ( e ω,t )0 F su ( g s θ ( e ω t )) ds = − ℓ Z F su dν. LT IN PINCHED NEGATIVE CURVATURE 23
Thus, combining with Theorem 5, the equality
Υ = h is equivalant to P ( F su , ν ) = h ν + Z F su dν = 0 , which holds if and only if ν is the equilibrium state for F su . (cid:3) The following theorem demonstrates how dynamical invariants and stochastic invariants are relatedto each other. We follow the argument in [22], but we complete the proof by showing the inequality h ≤ ℓh ν using the idea in [21] Theorem 5. If F BM admits an equilibrium state ν , then h = ℓh ν . In particular, if we denote the topological entropy of ( T M , ( g t )) by h top , then h ≤ ℓh top , and the equality holds if and only if the harmonic measure class and the visibility class coincide.Proof. Let x ∈ f M , δ ∈ (cid:0) , (cid:1) and < ε, ε ′ . We denote for each T > , C T := { e ω : d( e ω T , ( ℓT, e ω ∞ )) ≤ εT and µ T x { v : d ℓT (v , θ ( e ω, ∞ )) ≤ ε ′ } ≤ e − ( ℓh ν − ε ) T } , D T := { e ω :d( e ω T , ( ℓT, e ω ∞ )) ≤ εT and µ T x { v : d (cid:0) γ v ( ℓT ) , γ θ ( e ω, ∞ ) ( ℓT ) (cid:1) ≤ ε ′ } ≥ e − ( ℓh ν + ε ) T } . For every T large enough, P x ( C T ) ≥ δ for some ε ′ > by (2). Thus if we fix a sufficiently large T and choose E ⊂ f M with Card E = N ( x, T, − δ ) , P x { d( e ω T , E ) ≤ } ≥ − δ. We note that E ∞ := { θ ( e ω, ∞ ) : e ω ∈ C T , d( e ω T , E ) ≤ } has the µ T x -measure greater than δ and { γ θ ( e ω, ∞ ) ( ℓT ) : e ω ∈ C T , d( e ω T , E ) ≤ } is covered by balls on the sphere of radius ε ′ less than N ( x, T, − δ ) C εT . ( C is the maximal cardinal of covers for the intersection of the sphere of radius ℓT and ( ε + 1) T balls by ε ′ balls on the sphere.) Such ball O in the sphere of radius ε ′ is the set ofbase points of vectors in g ℓT V where V = { v : d ℓT (v , w) ≤ ε ′ } for some w . We conclude that sincesuch V has the µ T M -measure less than e − ( ℓh ν − ε ) T , δ ≤ µ T x ( E ∞ ) ≤ N ( x, T, − δ ) e − T [ ℓh ν − ε − ε log C ] . Thus we have ℓh ν ≤ lim T →∞ T log N ( x, T, − δ ) .Choose a smallest set E ⊂ f M such that d( e ω T , E ) ≤ for each e ω ∈ D T and a maximal ε ′ -separetedset F ⊂ { γ θ ( e ω, ∞ ) ( ℓT ) : e ω ∈ D T } . Since D T ⊂ { e ω : d( e ω T , E ) ≤ } , Card( E ) ≥ N ( x, T, P x ( D T )) and Card( F ) ≤ C ′ e ℓh ν T + εT . ( C ′ is the maximal number of overlappings.) For every f ∈ F if wedenote N ( f ) := { e ∈ E : ∃ e ω ∈ D T s.t. d( f, γ θ ( e ω, ∞ ) ( ℓT )) ≤ ε ′ , d( e, e ω T ) ≤ } . Since ∪ e ∈ N ( f ) B ( e, ⊂ B ( f, εT + ε ′ + 1) , there exists C ′′ > such that Card N ( f ) ≤ sup e ∈ E,f ∈ F vol ( B ( f, εT + ε ′ + 1))vol( B ( e, ≤ e C ′′ εT . Therefore, N ( x, T, P x ( D T )) ≤ Card( E ) ≤ exp( C ′′ εT )Card F ≤ e T [ ℓh ν +(1+ C ′′ ) ε ] . (cid:3) Proof of Theorem 2.
We conclude this section with the proof of Theorem 2. We begin with theproof of the integral equation for the foliated Laplacian ([33]): for every bounded function ϕ uniformly C on stable leaves,(21) Z h∇ log k , ∇ ϕ i d m Q = − Z ∆ s ϕd m Q . Consider the function Φ( y ) := R ∂ f M ϕ ( y, ξ ) dν y ( ξ ) = R ∂ f M ϕ ( y, ξ )k( x, y, ξ ) dν x ( ξ ) . Applying theLaplacian, since ∆ y k( x, y, ξ ) = 0 we have ∆Φ( y ) = Z ∂ f M k( x, y, ξ )∆ s ϕ ( y, ξ ) + 2 h∇ y ϕ ( y, ξ ) , ∇ y k( x, y, ξ ) i dν x ( ξ )= Z ∂ f M ∆ s ϕ ( y, ξ ) + 2 h∇ y ϕ ( y, ξ ) , ∇ y log k( x, y, ξ ) i dν y ( ξ ) . Thus integrating with respect to vol and using Green’s formula, Z T M ∆ s ϕ ( y, ξ ) + 2 h∇ y ϕ ( y, ξ ) , ∇ y log k( x, y, ξ ) i d m Q ( y, ξ ) = Z M ∆Φ d vol( x ) = 0 . From the integral formula (21) for the foliated Laplacian, it follows that σ = 2 Z T M k∇ log k( x, · , ξ ) + ∇ u k k d m Q = 2 Z k∇ log k k + k∇ u k k d m Q , since R ∆ s u k d m Q = 0 . Since R k∇ log k k d m Q = h (Proposition 3.1), σ − h = Z k∇ u k k d m Q ≥ , and the equality holds if and only if u k is constant. If u k is constant then −k∇ log k k + Z k∇ log k k d m Q = ∆ u k = 0 . Thus k∇ log k k = h is constant, which implies the asymptotic harmonicity of M : h ≤ ℓh top ≤ − ℓ Z F BM dν = ℓ Z h X, ∇ log k i dν ≤ ℓ (cid:12)(cid:12)(cid:12)(cid:12)Z k∇ log k k dν (cid:12)(cid:12)(cid:12)(cid:12) / = ℓ √ h ≤ h, Where X ( x, ξ ) := ( x, ξ ) . This completes the proof of Theorem 2. References [1] A. Ancona. Negatively curved manifolds, elliptic operators, and the martin boundary.
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