aa r X i v : . [ m a t h . F A ] J un Chernoff ’s theoremfor evolution families
Evelina Shamarova ∗ December 11, 2018
Abstract
A generalized version of Chernoff’s theorem has been obtained.Namely, the version of Chernoff’s theorem for semigroups obtainedin a paper by Smolyanov, Weizs¨acker, and Wittich [1] is generalizedfor a time-inhomogeneous case. The main theorem obtained in thecurrent paper, Chernoff’s theorem for evolution families, deals witha family of time-dependent generators of semigroups A t on a Banachspace, a two-parameter family of operators Q t,t +∆ t satisfying the re-lation: ∂∂ ∆ t Q t,t +∆ t (cid:12)(cid:12) ∆ t =0 = A t , whose products Q t i ,t i +1 . . . Q t k − ,t k areuniformly bounded for all subpartitions s = t < t < · · · < t n = t .The theorem states that Q t ,t . . . Q t n − ,t n converges to an evolutionfamily U ( s, t ) solving a non-autonomous Cauchy problem. Further-more, the theorem is formulated for a particular case when the gen-erators A t are time dependent second order differential operators. Fi-nally, an example of application of this theorem to a construction oftime-inhomogeneous diffusions on a compact Riemannian manifold isgiven. Keywords:
Chernoff’s theorem, evolution family, strongly continuoussemigroup, evolution families generated by manifold valued stochasticprocesses. ∗ Institute for mathematical Methods in Economics, Vienna University of TechnologyThis work was supported by the research grant of the Erwin Schr¨odinger Institute formathematical physics, and by the Austrian Science Fund (FWF) under START-prize-grantY328.Email:
[email protected] Chernoff ’s theorem for evolution families
Let A t , t ∈ [ S, T ] ⊂ R + ∪{ } , be generators of strongly continuous semigroupson a Banach space E . Let D ( A t ) denote the domain of A t . We assume thatthere exists a Banach space Y ⊂ ∩ t ∈ [ S,T ] D ( A t ), which is dense in E .Given a t ∈ ( S, T ], and an x ∈ Y , we consider a non-autonomous Cauchyproblem on the interval [ S, t ] with the final condition x : ( ˙ u ( s ) = − A s u ( s ) u ( t ) = x. (1)Let D [ S,T ] = { ( s, t ) : s t, s ∈ [ S, T ] , t ∈ [ S, T ] } . The evolution family U ( s, t ), ( s, t ) ∈ D [ S,T ] , solving the Cauchy problem (1) satisfies the relation: U ( s, r ) U ( r, t ) = U ( s, t ) (2)for all s r t (see [5], Chapter VI, paragraph 2). Theorem . Let A t be genera-tors of strongly continuous semigroups, Q t ,t , t , t > , be a two-parameterfamily of bounded operators E → E , and U ( s, t ) , S s t T , be anevolution family of operators with the generators A t (see [5], Chapter VI,paragraph 2). We assume that the following assumptions are fulfilled:1) there exists a Banach space Y ⊂ ∩ t ∈ [ S,T ] D ( A t ) which is dense in E , invariant under the action of U ( s, t ) for all ( s, t ) ∈ D [ S,T ] , i.e. U ( s, t ) Y ⊂ Y , and such that the Cauchy problem (1) is well-posed(backward solvable) for all x ∈ Y ;2) the function [ S, t ] → E , s ∂∂s U ( s, t ) x is continuous for all x ∈ Y and t ∈ [ S, T ] ;3) for any subinterval [ s, t ] ⊂ [ S, T ] , there exists a constant M ( s, t ) > such that for all sequences { s = τ < τ < · · · < τ k t } , k Q τ ,τ · · · Q τ k − ,τ k k M ( s, t ) ; For example, this assumption is fulfilled when Q τ,τ +∆ τ are contractions. ) for any subinterval [ s, t ] ∈ ( S, T ] , for any fixed x ∈ Y , Q τ − ∆ τ,τ − I ∆ τ U ( τ, t ) x → A τ U ( τ, t ) x, ∆ τ → uniformly in τ ∈ [ s, t ] .Then, for any subinterval [ s, t ] ⊂ [ S, T ] , for any sequence of partitions { s = t < t < · · · < t n = t } of [ s, t ] such that max ( t j +1 − t j ) → as n → ∞ , andfor all x ∈ E , Q t ,t . . . Q t n − ,t n x → U ( s, t ) x, n → ∞ . Proof.
First we consider the case s > S , i.e. [ s, t ] ⊂ ( S, T ]. We fix an arbitrary x ∈ Y . Using relation (2), we obtain: Q t ,t Q t ,t . . . Q t n − ,t n − U ( s, t )= n − X j =0 Q t ,t . . . Q t j − ,t j − ( Q t j − ,t j − U ( t j − , t j )) U ( t j , t ) . (4)Let δ n = max j ( t j − t j − ) be the mesh of the partition { s = t < t < · · · 1) such that U ( τ − ∆ τ, t ) x = U ( τ, t ) x + ∆ τ A τ − θ ∆ τ U ( τ − θ ∆ τ, t ) x. (7)Hence, U ( τ − ∆ τ, t ) x − U ( τ, t ) x ∆ τ − A τ U ( τ, t ) x = A τ − θ ∆ τ U ( τ − θ ∆ τ, t ) x − A τ U ( τ, t ) x → , ∆ τ → , where the right hand side converges to zero uniformly in τ ∈ [ s, t ] since thecontinuous function [ s, t ] → E, ζ A ζ U ( ζ , t ) is uniformly continuous.Thus, we have proved that Q t ,t . . . Q t n − ,t n x → U ( s, t ) x as n → ∞ foreach x ∈ Y where Y is dense in E . Note that by Assumption 3 of the the-orem, the bounded on E operators Q t ,t . . . Q t n − ,t n are bounded uniformlyin { t , t , . . . , t n } . Hence, for an arbitrary subinterval [ s, t ] ⊂ ( S, T ], the con-vergence Q t ,t . . . Q t n − ,t n x → U ( s, t ) x holds for all x ∈ E . Thus, we haveproved the theorem for the case s > S .Now we consider the case s = S . Let s N be a decreasing system of realnumbers such that lim N →∞ s N = s . For each fixed N and for all x ∈ E , wehave: Q t N ,t . . . Q t n − ,t n x → U ( s N , t ) x as n → ∞ (8)where s N = t N < t < · · · < t n = t is a partition of [ s N , t ]. Note that foreach fixed τ and for each fixed x , τ k ( Q τ − ∆ τ,τ − I ) x k E is bounded, whichfollows from convergence (3) in Assumption 4 if we set t = τ . By the Banach-Steinhaus theorem there exists a constant M τ > τ k Q τ − ∆ τ,τ − I k E → E < M τ . This implies that k Q τ − ∆ τ,τ − I k E → E tends to zero as ∆ τ → ε > 0, and find a δ > k Q s,s N − I k E → E < ε and k U ( s, s N ) − I k E → E < ε whenever s N − s < δ . By Assumption 3 of Theorem1, k Q s,s N Q s N ,t . . . Q t n − ,t n − Q s N ,t . . . Q t n − ,t n k E → E k Q s,s N − I k E → E k Q s N ,t . . . Q t n − ,t n k E → E M ( s, t ) ε. 4y continuity of U ( · , t ), k U ( s, t ) − U ( s N , t ) k E → E k U ( s, s N ) − I k E → E k U ( s N , t ) k E → E sup ξ ∈ [ S,t ] k U ( ξ, t ) k E → E ε Convergence (8) and two last estimates imply that for all x ∈ EQ t ,t . . . Q t n − ,t n x → U ( s, t ) x as n → ∞ . The theorem is proved. Lemma . Let Y ⊂ ∩ τ ∈ [ S,T ] D ( A τ ) be a Banach space, dense in E , let [ s, t ] ⊂ ( S, T ] be a closed interval. Further let B τ, ∆ τ : Y → E , τ ∈ [ s, t ] , ∆ τ ∈ (0 , s − S ) , be bounded operators, and let U ( ξ, τ ) , ( ξ, τ ) ∈ D [ S,T ] , be an evolutionfamily of operators with the generators A τ . We assume that the followingassumptions are fulfilled:1) for every y ∈ Y , k B τ, ∆ τ y k E is bounded uniformly in τ ∈ [ s, t ] and ∆ τ ∈ [ δ, s − S ] , where δ ∈ (0 , s − S ) is fixed arbitrary;2) U ( τ, t ) Y ⊂ Y for all τ ∈ [ s, t ] ;3) for each y ∈ Y , the mapping [ s, t ] → Y, τ U ( τ, t ) y , is continuous;4) for each fixed y ∈ Y , sup τ ∈ [ s,t ] k A τ y k E < ∞ ; 5) for each fixed y ∈ Y , lim ∆ τ → B τ, ∆ τ y = A τ y (9) where the convergence is uniform in τ ∈ [ s, t ] .Then, for every y ∈ Y , lim ∆ τ → B τ, ∆ τ U ( τ, t ) y = A τ U ( τ, t ) y (10) and the convergence is uniform in τ ∈ [ s, t ] . roof. Assumptions 4 and 5, along with Assumption 1, imply that for eachfixed y ∈ Y , k B τ, ∆ τ y k E is bounded uniformly in τ ∈ [ s, t ], and ∆ τ ∈ (0 , s − S ].By the Banach-Steinhaus theorem, k B τ, ∆ τ k Y → E are bounded uniformly in τ ∈ [ s, t ] and ∆ τ ∈ (0 , s − S ], i.e. there exists a constant K such that k B τ, ∆ τ k Y → E < K. We fix a y ∈ Y . The set { U ( τ, t ) y, τ ∈ [ s, t ] } (11)is a compact in Y due to the continuity of the mapping [ s, t ] → Y, τ U ( τ, t ) y . Next, we fix an arbitrary small ε > ε -net { y i } Ni =1 ⊂ Y for the compact (11). Furthermore, we find a small δ > 0, such that forall τ ∈ [ s, t ], for all ∆ τ ∈ (0 , δ ), and for all y i , 1 i N , k B τ, ∆ τ y i − A τ y i k E < ε. Let τ ∈ [ s, t ] be fixed arbitrary, and y i be such that k U ( τ, t ) y − y i k E < ε . Weobtain: k B τ, ∆ τ U ( τ, t ) y − B τ, ∆ τ y i k E < K k U ( τ, t ) y − y i k E < Kε. Taking the limit in the right hand side, as ∆ τ → 0, we obtain k A τ U ( τ, t ) y − A τ y i k E Kε. This implies: k B τ, ∆ τ U ( τ, t ) y − A τ U ( τ, t ) y k E k B τ, ∆ τ U ( τ, t ) y − B τ, ∆ τ y i k E + k B τ, ∆ τ y i − A τ y i k E + k A τ y i − A τ U ( τ, t ) y k E < (2 K + 1) ε. This proves that the limit (10) exists and is uniform in τ ∈ [ s, t ]. The lemmais proved. Theorem . Let A t , Q t ,t , U ( s, t ) , and Y be as in Theorem 1. Let us assume that Assumptions 1–3 of Theorem 1are fulfilled, and that for any subinterval [ s, t ] ⊂ ( S, T ] , Assumptions 3 and4 of Lemma 1 are fulfilled. We assume, that for any y ∈ Y , lim ∆ τ → Q τ − ∆ τ,τ − I ∆ τ y = A τ y here the convergence is uniform in τ running over closed subintervals [ s, t ] ⊂ ( S, T ] . Then, the statement of Theorem 1 holds true, i.e. for any subinterval [ s, t ] ⊂ [ S, T ] , for any sequence of partitions { s = t < t < · · · < t n = t } of [ s, t ] such that max ( t j +1 − t j ) → as n → ∞ , and for all x ∈ E , Q t ,t . . . Q t n − ,t n x → U ( s, t ) x, n → ∞ . Proof. Since we assume that Assumptions 1–3 of Theorem 1 are fulfilled,it suffices to prove that Assumption 4 of Theorem 1 is fulfilled. This willfollow from Lemma 1 if we prove that Assumptions 1 and 5 of this lemmaare fulfilled for the operators B τ, ∆ τ = Q τ − ∆ τ,τ − I ∆ τ . Assumptions 2–4 of Lemma1 clearly follow from those assumptions of Theorem 1 and Lemma 1 that areassumed here to be fulfilled. To prove Assumption 4 of Theorem 1, we fixan arbitrary closed interval [ s, t ] ⊂ ( S, T ], and a δ ∈ (0 , s − S ). Then, for∆ τ ∈ [ δ, s − S ], we obtain: (cid:13)(cid:13)(cid:13) Q τ − ∆ τ,τ − I ∆ τ (cid:13)(cid:13)(cid:13) E → E < M ( s, t ) + 1 δ where M ( s, t ) is the constant in Assumption 3 of Theorem 1. Assumption 5of Lemma 1 is obviously fulfilled. By Lemma 1,lim ∆ τ → Q τ − ∆ τ,τ − I ∆ τ U ( τ, t ) y = A τ U ( τ, t ) y and the limit is uniform in τ ∈ [ s, t ]. Applying Theorem 1 completes theproof of the theorem. The following result has been obtained in [11] (p. 489, Proposition 2.5): Proposition . Let { A t } be a stable family of pairwise commuting gener-ators of strongly continuous semigroups. Let us assume that there exists aspace Y ⊂ ∩ t ∈ [ S,T ] D ( A t ) which is dense in E , and let for all y ∈ Y , themapping [ S, T ] → E, t A t y be continuous. Then, ( R ts A r dr, Y ) is closableand its closure (which we still denote by R ts A r dr ) is a generator. Moreover,the the Cauchy problem (1) is well-posed and the evolution family solving (1) is given by U ( s, t ) = e R ts A r dr , s t . heorem . Let A t be a stablefamily of pairwise commuting generators of strongly continuous semigroups,and let Q t ,t , t , t > , be a two-parameter family of bounded operators E → E , such that Assumptions 2–4 of Theorem 1 are fulfilled. Then, for anysubinterval [ s, t ] ⊂ [ S, T ] , for any sequence of partitions { s = t < t < · · · Proposition 1 implies that Cauchy problem (1) is well-posed, and that U ( s, t ) = e R ts A r dr is the evolution family solving the Cauchy problem (1). Nowthe statement of the theorem follows immediately from Theorem 1. Let M be a C k -smooth compact manifold, and let A ( t, x ), A ( t, x ), . . . , A d ( t, x ), t ∈ [ S, T ], x ∈ M , be C k -smooth vector fields on M . This meansthat if f ∈ C j ( M ) and j > k , then A i ( t, · ) f ∈ C k ( M ) and if j k , then A i ( t, · ) f ∈ C j − ( M ) for all t ∈ [ S, T ]. Let us consider t -dependent secondorder differential operators: A t = 12 d X α =1 A α ( t, · ) ◦ A α ( t, · ) + A ( t, · ) (12)with the common domain C k ( M ) independent of t . In the space C k ( M ) weintroduce the norm: k f k k = k X | λ | =0 sup y sup x ∈ V | ∂ λ f ◦ ψ y ( x ) | (13)where { ( V, ψ y ) , y ∈ M } is an atlas covering M . The fact that k · k k defines anorm is proved in [9] (pp. 175-176). The space C k ( M ) with the norm k · k k becomes a Banach space. We denote it by Y .8iven a probability space (Ω , F , P ) with the filtration F t , and a d -dimensional F t -Brownian motion B αt , we consider the stochastic differentialequation: ( dX t = A α ( t, X t ) ◦ dB αt + A ( t, X t ) dtX s = x (14)where A α ( t, X t ) ◦ dB αt is the Stratonovich differential. We denote by E theexpectation relative to the measure P . The operators A t are generators ofdiffusions X t on M . Lemma . Let Y = ( C k ( M ) , k · k k ) where k > . Then, the solution ofCauchy problem (1) on the interval [ S, t ] with the generators (12) and withthe final condition u ( t, x ) = f ( x ) , f ∈ Y , x ∈ M , exists, it is unique, and isgiven by u ( s, x ) = E [ f ( X t ( s, x ))] where X t ( s, x ) is the solution of SDE (14) . Moreover, u ( s, x ) ∈ Y , ∂∂s u ( s, x ) ∈ Y for all s ∈ [ S, t ] , and the mappings [ S, t ] → Y, s u ( s, · ) , and [ S, t ] → E, s ∂∂s u ( s, · ) are continuous.Proof. Theorem 1.3 of Chapter 5 in [4] (p. 182) implies as a particular casethat u ( s, x ) = ( U ( s, t ) f )( x ) = E [ f ( X t ( s, x ))] (15)is a solution of the Cauchy problem (1). Here U ( s, t ), ( s, t ) ∈ D [ S,T ] , is theevolution family solving this Cauchy problem. Consider another evolutionfamily ˜ U ( τ, ξ ), ( τ, ξ ) ∈ D [ S,T ] , satisfying the relation U ( s, t ) = ˜ U ( T + S − t, T + S − s ). Evidently, there exists another SDE of type (14) with C k - smoothcoefficients, having a unique solution ˜ X ξ ( τ, x ), such that for all f ∈ Y , forall x ∈ M , ( ˜ U ( τ, ξ ) f )( x ) = E [ f ( ˜ X ξ ( τ, x ))] . Applying Ito’s formula gives:( U ( s, t ) f )( x ) = ( ˜ U ( S + T − t, S + T − s ) f )( x ) = E [ f ( ˜ X S + T − s ( S + T − t, x ))]= f ( x ) + Z S + T − s E [( A S + T − ζ f )( ˜ X ζ ( S + T − t, x ))] dζ (16)9here we have exchanged the symbol E for expectation with the integral in ζ by Fubini’s theorem. For the partial derivative in s we obtain: ∂∂s u ( s, x ) = ∂∂s ( U ( s, t ) f )( x ) = − E [( A S + T − s f )( ˜ X s ( S + T − t, x ))] . (17)Clearly, u ( s, · ) ∈ Y and ∂∂s u ( s, · ) ∈ Y . Also, relations (16) and (17) implythat the mappings [ S, t ] → Y, s u ( s, · ) and [ S, t ] → E, s ∂∂s u ( s, · ) arecontinuous. The lemma is proved. Theorem . Let A t , t ∈ [ S, T ] , be given by (12) , andlet D ( A t ) = Y for all t . Further, let Q t ,t , S t < t T , be a fam-ily of contractions on C( M ) . We assume that the following assumptions arefulfilled:1) the functions [ S, T ] → C( M ) , t A t f are continuously differentiablefor all f ∈ Y ;2) stochastic differential equation (14) has a unique solution X t ( s, x ); 3) for all f ∈ Y , lim ∆ τ → Q τ − ∆ τ,τ − I ∆ τ f = A τ f and the limit is uniform in τ running over closed intervals [ s, t ] ⊂ ( S, T ] .Then, for any subinterval [ s, t ] ⊂ [ S, T ] , for any sequence of partitions { s = t < t < · · · < t n = t } of [ s, t ] such that max ( t j +1 − t j ) → as n → ∞ , andfor all f ∈ C( M ) , the following convergence holds in C( M ) : ( Q t ,t . . . Q t n − ,t n f )( · ) → E [ f ( X t ( s, · ))] , n → ∞ . Proof. Let [ s, t ] ⊂ ( S, T ] be fixed. We would like to apply Theorem 2. To thisend, we have to verify Assumptions 1 – 3 of Theorem 1 and Assumptions2 and 3 of Lemma 1. Assumption 1 of Theorem 1 follows from the paper[12] by Kato. The paper [12] guaranties existence and uniqueness of thesolution of the Cauchy problem (1) if the following assumptions are fulfilled: Sufficient conditions under which (14) has a unique solution can be found for examplein [4] and [13] D ( A t ) = Y for all t ∈ [ S, T ], and Y is dense in E ; 2) the functions t A t f are continuously differentiable. Due to this result, Assumption 1 of Theorem1 is fulfilled. Let U ( s, t ) be the evolution family solving the Cauchy problem(1), and let u ( s, x ) denote the solution of (1) with the final condition f ( x )at time t . Assumption 2 of Theorem 1 is fulfilled by Lemma 2. Assumption3 of Theorem 1 is fulfilled since Q t ,t are contractions. Assumptions 2 and3 of Lemma 1 follow immediately from Lemma 2. Now the statement of thetheorem is implied by Theorem 2. Below, we describe a construction of a time-inhomogeneous Markov processon a compact Riemannian manifold using Theorem 4. Let M be a compactRiemannian manifold without boundary isometrically imbedded into R m , anddim M = d . Let B t be a Brownian motion on R m starting at the origin, andlet ϕ : [0 , → M be a two times continuously differentiable (non-random)function such that ϕ (0) = x . We consider the process W t = B t + ϕ ( t ). Let W ϕ be its distribution, P ϕ ( t , z, t , A ) be its transition probability. Clearly, P ϕ ( t , z, t , A ) = P ( t , z − ϕ ( t ) , t , A − ϕ ( t ))= 12 π ( t − t ) m Z A e − −| z − y − ( ϕ ( t − ϕ ( t | t − t dy (18)where P corresponds to the case when ϕ is equal to zero identically.Let U ε ( M ) be the ε -neighborhood of M , and let W xε,t be the distributionof the process which is conditioned to take a value in U ε ( M ) at time t .Specifically, we define a measure W xε,t by the following expression on theright hand side: Z C([0 ,t ] , R m ) f ( ω ) W xε,t ( dω ) = R C([0 ,t ] , R m ) I { ω : ω ( t ) ∈ U ε ( M ) } ( ω ) f ( ω ) W xϕ ( dω ) W xϕ { ω : ω ( t ) ∈ U ε ( M ) } . (19)Let P xε,t ( · , · , · , · ) be the transition probability for the distribution W xε,t . By1118) and (19), P xε,t ( · , · , t, · ) is given by Z R m P xε,t ( s, z, t, dy ) g ( y ) = Z C([0 ,t ] , R m ) g ( ω ( t )) W xε,t ( dω )= R U ε ( M ) e − −| z − y − ( ϕ ( s ) − ϕ ( t )) | t − s ) g ( y ) dy R U ε ( M ) e − −| z − ¯ y − ( ϕ ( s ) − ϕ ( t )) | t − s ) d ¯ y (20)where g : R m → R is bounded and continuous. Obviously, as ε → 0, the limitof the right hand side exists. Hence, the weak limit P [ s,t ] of the measures P xε,t ( s, · , t, · ) exists and equals Z R m P [ s,t ] ( z, dy ) g ( y ) = R M e − | z − y − ( ϕ ( s ) − ϕ ( t )) | t − s ) g ( y ) λ M ( dy ) R M e − | z − ¯ y − ( ϕ ( s ) − ϕ ( t )) | t − s ) λ M ( d ¯ y )= Z M q ϕ ( s, z, t, y ) g ( y ) λ M ( dy )where λ M is the volume measure on M , and q ϕ ( s, z, t, y ) = e − | z − y − ( ϕ ( s ) − ϕ ( t )) | t − s ) R M e − | z − ¯ y − ( ϕ ( s ) − ϕ ( t )) | t − s ) λ M ( d ¯ y ) . Given an interval [ s, t ], the family of functions p ϕ ( t , z, t , y ) = 12 π ( t − t ) m e − | z − y − ( ϕ ( t − ϕ ( t | t − t (21) s < t < t < t , together with the function q ϕ ( t , z, t, y ), t < t , buildsa family of transition densities that defines the distribution of a Markovprocess on [ s, t ] conditioned to take a value on M at time t .Consider a partition P = { s = t < t < · · · < t n = t } . For each partitioninterval [ t i , t i +1 ], for each pair of points ξ and τ such that t i < ξ < τ t i +1 ,and for each Borel set A ⊂ R m , we define Q ( ξ, z, τ, A ) = (R A p ϕ ( ξ, z, τ, y ) dy, τ < t i +1 , R A ∩ M q ϕ ( ξ, z, τ, y ) λ M ( dy ) , τ = t i +1 . (22)12ext, we add more points to the partition P to obtain a partition P ′ = { s = ξ < ξ < · · · < ξ N = t } containing P . The family of measures Q ( s, x, t, A ) = Z R m Q ( s, x, ξ , dx ) Z R m Q ( ξ , x , ξ , dx ) . . . Z R m Q ( ξ N − , x N − , ξ N − , dx N − ) Q ( ξ N − , x N − , t, A )is a family of transition probabilities for a Markov process starting at thepoint x ∈ M at time s , and conditioned to take values on M at all points ofthe partition P .We apply Theorem 4 to a subfamily of the family Q ( · , · , · , · ). Specifi-cally, we investigate weak convergence of the family q P ( s, x, t, y ) = Z M q ϕ ( s, x, t , x ) λ M ( dx ) Z M q ϕ ( t , x , t , x ) λ M ( dx ) . . . Z M q ϕ ( t n − , x n − , t n − , x n − ) q ϕ ( t n − , x n − , t, y ) λ M ( dx n − ) . This family is a subfamily of Q ( · , · , · , · ) by definition (22) of the family Q . We consider the following two parameter family of contractions C( M ) → C( M ): ( Q t i ,t i +1 f )( · ) = Z M q ϕ ( t i , · , t i +1 , y ) f ( y ) λ M ( dy ) . (23) Theorem . As the mesh of P tends to zero, the following convergence holdsin C( M ) : Z M q P ( s, · , t, y ) g ( y ) λ M ( dy ) → Z M p ( s, · , t, y ) g ( y ) λ M ( dy ) (24) where g ∈ C( M ) , p ( s, x, t, y ) is the transition density function of the processgenerated by A s = ( ϕ ′ ( s ) , ∇ M )) R m − 12 ∆ M . (25) Lemma . The A s given by (25) generate contraction semigroups on C( M ) .Moreover, each A s is the generator of a diffusion X ( τ ) on M which is thesolution of the following SDE: ( dr ( τ ) = ˜ L α ( r ( τ )) ◦ dw αs,ϕ ,r (0) = r, (26)13 here r ( τ ) = ( X i ( τ ) , e iα ( τ )) , { e α ( τ ) } is a basis in the tangent space at thepoint X ( τ ) , ˜ L α are canonical horizontal vector fields [13], w αs,ϕ ( τ ) = ϕ ′ ( s ) α τ + B α ( τ ) , B α ( τ ) is a Brownian motion in R d .Proof. Let r ( τ ) = ( X i ( τ ) , e iα ( τ )) be the solution of (26). We find the gener-ator of X ( τ ). Consider the function f ( r ) = f ( x ) for r = ( x, e ). We have: f ( X ( τ )) − f ( X (0)) = f ( r ( τ )) − f ( r (0))= τ Z ( ˜ L α f )( r ( ξ )) ◦ dw αs,ϕ ( ξ )= τ Z ˜ L α f ( r ( ξ )) dB α ( ξ ) + τ Z ˜ L α f ( r ( ξ )) ϕ ′ ( s ) α dξ + 12 τ Z d X α =1 ˜ L α ( ˜ L α f )( r ( ξ )) dξ. The definition of the generator of a process gives: A s f = d X α =1 ( ˜ L α f, ϕ ′ ( s ) α ) R d + 12 d X α =1 ˜ L α ( ˜ L α f ) . Since f ( r ) = f ( x ), i.e. does not depend on e , then the scalar product in thefirst term of the right hand side is well-defined, and˜ Lf = ∇ M f by definition of ˜ L . Further, it was shown in [13] (Chapter V, paragraph 3)that d X α =1 ˜ L α ( ˜ L α f ) = − ∆ M f . Thus, we have proved that A s f = ( ϕ ′ ( s ) , ∇ M f ) R d − 12 ∆ M f = ( ϕ ′ ( s ) , ∇ M f ) R m − 12 ∆ M f . Since A s is a generator of a diffusion on M , A s generates a contractionsemigroup on C( M ). The lemma is proved.14 roof of Theorem 5. We apply Theorem 4 to the generators (25) and thetwo-parameter family of contractions( Q t ,t f )( x ) = Z M q ϕ ( t , x, t , y ) f ( y ) λ M ( dy ) . Assumption 1 of Theorem 4 is fulfilled by continuity of ϕ ′′ ( t ) which we haveassumed. Assumption 2 of Theorem 4 is fulfilled by Theorem 2.1, p. 152,from the book [4], where the authors have considered a more general case ofa manifold in a Banach space. We show that Assumption 3 of Theorem 4 isfulfilled too. We have to prove thatlim ∆ τ → τ (cid:0)Z M q ϕ ( τ − ∆ τ, y, τ, z ) g ( z ) λ M ( dz ) − g ( y ) (cid:1) = ( ϕ ′ ( τ ) , ∇ M g ( y )) R m − 12 ∆ M g ( y ) , (27)and that the limit is uniform in τ . Introduce the notation: ∆ ϕ τ = ϕ ( τ ) − ϕ ( τ − ∆ τ ). Lemma . Let g ∈ C ( M ) . There exist a δ > , a constant K g > , and afunction R : [0 , δ ] × M × C ( M ) → R satisfying: | R (∆ τ, · , g ) | < K g ∆ τ , (28) and such that for all y ∈ M the following relation holds: R M g ( z ) e − | z − y − ∆ ϕτ | τ λ M ( dz ) R M e − | z − y − ∆ ϕτ | τ λ M ( dz ) = g ( y ) + (∆ ϕ τ , ∇ M g ( y )) R m − ∆ τ M g ( y )+∆ τ R (∆ τ, y, g ) . (29) Proof. We find a U ε ( M ), the ε -neighborhood of M , where the normal spaces N y and N y do not intersect each other for each pair of points y ∈ M and y ∈ M . Hence, each y ∈ U ε ( M ) can be uniquely presented as y = z + tn ( z ),where z ∈ M , n ( z ) ∈ N z , and | n ( z ) | = 1. Let P M : U ε ( M ) → M , z + tn ( z ) z , t ∈ R , be the projection on M . For an arbitrary u ∈ R m , | u | < ε , and a y ∈ M , we define: u M ( y ) = P M ( y + u ) − u, u ⊥ ( y ) = u − u M ( y ) . 15e have: R M g ( z ) e − | z − y − ∆ ϕτ | τ λ M ( dz ) R M e − | z − y − ∆ ϕτ | τ λ M ( dz ) = R M g ( z ) e − | z − y − (∆ ϕτ ) M ( y ) | τ e ( z − y, (∆ ϕτ ) ⊥ ( y ))∆ τ λ M ( dz ) R M e − | z − y − (∆ ϕτ ) M ( y ) | τ e ( z − y, (∆ ϕτ ) ⊥ ( y ))∆ τ λ M ( dz ) . (30)We will need the following formula (see [1] and [2]): R M e − | z − y | t h ( z ) λ M ( dz ) R M e − | z − y | t λ M ( dz ) = h ( y ) − t M h ( y ) + t ¯ R ( t, y, h ) (31)where | ¯ R ( t, y, h ) | < K k h k t , K is a constant, and the norms k · k aredefined by (13) for k = 3. Dividing the numerator and denominator in theright hand side of (30) by R M e − | z − y − (∆ ϕτ ) M ( y ) | τ λ M ( dz ), and applying (31), weobtain: R M g ( z ) e − | z − y − ∆ ϕτ | τ λ M ( dz ) R M e − | z − y − ∆ ϕτ | τ λ M ( dz )= g (cid:0) y + (∆ ϕ τ ) M ( y ) (cid:1) − ∆ τ ∆ M (cid:0) g e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ (cid:1) ( y ) + ∆ τ R − ∆ τ (cid:0) ∆ M e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ (cid:1) ( y ) + ∆ τ R (32)where R and R are short-hand notations for ¯ R (∆ τ, y +∆ ϕ τ , g e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ )and ¯ R (∆ τ, y + ∆ ϕ τ , e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ ), respectively. They can be estimated asfollows: | R | < ˜ K k g k k e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ k (∆ τ ) , | R | < K k e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ k (∆ τ ) . We show that (cid:0) ∆ M e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ (cid:1) ( y ) and k e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ k are bounded in τ and ∆ τ . We consider local normal charts ψ ( ¯ ξ ) = ψ ( ξ , . . . ξ d ) at the point y .We obtain: ∂∂ξ i e ( ψ (¯ ξ ) − y, (∆ ϕτ ) ⊥ ( y ))∆ τ = e ( ψ (¯ ξ ) − y, (∆ ϕτ ) ⊥ ( y ))∆ τ (cid:16) ∂∂ξ i ψ ( ¯ ξ ) , (∆ ϕ τ ) ⊥ ( y )∆ τ (cid:17) R m . (33)This formula makes obvious the resulting expression upon taking twofurther derivatives, and thus, it shows that (cid:0) ∆ M e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ (cid:1) ( y ) and16 e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ k are bounded in τ and ∆ τ if and only if (∆ ϕ τ ) ⊥ ( y )∆ τ is boundedin τ and ∆ τ . The latter fact holds by existence of the limit:lim ∆ τ → (∆ ϕ τ ) ⊥ ( y )∆ τ = Pr N y ϕ ′ ( τ ) (34)where Pr N y is the orthogonal projection onto N y , the normal space at y .Now we can apply the short time asymptotic in ∆ τ to the denominator atthe right hand side of (32) while using the relation ∆ M (cid:0) g e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ (cid:1) = e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ ∆ M g − ∇ M e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ , ∇ M g ) R m + g ∆ M e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ . Weobtain: R M g ( z ) e − | z − y − ∆ ϕτ | τ λ M ( dz ) R M e − | z − y − ∆ ϕτ | τ λ M ( dz ) = g ( y )+ (cid:0) (∆ ϕ τ ) M ( y ) , ∇ M g ( y ) (cid:1) R m − ∆ τ M g ( y )+ ∆ τ (cid:16) ∇ M e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ (cid:12)(cid:12)(cid:12) y , ∇ M g ( y ) (cid:17) R m + ∆ τ ˜ R (∆ τ ) (35)where | ˜ R (∆ τ ) | < ˜ K k g k ∆ τ , ˜ K is a constant. Formulas (33) and (34) imply: (cid:16) ∇ M e ( · − y, (∆ ϕτ ) ⊥ ( y ))∆ τ (cid:12)(cid:12)(cid:12) y , ∇ M g ( y ) (cid:17) = (cid:16) (∆ ϕ τ ) ⊥ ( y )∆ τ , ∇ M g ( y ) (cid:17) . Substituting this in (35), we obtain (29). The lemma is proved.We continue the proof of Theorem 5. Lemma 4 easily implies the con-vergence in (27). This convergence is uniform in τ . Indeed, (28) implies that R (∆ τ, · , g ) converges to zero uniformly in τ . Further, we have: (cid:12)(cid:12)(cid:12) ∆ ϕ τ ∆ τ − ϕ ′ ( τ ) (cid:12)(cid:12)(cid:12) 12 sup θ ∈ [0 , | ϕ ′′ ( τ + θ ∆ τ ) | ∆ τ, and hence, ∆ ϕ τ ∆ τ converges to ϕ ′ ( τ ) uniformly in τ . Thus, we have verified allthe assumptions of Theorem 4. Finally, we note that since p ( · , · , · , · ) is thetransition density for the diffusion process X t ( s, x ), and hence Z M p ( s, x, t, y ) f ( y ) λ M ( dy ) = E [ f ( X t ( s, x ))] . 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