Stochastic Analysis on Product Manifolds: Dirichlet Operators on Differential Forms

Journal of Functional Analysis 176, 280316 (2000) doi:10.1006jfan.2000.3629, available online at http:www.idealibrary.com on

Stochastic Analysis on Product Manifolds: Dirichlet Operators on Differential Forms Sergio Albeverio Institut fur Angewandte Mathematik, Universitat Bonn, Wegelerstr. 6, D 53115 Bonn, Germany; SFB 256, Bonn, Germany; BiBos Research Centre, Bielefeld, Germany; CERFIM, Locarno, Switzerland ; Acc. Arch., USI

and Alexei Daletskii and Yuri Kondratiev Institut fur Angewandte Mathematik, Universitat Bonn, Wegelerstr. 6, D 53115 Bonn, Germany; SFB 256, Bonn, Germany; BiBos Research Centre, Bielefeld, Germany; Institute of Mathematics, Kiev, Ukraine Communicated by L. Gross Received March 8, 1999; revised May 9, 2000; accepted May 17, 2000

We define a de Rham complex over a product manifold (infinite product of compact manifolds), and Dirichlet operators on differential forms, associated with differentiable measures (in particular, Gibbs measures), which generalize the notions of Bochner and de Rham Laplacians. We give probabilistic representations for corresponding semigroups and study properties of the corresponding stochastic dynamics.  2000 Academic Press

Contents. Introduction. Setting. Differential forms and the de Rham complex. Dirichlet operators on differentiable forms. 4.1. Dirichlet forms and stochastic dynamics on functions. 4.2. Dirichlet forms and Dirichlet operators in spaces of differential forms, and associated stochastic dynamics. 5. Probabilistic representations of semigroups. 5.1. Stochastic differential equations on product manifolds. 5.2. Parallel translation and diffusions on tensor bundles. 5.3. Probabilistic representations of semigroups. 6. Stochastic dynamics for lattice models associated with Gibbs measures on product manifolds. 6.1. Gibbs measures on product manifolds. 6.2. Stochastic dynamics on functions. 6.3. Stochastic dynamics in the de Rham complex. 7. Appendix. 7.1. Differentiable structures on product manifolds. 7.2. Existence and uniqueness of solutions for infinite systems of SDE.

1. 2. 3. 4.

280 0022-123600 35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved.

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

281

1. INTRODUCTION Various questions of stochastic analysis on product manifolds (that is, on infinite products of compact manifolds) have received a strong interest in recent times. This interest is strongly motivated by applications to models of statistical mechanics. Various aspects of the study of Dirichlet operators associated with Gibbs measures on product manifolds, and corresponding semigroups, were developed in, e.g., [HS1, HS2, SZ1, SZ2, AKR2, ADKR]; see also [A, AKR3] for a detailed review of the literature. In [Be1, Be2, BeSC] (see also references therein), some questions of potential analysis (properties of heat semigroups, harmonic functions) on product manifolds were considered. Let us remark that in the case of a linear spin space, the stochastic dynamics associated with a Gibbs measure can be constructed in different ways (Dirichlet forms method, direct operator construction of the Markov semigroup with the given generator, and stochastic differential equations (SDE) approach). In the case of compact spins, the first two methods are well-developed. In particular, a direct construction of the semigroup which uses the explicit form of the generator is given in [SZ1, SZ2]. However, a construction of the corresponding process by means of SDE theory (``Glauber dynamics'') meets certain difficulties and requires a development of the relevant analytic framework. In the simplest case of the infinite dimensional torus this approach was realized in [HS1]. The case of nontrivial compact manifolds was considered in the works [ADK1ADK4]. Let us remark that the absence on such product manifolds of a proper structure of a smooth Hilbert or Banach manifold makes it impossible to use the existing theory of SDE on infinite dimensional manifolds [BDa, BrE]. Therefore the first step towards the realization of a ``stochastic quantization'' program on product manifolds is the development of the necessary differential-geometrical framework and relevant theory of SDE. The discussion of these questions, and application of the developed technique to the construction and study of the stochastic dynamics associated with differentiable measures on product manifolds, is given in [ADK4] (see also [ADK3]). Stochastic differential equations on product manifolds, associated with an arbitrary connection on the initial single manifold, were studied in [G1Mo]. Let us note that the Dirichlet operator of a differentiable measure, which is actually an infinite dimensional generalization of the LaplaceBeltrami operator, has a natural supersymmetric extension. Namely, we can consider Dirichlet operators on differential forms over a product manifold M, extending the notions of Bochner and de Rham Laplacians. The study of the latter operators and corresponding semigroups on finite dimensional manifolds was the subject of many works and has led to deep results on the

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ALBEVERIO, DALETSKII, AND KONDRATIEV

border of stochastic analysis, differential geometry and topology, and mathematical physics; see, e.g., [E3, CFKSi]. Dirichlet forms and processes in connection with non-commutative C*-algebras were considered in, e.g., [Gr, AH-K, DavLin]. In an infinite dimensional situation, such questions were discussed in the flat case in [Ar1, Ar2, ArM, AK]. A regularized heat semigroup on differential forms over the infinite dimensional torus was studied in [BeLe]. The case of loop spaces was considered in [JLe, LeRo]. Laplace operators in abstract (finite) Hilbert complexes were considered in [BrLes]. The study of such questions on product manifolds seems to be more realistic at the moment, and at the same time it reflects the non-trivial interplay of measure theory and differential geometry (which is essential in an infinite dimensional setting). The present paper represents a step in this direction. The structure of it is as follows. In Section 2 we introduce main notations and geometrical objects, referring for more detailed discussion to Appendix 1. Section 3 is devoted to the definition of the de Rham complex over M. In Section 4 we introduce BochnerDirichlet and de RhamDirichlet operators in the de Rham complex and prove an analog of the Weitzenbock formula. Then we formulate the main result which characterizes Markov and hypercontractivity properties of the corresponding semigroups. Section 5 is devoted to the development of a probabilistic technique needed for the proof of this theorem. In particular, we consider SDE on tensor bundles over M and give with their aid probabilistic representations of semigroups. Section 6 is devoted to a discussion of the obtained results in the framework of Gibbs measures. In order to make the paper more self-contained, we give the proof of the existence and uniqueness of solutions to SDE on M in Appendix 2.

2. SETTING Let A, B be Banach spaces and H be a Hilbert space. We will use the following general notations:  ( } , } ), pairing of A and A$, A$ being the dual space;  ( } , } ) H , the scalar product in H;  & } & A , the norm in A;  L(A, B), the space of bounded linear operators A Ä B;  L(A)#L(A, A);  Ln(A, B), the space of bounded n-linear operators A n Ä B;  HS(H, B), the space of HilbertSchmidt operators H Ä B.

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

283

Let M be a compact connected smooth N-dimensional manifold. Let us assume that M is equipped with the Riemannian structure given by the operator field G(x): T x M Ä T* x M, ( } , } ) Tx M =( G(x) } , } ). The distance on M corresponding to this Riemannian structure will be denoted by \. Let us consider the integer lattice Z d, d1, and define the space M, which is an infinite product of manifolds M k =M, M#M Z := _ M k % x=(x k ) k # Zd . d

(1)

k # Zd

M is endowed with the product topology. Given 4/Z d M % x [ x 4 =(x k ) k # 4 # M 4

(2)

is the natural projection of M onto M 4. We denote by { k ,(x) the derivative of the function resp. the LeviCivita covariant derivative of a vector or tensor field , on M w.r.t. the variable x k at point x # M (which will always be identified with a linear functional on T xk M). Thus for an n-times differentiable w.r.t. x k function , we have { nk ,(x) # Ln(T xk M, R 1 ). The symbol 2 k , :=Tr { 2k , will denote the corresponding Laplace operator. We will use the notations {u :=({ k u) k # Zd , 2 := k # Zd 2 k . Let 0 be the family of all finite subsets of Z d. We will denote by FC m(M) the space of m-times continuously differentiable real-valued cylinder functions on M, FC m(M) := . C m(M 4 ).

(3)

4#0

We also define the space FC m(M Ä TM) of m-times differentiable cylinder vector fields on M with both domain and range consisting of cylinder elements, FC m(M Ä TM) := . C m(M 4 Ä TM 4 ),

(4)

4#0

where C m(M 4 Ä TM 4 ) is the space of m-times differentiable vector fields on M 4. Let us remark that for u # FC m(M) we have {u # FC m&1 (M Ä T M). We will use the notations (X(x), Y(x)) x = : (X k (x), Y k (x)) Tx k # Zd

div X= : div k X k , k # Zd

div k meaning the divergence with respect to x k .

k

M

, (5)

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ALBEVERIO, DALETSKII, AND KONDRATIEV

Remark 2.1. The assumption that all M k coincide (i.e., M k is the k th copy of a fixed manifold M) is made just for simplicity. We can indeed study by the same methods the case of different compact manifolds M k . The space M has a Banach manifold structure with the Banach space l (Z d Ä R N ) of bounded sequences Y=(Y k ) k # Zd , Y k # R N, equipped with the norm &Y&  := sup &Y k & RN ,

(6)

k # Zd

as the model. However, this norm being not smooth, one gets difficulties in using this manifold structure for the purposes of stochastic analysis. The way of overcoming this difficulty was proposed in [ADK3, ADK4]. In these works, an analog of Riemannian structure on M was introduced. On a heuristic level, the tangent space to M at the point x can be identified with the space _k # Zd T xk M. In order to define a differentiable structure on M, it is natural to consider some Hilbert subspace of _k # Zd T xk M. Let l 1+ :=l 1(Z d Ä R 1+ ) be the space of summable sequences p=( p k ) k # Zd of positive numbers. For a fixed p # l 1+ let us define the space

{

T p, x = X #

_

k # Zd

T x k M : 7p k &X k & 2Tx

k

M

=

< ,

(7)

.

(8)

equipped with the natural scalar product (X, Y) p, x = : p k (X k , Y k ) Tx k # Zd

k

M

The scalar product (X, Y) p, x in the spaces T p, x will play the role of a Riemannian-like structure for M. The space M equipped with this structure will be denoted by M p . The bundle over M p with fibres T p, x will be called the tangent bundle of M p and denoted by TM p . The fibres T p, x will be denoted by T x M p . The bundle T M p is not the tangent bundle to M p in the proper sense. Nevertheless T M p gives us the possibility to define analogues of various differentiable structures on M p . In [ADK3, ADK4], the spaces C n(M p ) of differentiable functions on M, resp. C n(M p Ä TM p ) of differentiable vector fields over M were introduced. For convenience, we give these definitions and the main facts concerning differentiable structures on M in the Appendix. We will use the notation T M :=T M 1 , where 1 is the weight sequence with elements 1 k =1.

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STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

The space M p possesses the metric \ p defined by \ p(x, y) 2 = : \(x k , y k ) 2 p k ,

(9)

k # Zd

which makes it a complete metric space; see the Appendix. Let + be a probability measure on M differentiable in the sense that the following integration by parts formula holds true: for any u # FC 1(M), and any vector field X # FC 1(M Ä T M)

|

: ({ k u(x), X k (x)) Tx

k # Zd

k

M

|

d+(x)=& ; +X u(x) d+(x),

(10)

with some ; +X # L 2(M, +). ; +X is called the logarithmic derivative of + in the direction X. We assume that ; +X is given by ; +X(x)= : ((; +k (x), X k (x)) Tx k # Zd

k

M

+div X k (x)),

(11)

where ; +(x)=( ; +k (x)) # C 1(M p Ä TM p ) for some weight sequence p # l 1+ . We will call ; + the (vector) logarithmic derivative of +.

3. DIFFERENTIAL FORMS AND THE DE RHAM COMPLEX The simplest differentiable forms over M are the forms with both image and domain consisting of cylinder elements. The space of m-times differentiable forms of the order n of such type will be denoted by F0 m n . We will use the notation F0 n := m F0 m . Each v # F0 has the form n n v(x)=

: k1, ..., kn # Zd

v k1, ..., kn(x),

(12)

where v k1, ..., kn(x) # T xk M 7 } } } 7 T xk M, and the sum is finite. 1 n Actually, each | # F0 n can be regarded as a differential form on the manifold M 4 for some finite 4/Z d. Let + be a differentiable measure on M with the logarithmic derivative ;. We introduce spaces of integrable forms L s+ 0 n as the completions of F0 n w.r.t. the norm & } & s given by &v& ss :=

|_

: k1, ..., kn # Zd

(v k1, ..., kn(x), v k1, ..., kn(x)) Tx

M 7 } } } 7 Tx M k1 kn

&

s2

d+(x). (13)

286

ALBEVERIO, DALETSKII, AND KONDRATIEV

In order to define the de Rham complex over M, we need spaces of smooth forms. Given a Hilbert space K, we define the space  n K which is the n-fold antisymmetric tensor power of the Hilbert space K. We introduce in the standard way the exterior multiplication  :  n K_ m K Ä  n+m K, (14) (n+m)! AS n+m(, ), , 7  := n! m ! AS n+m : } n+m K Ä  n+m K being the antisymmetrization operator, and creation resp. annihilation operators a*(k):  n K Ä  n+1 K,

k # K, (15)

a*(k) , :=k 7 , resp. a(k):  n+1 K Ä  n K

(16)

(the adjoint operator). Let us define the bundle  n T M with fibres  n T x M. Sections of this bundle can be considered as differential forms over M (we identify the n space T * m M with T m M). Each section | of  T M can be represented as a sum of sections with cylinder images, |(x)=

: k1, ..., kn # Zd

| k1, ..., kn(x),

| k1, ..., kn(x) # T xk M 7 } } } 7 T xk M, n

1

(17)

converging at each x in the norm of  n T x M. We define the (covariant) derivative {| as an infinite block-operator matrix, {|(x) :=({ i | k1, ..., kn(x)) i, k1, ..., kn # Zd ,

(18)

where { i | k1, ..., kn(x) # L(T xi M, T xk M 7 } } } 7 T xk M) (if it exists); cf. the 1 n Appendix. We can now define the space 0 1n(M p ) of differentiable n-forms, requiring that

\

+

{|(x) # L T x M p ,  n T x M ,

(19)

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STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

and is bounded uniformly in x # M. Similarly we define the spaces 0 m n (M p ) of m-times differentiable n-forms. We introduce the exterior differential m&1 dn : 0 m n (M p ) Ä 0 n+1 (M p ),

(20)

d n |(x)=(n+1) AS n+1({|(x)),

(21)

setting

where AS n+1 : } n+1 T x M Ä  n+1 T x M is the antisymmetrization operator, and {|(x) # L(T x M p Ä n T x M) is identified with the element of ( n T x M)T x M, which is possible because the embedding T x M/T x M p is of HilbertSchmidt class [BK]. Let us remark that the operator d n obviously m&1 preserves the space of cylinder forms, d n : F0 m n Ä F0 n+1 . This definition can be given also using the coordinate representation (17). Indeed, each | k1, ..., kn can be locally represented as | k1, ..., kn(x)=: k1, ..., kn(x) h xk 7 } } } 7 h xk , n

1

h xk # T xk M, j

j

: k1, ..., kn # C m(M p Ä R 1 ). We have as usual d n | k1, ..., kn(x)=d: k1, ..., kn(x) 7 h xk 7 } } } 7 h xk , 1

n

(22)

d n |=: d| k1, ..., kn , where d: k1, ..., kn is the 1-form defined by the derivative of the function : k1, ..., kn . Let us observe that d n can be considered as an unbounded operator 2 2 L 2+ 0 n Ä L 2+ 0 n+1 . We denote by d * the adjoint n : L + 0 n+1 Ä L + 0 n operator. Proposition 3.1. Let us assume that ; + # C 1(M p Ä T M p ). Then the operator d * n is densely defined, its domain of definition contains the space 1 F0 n+1 , and for | # F0 n+1 we have d * n | # 0 n(M p ). Proof. Let us fix x # M and choose local coordinates e (k, k) , around each x k # M, k # Z d, k=1, ..., N. Let us denote by { (k, k) resp. a (k, k) the corresponding to e (k, k) covariant derivative resp. annihilation operator. As we already observed, any | # F0 n+1 can be considered as a differential form on the compact manifold M 4 for some finite 4/Z d. Obviously the definition of exterior differential d n given above is compatible in this case

288

ALBEVERIO, DALETSKII, AND KONDRATIEV

with the usual definition of exterior differential on M 4. We have therefore locally d|=(a k, k )* { k, k |,

(23)

where the sum is finite (here and in what follows we use the summation rule w.r.t. lower-upper coinciding indexes). By standard arguments [CFKSi] and the integration by parts formula we have d *|=&a k, k({ k, k +; +k, k ) |,

(24)

where ; +k, k is the corresponding component of ; +. It is easy to see that | 1 :=a k, k{ k, k | # F0 n , and | 2 :=a k, k; +k, k | # 0 1n(M p ). K In what follows, we will also use the space L s(M Ä  n TM q , +) of integrable sections of the bundle  n T M q (which is defined similarly to n T M), with the norm given by ( &} & sn TxMq d+(x)) 1s, and the space n 0m n (M p, q ) of differentiable sections |: M p Ä  T M q with the derivatives bounded uniformly in x in the norms of the space Lm(T x M p ,  n T x M q ).

4. DIRICHLET OPERATORS ON DIFFERENTIABLE FORMS In this section we define Dirichlet forms and Dirichlet operators in spaces of differential forms over M. Let + be a differentiable measure on M such that the corresponding logarithmic derivative ; + belongs to C 1(M p Ä T M p ) for some weight sequence p # l 1+ . We first recall a construction of the classical Dirichlet form associated with +, and of the corresponding stochastic dynamics. 4.1. Dirichlet Forms and Stochastic Dynamics on Functions For u, v # FC 2(M) we define the classical pre-Dirichlet form E+ associated with +, E+(u, v)= 12

| : ({ u(x), { v(x)) k

k

k

Tx M k

d+(x).

(25)

Obviously it has a generator H + acting in L 2(M, +) on the domain FC 2(M) as H + u(x)=& 12 : 2 k u(x)& 12 : ( ; k (x), { k u(x)) Tx k

k

k

M

.

(26)

Our goal is to construct a Markov process on M such that its generator coincides with H + on FC 2(M). Such a process is called some times the

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

289

stochastic dynamics associated with +. One possibility to proceed in constructing the stochastic dynamics is given by the theory of Dirichlet forms. Indeed, the pre-Dirichlet form E+ is closable. Its closure defines the classical Dirichlet form given by + (which will be denoted also by E+ ; for this concept see, e.g., [AR]). We can consider the semigroup T +(t) in L 2(M, +) associated with its generator and construct the corresponding process as described in [AR]. Another approach (which gives in our case better control on properties of the stochastic dynamics) is based on the SDE theory. In the case where the relevant SDE has ``nice coefficients'' this can be solved and the so constructed process (sometimes called ``Glauber dynamics'') coincides (in the sense of having the same transition semigroup) with the stochastic dynamics process. In our framework the corresponding process ! can be obtained as the Brownian motion on M with the drift ; +. It was constructed as the strong solution of certain SDE on M in [ADK3, ADK4]. For convenience of the reader we give this construction in Subsection 5.1 (Example 5.1). Let us consider the corresponding Markov semigroup T !(t) in the space C(M) of continuous functions on M, T !(t) u(x) :=E(u(! x (t))),

(27)

where ! x (0)=x. Each operator T !(t) is symmetric in L 2(M, +). Moreover, the semigroup T !(t) can be uniquely extended to a strongly continuous t semigroup T ! (t) of symmetric contraction operators in L 2(M, +) (cf., e.g., [Fuk, pp. 2728]). We have the following result [ADK3, ADK4]. Theorem 4.1. There exists a unique Markov process ! x with values in M such that the associated semigroup T !(t) u(x) :=E(u(! x (t))) acts in the space C(M) of continuous functions on M, and its generator H ! coincides with &H + on FC 2(M). This process is given by Brownian motion on M (constructed in Example 5.1) with the drift ; +. If ; + # C 3(M p Ä TM p ), then H + is an essentially self-adjoint operator on FC 2(M). In this case we have t! T (t)=T +(t) for all t. Proof. The existence of the process ! such that H !u=&H + u for u # FC 2(M), and the Feller property of the corresponding semigroup T !(t), follow directly from Theorem 5.1. If ; + # C 3(M p Ä TM p ), Corollary 5.1 implies that the semigroup T !(t) leaves invariant the space C 2(M p Ä R 1 ), which obviously is contained in the domain of H + . This implies essential self-adjointness of H + on C 2(M p Ä R 1 ); see, e.g., [DaF, Theorem IV.1.4]. It can be shown by an argument similar to the one given in [AKR1], that the closure of (H + , C 2(M p Ä R 1 )) coincides with the

290

ALBEVERIO, DALETSKII, AND KONDRATIEV

closure of (H + , FC 2(M)). In this case the L 2-semigroup with the generator t (H + , FC 2(M)) is unique, which implies that T ! (t)=T +(t) for all t. K 4.2. Dirichlet Forms and Dirichlet Operators in Spaces of Differential Forms, and Associated Stochastic Dynamics In this section we define Dirichlet operators on differential forms over M, generalizing the notion of Bochner and de Rham Laplacians. We introduce two Dirichlet forms on L 2+ 0 n . For u, v # F0 n we define the pre-Dirichlet form E B+ resp. E R + associated with +, setting

|

E B+(u, v) := : ({ k u(x), { k v(x)) d+(x),

(28)

k

resp. E R+(u, v) :=(du, dv) L2+ 0n+1 +(d *u, d *v) L2+ 0n&1 .

(29)

Analogously to the classical pre-Dirichlet form E+ , the forms E B+ resp. E R + 2 have generators H B+ resp. H R + acting in L 0 n on the domain F0 n as H B+ u(x)=&: 2 k u(x)&: ( ; k (x), { k u(x)) n Tx M  Tx k

k

k

M

,

(30)

resp. H R+ =dd *+d *d,

(31)

and are therefore closable. We preserve the same notations for their closures resp. generators of the closures. Remark 4.1. We call the bilinear forms E B+ and E R+ ``Dirichlet forms'' in order to outline the similarity of their construction and the construction of the classical Dirichlet form E+ . We don't discuss any contractivity properties of E B+ and E R+ which would make them Dirichlet forms in the sense of [Gr, AH-K, DavLin]. Let us observe that in the case of a single manifold M and + a Riemannian volume measure on it, the operator H B resp. H R + coincides with the Bochner resp. de Rham Laplacian; see, e.g., [E3, CFKSi]. We will call them the BochnerDirichlet resp. the de RhamDirichlet operator associated with +. Our next goal is to find a relation between the operators H R+ and H B+ . ..., N Let (e (k, j) ) j=1, ..., N be a orthonormal basis in T xk M. Then (e (k, j) ) kj=1, # Zd is obviously the orthonormal basis for T x M q (for any weight sequence q). We set a* (k, j) :=a*(e (k, j) ), a (k, j) :=a(e (k, j) ).

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

291

Let us introduce the operator : [R ijsl (x k ) a*(k, i) a (k, j) a*(k, s) a (k, l ) ]

R n(x)= : k # Zd

(32)

i, j, s, l

in  n T x M q , where R ijsl is the curvature tensor on M. Lemma 4.1. q arbitrary. Proof.

R n(x) is a bounded operator in each space  n T x M q ,

We have n Tx Mq=

 k1, ..., kn # Zd

- q k1 } } } q kn T xk M 7 } } } 7 T xk M. 1

n

(33)

The restriction R k1, ..., kn of R n(x) to the space T xk M 7 } } } 7 T xk M is welln n defined because in this case only a finite number of terms of the series (32) is not equal to zero, and its norm is bounded uniformly in k 1 , ..., k n # Z d, which implies the result. K Because of the assumptions on ; +, we have {; +(x) # L(T x M p ). Because of the symmetry of the LeviCivita connection, the operator {; +(x) is symmetric w.r.t. the scalar product of T x M, and therefore belongs also to L(T x M p &1 ), where p &1 :=( p &1 k ) k # Zd . Let us define the operator [{; +(x)] 7n :={; +(x) 1  } } } 1 +1  {; +(x)  } } }  1+ } } } +1  } } }  1 {; +(x)

(34)

in  n T x M. We have [{; +(x)] 7n # L( n T x M p &1 ) and [{; +(x)] 7n # L( n T x M p ). The following result is an extension of the classical Weitzenbock formula. Theorem 4.2. For u # F0 n H R+ u(x)=H B+ u(x)+R n(x) u(x)&[{; +(x)] 7n u(x).

(35)

Proof. Let us first give the proof in the case of a single manifold M, which is a modification of the well-known proof (see, e.g., [E3]) to the case of a differentiable measure d+(m)=\(m) dm instead of the Riemannian volume measure dm. Let us choose the normal coordinates (e 1 , ..., e N ) around m # M. We denote by { i resp. (a i )* resp. a i the corresponding e i covariant derivative resp. creation resp. annihilation operator. As usual, [ } , } ] resp. [ } , } ]

292

ALBEVERIO, DALETSKII, AND KONDRATIEV

denotes the commutator resp. anticommutator of operators. The following formula is well known, d=(a i )* { i

(36)

(here and in what follows we use the summation rule w.r.t. lower-upper j indexes). It is well known that in the case of \#1 we have { *=&{ i i &1 ji , k where 1 ji are Cristoffel symbols of the LeviCivita connection. Then the integration by parts formula shows that in the general case + j { *=&{ i i &; i &1 ji ,

(37)

where ; +i is the corresponding component of the vector logarithmic derivative ; +. In normal coordinates we have 1 jji(m)=0, and therefore + {* i v(m)= &{ i v(m)&; i (m) v(m),

[{ i , a j ] v(m)=[{ i , (a j )*] v(m)=0.

(38)

Applying (36) we obtain i dd*=(a j )* { j { * i a , i j d *d={ * i a (a )* { j .

(39)

We have i , (a j )*] { j v(m)+(a j )* [{ j , { i*] a iv(m) (dd *+d *d ) v(m)={ *[a i ij j i ={ * i g { j v(m)&(a )* [{ j , { i ] a v(m)

&(a j )* [{ j , ; +i ] a iv(m),

(40)

where g ij is the metric tensor. By the definition, ij B {* i g { j =H + .

(41)

We have also (a j )* [{ i , { j ] a iv(m)=(a j )* a iR(e i , e j ) 7 v(m) =R ikjl (a i )* a k(a j )* a lv(m),

(42)

and (a j )* [{ j , ; +i ] a i =(a j )* ({ j ; +i ) a i =({; + ) 7, from which the assertion follows.

(43)

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

293

We can pass to the case of M just by replacing the indexes i, j, k, l # [1, ..., N] of the summation by the multi-indexes (i, i), (j, j), (k, k), (l, l ) # Z d_[1, ..., N]. K Let us introduce the operator R +n(x) # L( n T x M p ), R +n(x)=R n(x)&[{; +(x)] 7n.

(44)

Formula (35) implies that R +n(x) does not depend on the choice of the basis in (32). Both H B+ resp. H R+ are non-negative self-adjoint operators in L 20 n and generate therefore strongly continuous contraction semigroups T B+(t) resp. T R+(t) in it. The following result characterizes the properties of these semigroups. Theorem 4.3. Let us assume that ; + # C 4(M p Ä T M p ). Then: (1)

both H B+ and H R+ are essentially self-adjoint operators on FC 2(M);

(2) for any weight sequence q the semigroup T B+(t) leaves invariant the space C(M Ä  n T M q ), and for v # C(M Ä  n T M q ) we have &T B+(t) v(x)& n Tx Mq T +(t) &v& n T Mq (x);

(45)

(3) for the weight sequence q#p resp. q#1 resp. q#p &1 the semigroup T R+(t) leaves invariant the space C(M Ä  n T M q ), and for v # C(M Ä  n T M q ) we have &T R+(t) v(x)& n Tx Mq e &trq T +(t) &v& n T Mq (x),

(46)

where r q is such that r q } (h, h) n Tx Mq (R +n(x) h, h) n Tx Mq

(47)

for each h #  n T x M p &1 and x # M; Proof. The proof will be given in Subsection 5.3, using probabilistic K representations of the semigroup T B+(t) and T R + (t). Let us recall that a semigroup T(t) (acting on functions on M) is called hypercontractive, if for all 1
(48)

294

ALBEVERIO, DALETSKII, AND KONDRATIEV

and T(t) is a contraction when s&1 e &2t* _&1

(49)

for all t>0 and some *>0. Formula (45) emphasizes a certain Markov property of T ;+(t) and applies its hypercontractivity provided T + is hypercontractive. An analogous state+ ment for T R + (t) requires positivity of R n . We have the following result. Theorem 4.4. Let us assume in the framework of Theorem 4.3 r q >0. Then: (1) the semigroup T R + (t) is Markov in the sense that for q#p resp. q#1, q#p &1 resp. q#1 &T R+(t) v(x)& n Tx Mq T +(t) &v& n T Mq (x);

(50)

s n (2) the semigroup T R + (t) is contractive in each L (M Ä  T M q , +), where q#p &1 resp. q#1;

(3) if T +(t) is hypercontractive, then T R+(t) is also hypercontractive (with the same *), in the sense that

\

+

\

+

T R+(t): L s M Ä  n T M q , + Ä L _ M Ä  n T M q , + ,

(51)

where q#p &1 resp. q#1, and T R + (t) is a contraction when (49) holds. Proof. Part (1) follows from formula (46); (2) and (3) are immediate corollaries of (1). K The following gives the lower bound for R n(x) in terms of the spectrum of the curvature operator R (2) of the manifold M. Let _(R (2)(m)) be the lowest eigenvalue of R (2)(m), m # M. Proposition 4.1.

Let n=N } k+n 0 for some k0, 1n 0 N. Then R n(x) 12 b(x)(N&n 0 ) n 0 I,

(52)

where b(x)=inf k # Zd _(R (2)(x k )), as operators in  n T x M q for any weight sequence q.

295

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

Proof. It is enough to prove (52) on the dense set of cylinder elements | of  n T x M p . Any such | belongs to  n1 T xk M 7 } } } 7ns T xk M for 1 s some k 1 , ..., k s # Z d, n 1 , ..., n s # Z + , n 1 + } } } +n s =n. Then we have R n(x) |=R kn11(x k1 ) |+ } } } +R knss(x ks ) |,

(53)

where

\

R kk(x k ) := : [R ijsl (x k )(a ik )* a kj (a sk )* a lk ] # L  k T xk M i, j, s, l

+

(54)

is the Weitzenbock operator associated with the single manifold M. It is well known (see, e.g., [CFKSi]) that R kk(x k ) 12 _(R (2)(x k ))(dim M&k) kI. Therefore (R n(x) |, |) n Tx Mp =(R kn11(x k1 ) |, |) n Tx Mp + } } } +(R kn11(x ks) |, |) n Tx Mp  12 _(R (2)(x k1 ))(dim M&n 1 ) n 1(|, |) n Tx Mp + } } } + 12 _(R (2)(x ks ))(dim M&n s ) n s(|, |) n Tx Mp  12 b(x)(dim M&n 0 ) n 0(|, |) n Tx Mp . K Remark 4.2.

(55)

As an immediate corollary of formula (35) we have

\x # M R +n(x)cI, c>0 (as operators in  n T x M) (56)

R +

(H |, |) L2+ 0n c(|, |) L2+ 0n , | # 0 n . If n=1, the latter estimate implies that H + has the spectral gap (0, c). Indeed, for f # C 1(M p Ä R 1 ) such that df{0 (H R+ df, df ) L2+ 01 =(d * d 2f +dd* df, df ) L2+ 01 =(dd* df, df ) L2+ 01 =(d* df, d * df ) L2+ 01 =(H + f, H + f ) L2(M, +) .

(57)

From (56) we have (H + f, H + f ) L2(M, +) c(df, df ) L2+ 01 =(H + f, f ) L2(M, +) , from which the assertion follows.

(58)

296

ALBEVERIO, DALETSKII, AND KONDRATIEV

Remark 4.3. If the Dirichlet form E+ is irreducible (i.e., E+(u, u)=0 O u=const), then the statements of Theorem 4.4 are valid under the condition R +n(x)0 for each x # M, and R +n(x)>0 for x belonging to a set of positive measure.

5. PROBABILISTIC REPRESENTATIONS OF SEMIGROUPS The aim of this section is to obtain probabilistic representations of the semigroups T B+(t) and T R+(t), and to prove with their aid Theorem 4.3. First we recall the construction of the diffusion process in M, which gives the stochastic dynamics associated with the classical Dirichlet form E+ . This process was constructed in [ADK3, ADK4] as the strong solution to a SDE on M (i.e., an infinite system of SDE on M). 5.1. Stochastic Differential Equations on Product Manifolds Let a: M Ä T M be a vector field on M, a(x)=(a k (x)) k # Zd . Let us consider the system of SDE d! k (t)=a k (!(t)) dt+A(! k (t)) b dw k (t),

k # Z d,

(59)

in the Stratonovich form on M, where w k , k # Z d, are independent Wiener processes in a given Euclidean space R n, and A is a smooth operator field on M, A(m) # L(R n, T m M). We will also write this system in the form of one SDE d!(t)=a(!(t)) dt+A(!(t)) b dw(t)

(60)

on M, where w is the cylinder Wiener process in K :=l 2(Z d Ä R n ), and A(x) is the block-diagonal operator K Ä T x M with diagonal blocks A kk(x)=A(x k ). It is easy to see that A(x) # HS(K, T x M p ) for each p # l 1+ . Remark 5.1. For the purposes of this paper we need just the SDE with the ``diagonal'' diffusion operator (as above). However, the more general equations with the diffusion term given by an arbitrary 2-times differentiable operator field A(x) # HS(K, T M p ), where K is an arbitrary Hilbert space, can be studied by similar methods [ADK3, ADK4]. Theorem 5.1 [ADK3, ADK4]. Let a # C 1(M p Ä T M p ). Then the SDE (60) has a unique solution ! x (t) for any initial data x # M. ! x (t) depends continuously (in the mean square sense) on the initial data x. Proof.

See Appendix 2.

K

297

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

Many properties of the process !(t) are similar to those of the solutions of SDE on Banach spaces and finite dimensional manifolds. In particular, !(t) is a Markov process with continuous paths. It generates a Markov semigroup T !(t) in the space of bounded measurable functions u on M by the formula T !(t) u(x)=E(u(! x (t))).

(61)

Square mean continuous dependence of ! x (t) on x implies the Feller property of T ! (that is, T !(t) operates in the space C(M)). The Ito formula can be obviously applied to the functions of the class FC 2(M). This allows us to specify the generator H ! of the semigroup T !(t) for u # FC 2(M), H !u(x)= 12 Tr(A(x)* { 2u(x) A(x))+( a(x)+c A (x), {u(x)) x ,

(62)

c A (x)= 12 Tr({A(x) A(x)). Our next goal is to study the differentiability of the mapping M % x [ ! x (t) # M.

(63)

We give the following definition. Definition 5.1. We say that the random mapping .: M Ä M is p-differentiable in the mean square sense, if for any h # FC 1(M Ä T M) there exists a (random) vector field ' n with values in TM p s.t. for any Hilbert space K and any mapping f # FC 1(M Ä K) we have at each x#M

"

E {f (.(x)) ' h(.(x))&

f (.(x = ))& f (.(x)) =

"

2

Ä 0,

= Ä 0,

(64)

K

where x = means the shift of x along the integral curve of the vector field h at time =. ' h is called the derivative of . in the direction h. We will use the notation !$x (t) h(x) for the derivative ' h(x) of ! x (t) in the direction h at point x. Proposition 5.1.

Let us assume that a # C 2(M p Ä T M p ).

(65)

298

ALBEVERIO, DALETSKII, AND KONDRATIEV

Then the solution ! x (t) of the Eq. (60) is p-differentiable with respect to the initial data x in the mean square sense and the following estimate holds true for some constant C and any h # FC 1(M p Ä TM p ), E &!$x (t) h(x)& T! (t) Mp e tC &h(x)& Tx Mp . x

Proof.

(66)

The proof will be given in Appendix 2. K

Corollary 5.1. If a # C n+1(M p Ä TM p ), then the solution ! x (t) of Eq. (60) is n-times p-differentiable with respect to the initial data x in the mean square sense. In this case the semigroup T !(t) leaves invariant the space C n(M p ). Example 5.1. Brownian Motion with Drift. Let A be such that for each m#M G &1(m)=A(m) A*(m),

(67)

where G is the metric tensor of M. Let us remark that for each metric G the operator field A satisfying (67) exists for some n (see, e.g., [E3]). The solution to system (59) defines in this case the Brownian motion on M with the drift :, : k(x)=a k(x)+c A(x k ).

(68)

The corresponding generator H ! has on FC 2(M) the form H !u(x)= 12 2u(x)+( :(x), {u(x)) x .

(69)

For instance, we can choose R n such that M/R n isometrically (cf. (116)), and set A(m) equal to the orthoprojector P(m): R n Ä T m M. In this case we have c A #0, and the SDE d!(t)=a(!(t)) dt+P(!(t)) b dw(t)

(70)

describes the Brownian motion on M with drift a. 5.2. Parallel Translation and Diffusions on Tensor Bundles Our goal is to construct the parallel translation of differential forms along solutions of (70) and to consider SDE on differential forms. We use the general scheme of [E3]. For this, we need an analog of the LeviCivita connection on M.

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

299

Let OM be the orthonormal frame bundle over M. The space OM has obviously the structure of a compact manifold which fits into the framework of previous sections. We define the product manifold O M := _ OM. k # Zd

(71)

This space will play the role of orthonormal frame bundle over M, with fibres O x M := _ O xk M. k # Zd

(72)

We denote by ?: O M Ä M the corresponding projection. Given the LeviCivita connection on OM, we equip O M with the product connection. This means the following. Let us consider the tangent bundle T(OM) p to O M. By the definition, for each z # O M T s(O M) p =  - p k T zk(OM).

(73)

k # Zd

We define the horizontal tangent space HT z (O M) p resp. the vertical tangent space VT z (O M) p as HT z (O M) p :=  - p k HT zk(OM), k # Zd

(74)

VT z (O M) p :=  - p k VT zk(OM), k # Zd

where HT zk(OM) resp. VT zk(OM) is the horizontal resp. vertical tangent space of OM associated with the given connection. We have obviously the decomposition T z (O M) p =HT z (O M) p Ä VT z (O M) p .

(75)

Let us observe that the corresponding covariant derivative of a vector field X # C 1(M p Ä T M p ) coincides with the derivative {X defined in the Appendix. @ the horizontal lift of the vector field X over M to O M. We denote by X It follows directly from our definition of HT z (O M) p that @(z)) =Y @k (z k ), (X k

(76)

@k is the horizontal lift of the vector field Y k :=X k (?(z 4 )) over M, where Y @ # C m((O M) p Ä T(OM) p ) 4=Z d "[k]. We have then obviously that X provided X # C n(M p Ä T M p ).

300

ALBEVERIO, DALETSKII, AND KONDRATIEV

Let now !(t) be the Brownian motion with the drift : # C 1(M p Ä TM p ) described by SDE (70). We consider its horizontal lift #(t) # O M defined by the SDE d#(t)=a^(#(t)) dt+P (#(t)) b dw(t),

(77)

@h, with initial data #(0) such that ?(#(0))=!(0), where P (z) h :=P(z) which is by Theorem 5.1 uniquely solvable. As in the case of a single manifold M (see [E3]) we have ?(g(t))=!(t). Given n1 and a weight sequence q of positive numbers (not necessarily decreasing), we consider the bundle V := n TM q over M. Let v # V x and define by r(g) v the natural action of g # O(T x M) := _k # Zd O(T xk M) on V x , O(T m M) being the space of orthogonal linear transformations of T m M. We can now define the parallel translation P !(t) : V !(0) Ä V !(t)

(78)

along the solution !(t) of (70) as P !(t) v=r(#(t) #(0) &1 ) v,

(79)

where #(t) solves (77). The transformation P !(t) is obviously orthogonal. Let J be a continuous operator field on M, J(x) # L(V x ). We define the mapping J: O M Ä L(V !(0) ) by the formula &1 J(?(z)) r(zz &1 J(z) :=r(zz &1 0 ) 0 ),

z 0 =#(0),

(80)

and consider the equation d'(t)=J(#(t)) '(t) dt

(81)

in V !(0) . This equation is uniquely solvable for any initial data by general theory of SDE in Hilbert space (see, e.g., [DaF]), and defines a multiplicative functional of the process #(t) and consequently of !(t). Let V$ be the dual bundle with fibres (V$) x :=(V x )$ (which can be naturally identified with the bundle  n TM q &1 ). Let us consider the semigroup T !, J (t) acting in the space C(M Ä V$), which is defined by the expression ( T !, J (t) w(x), y) =E( w(! x (t)), P !x(t) ' y(t)),

(82)

where ' y(t) is the solution to (81) with the initial condition y, and denote by H !, J its generator.

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

Proposition 5.2.

301

For u # FC 2(M Ä V$) we have

H !, Ju(x)= 12 2u(x)+(:(x), {u(x)) x +J*(x) u(x).

(83)

Proof. Let us first assume that :=0 and J=0. This case reduces obviously to the case of a single manifold, and the corresponding generator is equal to 12 2 by the finite dimensional theory [E3]. In the general case, applying the Ito formula to the function 8(z, v)=( u(?(z)), r(zz &1 0 ) v) on OM_V x , we obtain the additional first-order term ( { z 8(z, v), :^(z)) z +( { v 8(z, v), J(z) v) v &1 =(( {u(?(z)), :(?(z))), r(zz &1 0 ) v) +( u(?(z)), J(z) r(zz 0 ) v),

which implies the result. Proposition 5.3.

(84)

K

The semigroup T !, J (t) satisfies the estimate

&T !, J (t) v(x)& (V$)x e tcT !(t) &v& V$ (x),

(85)

v # C(M Ä V$), where c is such that cIJ(x) for each x, as operators in V x . Proof.

We have &'(t)& 2Vx &'(0)& 2Vx +2c

|

t 0

&'({)& 2Vx d{,

(86)

and by Gronwall's inequality &'(t)& Vx e tc &'(0)& Vx .

(87)

Orthogonality of the parallel translation implies that &P !(t) '(t)& V!(t) e tc &'(0)& Vx , which together with (82) implies the result.

(88)

K

Remark 5.2. Let us consider the bundle W := n TM q1 , where the weight sequence q 1 is such that q 1k
(89)

302

ALBEVERIO, DALETSKII, AND KONDRATIEV

which implies that T !, J (t) leaves invariant the space C(M Ä W$), and for v # C(M Ä W$) we have &T !, J (t) v(x)& (W$)x e tc1T !(t) &v& W$(x).

(90)

5.3. Probabilistic Representations of Semigroups Our next goal is to construct probabilistic representations for Bochner and de Rham semigroups and to obtain with their aid a proof of Theorem 4.3. Let us observe that on F0 n we have H B+ =&H !,

(91)

where H ! is the generator of the parallel translation along the paths of the ``stochastic dynamics'' process ! associated with +, defined by SDE (70). If (&H B+ , F0 n ) is essentially self-adjoint, the semigroup T B+(t) has a simple probabilistic interpretation. Indeed, in this case T B+(t) is the unique semigroup with the generator (&H B+ , F0 n ), and therefore for | # C(M Ä  n TM) we have T B+(t) |(x)=E(P * t |(! x (t))).

(92)

We want to obtain a probabilistic representation for the semigroup T R+ associated with H R+ . Let us observe that for | # F0 n we have +

H !, &Rn |=&H R+ |.

(93)

+

The corresponding semigroup T !, &Rn (t) leaves invariant the space C(M Ä  n TM p &1 ), and coincides on this space with T R+ , provided H R+ is essentially self-adjoint on F0 n . Proof of Theorem 4.3. (1) If ; + # C 4(M p Ä T M p ), then both semi+ groups T !(t), T !, &Rn (t) leave invariant the space 0 m n (M p, p &1 ), which obviously contains in the domain of H B+ and H R+ . This implies essential self-adjointness of H B+ and H R+ on 0 2n(M p, p &1 ); see, e.g., [DaF, Theorem IV.1.4]. It can be shown by a similar argument as in [AKR1], 2 that the closure of (H B+ , 0 2n(M p, p &1 )) resp. of (H R + , 0 n(M p, p &1 )) coincides B R with the closure of (H + , F0 n ) resp. of (H + , F0 n ). (2)

This follows from Proposition 5.3.

(3)

This follows from Proposition 5.3 and Remark 5.2.

K

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

303

6. STOCHASTIC DYNAMICS FOR LATTICE MODELS ASSOCIATED WITH GIBBS MEASURES ON PRODUCT MANIFOLDS 6.1. Gibbs Measures on Product Manifolds Let us recall the definition of a Gibbs measure on the Borel _-algebra B(M). Let us consider a family of potentials U=(U 4 ) 4 # 0 , U 4 # C(M 4 ). Let 0(k) be the family of all sets 4 # 0 containing the point k # Z d. We will assume (U1)

sup |U 4(x)| <

:

(94)

4 # 0(k) x # M

for any k # Z d. For any 4 # 0 we introduce the energy of the interaction in the volume 4 with fixed boundary condition ! # M as V 4(x 4 | !)=

:

U 4$( y),

(95)

4$ & 4{<

where y=(x 4 , ! 4c ) # M, 4 c =Z d "4. We define the corresponding Gibbs measure in the volume 4 with boundary condition ! as the measure on B4 :=B(M 4 ) d+ 4(x 4 | !)=

1 e &V4(x4 | !) dx 4 , Z 4(!)

(96)

where dx 4 =} k # 4 dx k is the product of the Riemannian volume measures dx k on M k and Z 4(!)=

|

M4

e &V4(X4 | !) dx 4 .

(97)

These measures are well-defined for any finite volume 4 and all boundary conditions ! # M. For any f # FC(M) we put

|

(E 4 f )(!)= f (x 4 , ! 4c ) d+ 4(x 4 | !).

(98)

304

ALBEVERIO, DALETSKII, AND KONDRATIEV

Definition 6.1. A probability measure + on B(M) is called a Gibbs measure (for a given U) if

|E

4

|

f d+= f d+

(99)

for each 4 # 0 and any f # FC(M). Remark 6.1. Condition (99) is equivalent to the assumption that + 4( } | !) is the conditional measure associated with + under the condition ! 4c . Remark 6.2.

Heuristically + can be given by the expression d+(x)=

1 &E(x) dx, e Z

E(x)= : U 4(x),

(100)

4#0

where dx=  k dx k is the product of the Riemannian volume measures on M k . Let G(U) be the family of all such Gibbs measures. G(U) is non-empty under the condition (94); see, e.g., [G, EFS]. Let us now assume that the family of potentials U satisfies (in addition to (U1)) the condition (U2)

U 4 # C 1(M 4 ) for each 4, and sup : |||{ k U 4 ||| TM <,

(101)

k # Zd 4 # 0

where |||{ k U 4 ||| TM :=sup x # M &{ k U 4(x)& Tx M . k The next statement shows that any Gibbs measure & # G(U) can be completely characterized by its logarithmic derivative. This is proved first in the special situation of path measures in [RZ]. The case of compact spins with finite range of interactions was considered in [AAnAnK]. Theorem 6.1 [ADKR]. The following conditions are equivalent: (i)

the measure & belongs to the class G(U);

(ii) the measure & is differentiable and its vector logarithmic derivative ; + #(; k ) k # Zd is given by ; k (x)=&{ k V k (x), V k (x)= 4 # 4(k) U 4(x). 6.2. Stochastic Dynamics on Functions Let us fix some + # G(U). We assume that, in addition to (U1), (U2), the following condition holds:

305

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

(U3)

U 4 # C 2(4) for each 4, and there exists C< such that sup

: |||{ j { k U 4 ||| TM  TM
:

k # Zd j # Zd

(102)

4#0

where |||{ j { k U 4 ||| TM  TM :=sup x # M &{ j { k U 4(x)& Tx M  Tx j

k

M

.

Remark 6.3. In the case of interactions of finite range the conditions (U1), (U2), and (U3) are obviously fulfilled. Obviously under the conditions (U1), (U2), (U3) we have ; + # Vect 1(M)

(103)

(cf. Definition 7.4), and by Proposition 7.1 there exists a weight sequence p # l 1+ such that ; + # C 1(M p Ä T M p ). Therefore Theorem 4.1 is applicable to this case. Moreover, the essential self-adjointness of H + on FC 2(M) can be proved without the additional condition of 3-times differentiability of ; + [ADKR]. We have the following result. Theorem 6.2 [ADK3, ADK4]. Let the family of interactions U satisfy (U1), (U2), (U3), and + # G(U). Then there exists a unique Markov process ! x with values in M such that its generator H ! coincides with &H + on FC 2(M). H + is an essentially self-adjoint operator on FC 2(M). The semigroup T +(t) acts in the space C(M Ä R 1 ) of continuous functions on M and for u # C(M Ä R 1 ) has the form T +(t) u(x)=E(u(! x (t))). This process is given by Brownian motion on M (constructed in Example 5.1) with a=; +. 6.3. Stochastic Dynamics in the de Rham Complex We see that any Gibbs measure + # G(U) under conditions (U1)(U3) fits the framework of Subsection 4.2, and we have the Weitzenbock representation (35) for de RhamDirichlet operator H R+ . However, Theorem 4.4 requires additional smoothness of potentials. Let us assume that U 4 # C 5(4) for each 4, and sup

:

k # Zd

j1, j2 # Zd

: |||{ j1 { j2 { k U 4 ||| TM  TM <, 4#0

}}} sup

:

k # Zd

j1, ..., j4 # Zd

(104)

: |||{ j1 } } } { j4 { K U 4 ||| TM  TM <, 4#0

where |||{ j1 } } } { jm { k U 4 ||| }m+1 TM := sup &{ j1 } } } { j m { k U 4(x)& Tx x#M

j1

M  } } }  Tx

jm

M  Tx M k

.

306

ALBEVERIO, DALETSKII, AND KONDRATIEV

This condition implies obviously that ; + # Vect 4(M),

(105)

and there exists a weight sequence p # l 1+ such that ; + # C 4(M p Ä T M p ). We summarize our discussion in the following Theorem 6.3. Let the family of interactions U satisfy (U1), (U2), (U3), (104), and + # G(U). Then there exists a weight sequence p # l 1+ such that the statements of Theorems 4.3 and 4.4 hold true. Remark 6.4. Let us observe that, under (U3), we have &{; +(x)& L(Tx Mp &1) C, and therefore &[{; +(x)] 7n& L(n Tx Mp &1 ) nC. Then (52) implies that for n=dim M } k+n 0 we have R +n(x) 12 b(dim M&n 0 ) n 0 &nC,

(106)

where b=inf x # M b(x). Remark 6.5. An application of an approximation technique similar to the one developed in [ADK3, AKR2] gives the possibility to prove the essential self-adjointness of operators H B+ and H R + on F0 n only under the conditions (U1), (U2), (U3), and therefore to avoid the additional condition (104) in Theorem 6.3.

7. APPENDIX 7.1. Differentiable Structures on Product Manifolds The bundle T M p is not the tangent bundle to M p in the proper sense. Indeed, the space Hp :=l 2, p(Z d Ä R N ), to which the spaces T x M p are isomorphic, cannot be considered as the model of M p . Let B(R) be the ball of radius R in Hp and b(r) be the ball of radius r in R N. Then obviously for each fixed j # Z d B(R)# _ b(r k ), k # Zd

(107)

where r j =(1- p j ) R and r k =0, k{j, and for any R there exists j such that r j is so big that b(r j ) cannot in general be contained in the image of any coordinate map of M. Nevertheless T M p gives us the possibility to define analogues of various differentiable structures on M p . Let f be a mapping M Ä B, where B

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

307

is a Banach space. We assume that all partial derivatives { k f (x) # L(T xk M k , B), k # Z d, exist, and introduce the derivative of f as {f (x) :=({ k f (x)) k # Zd .

(108)

{f (x) generates in a natural way (via the Riemannian structure of M) a linear B-valued functional on T x M. Definition 7.1.

We say that f is p-differentiable at x, if {f (x) # L(T x M p , B).

(109)

The space of continuous mappings M Ä B which are p-differentiable at each point with the derivatives bounded uniformly in x will be denoted by C 1(M p Ä B). We will use the notation C 1(M p ) :=C 1(M p Ä R 1 ). Let us introduce the notion of p-differentiability of vector fields !: M Ä T M. Definition 7.2. We say that !: M Ä T M is p-differentiable at x if all partial (covariant) derivatives { k ! j (x) # L(T xk M k , T xj M j ) exist, and {!(x) :=({ k ! j (x)) k, j # Zd # L(T x M p ).

(110)

Remark 7.1. According to the notations introduced above, { k ! j means the covariant derivative in the case k=j and ordinary derivative if k{j. The space of all continuous vector fields with values in T M p which are p-differentiable at each point with the derivatives bounded uniformly in x will be denoted by C 1(M p Ä T M p ). Similarly the notion of p-differentiability of tensor fields over M can be introduced. In particular, we can define the covariant derivative { nf of order n of a function f: M Ä R 1. Definition 7.3. We say that f is n-times p-differentiable at x, if all partial (covariant) derivatives of f up to order n exists, and { nf (x) # Ln(T x M p , R 1 ).

(111)

Let us now assume that the function f is n-times p-differentiable at each point with n th derivatives bounded in the corresponding Ln -norms uniformly in x, and the derivatives of order kn&1 continuous in the following sense: for any h # FC(M Ä T M) we have { kf (x)(h(x), ... , h(x)) # C(M).

(112)

308

ALBEVERIO, DALETSKII, AND KONDRATIEV

The space of such functions will be denoted by C n(M p ). The spaces C n(M p Ä B) and C n(M p Ä TM p ) can be defined similarly. Remark 7.2.

It is easy to see that for any n # N and each p # l 1+ we have FC n(M Ä TM)/C n(M p Ä TM p ).

FC n(M)/C n(M p ),

(113)

The space M p possesses the metric \ p associated in the standard way with the scalar product (8) in the fibres of the tangent bundle TM p , \ p(x, y) 2 =

inf #: #(0)=0, #(1)=1

|

1

(#* (t), #* (t)) p, #(t) dt,

# # C 1([0, 1] Ä M p ).

0

(114)

It is easy to see that \ p(x, y) 2 = : \(x k , y k ) 2 p k .

(115)

k # Zd

A convenient technical tool for the investigation of various objects on M is its embedding into a Hilbert space. It is well known that for K big enough there exists an isometric embedding M/R K.

(116)

M p /Xp :=l 2, p(Z d Ä R K ),

(117)

Then obviously

where the Hilbert space l 2, p(Z d Ä R K ) is defined in a similar way as Hp . Usually the exact choice of the dimension K does not play any role. It is easy to see that the embedding M p /Xp is isometric, that is, T x M p /Xp

(118)

isometrically for each x # M. The metric structures of M p and Xp are equivalent in the sense that \ p(x, y)c 1 &x& y& Xp c 2 \ p(x, y)

(119)

for some constants c 1 , c 2 {0. Nevertheless M p is not a submanifold of Xp . For instance M p does not in general possess a tubular neighborhood in Xp (the Z d-power of a tubular neighborhood of M in R n is not an open set in Xp ). Therefore it is not clear whether any differentiable function on M p can be extended to a differentiable function on Xp .

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

309

The following statements (which are proven in [ADK4]) play an important role in our considerations. v The convergence in any M p coincides with the component-wise convergence. v The topology on M p generated by the metric \ p coincides with the product topology. v The metric space M p is complete. v Let B be a Banach space and f # C 1(M p Ä B). Then f satisfies Lipshitz condition & f (x)& f ( y)& B C\ p(x, y),

(120)

with C=sup x # M &{f (x)& L(Tx Mp , B) . v Any vector field of the class C 1(M p Ä T M p ) has a global flow on M. A natural way of constructing differentiable vector fields over M p is the component-wise construction. Obviously any (formal) vector field a: M Ä TM

(121)

is given by its components a j : M Ä TM j , a j (x) # T xj M j . Definition 7.4. We say that the vector field a: M Ä TM belongs to the class Vect n(M) if each component a j is n-times differentiable and \ x # M sup &a k (x)& Tx M <,

(122)

: |||{ k a j ||| 1 <,

(123)

k # Zd

sup j # Zd

k # Zd

b sup

sup

:

j # Zd

k1, ..., kn&1 # Z

k # Zd

|||{ k { k1 } } } { kn&1 a j ||| n <,

(124)

where |||X||| l =sup x # M &X(x)& l , and & } & l means the corresponding norm in the space of bounded l-linear operators T x1 M  } } }  T xl M Ä T xj M. Proposition 7.1. Let a # Vect n(M). Then, for some weight sequence p # l 1+ , we have a # C n(M p Ä T M p ). Proof. We remark first that (122) implies that for any p # l 1+ we have a # C(M Ä TM p ).

310

ALBEVERIO, DALETSKII, AND KONDRATIEV

Let us consider the infinite matrix R=(R kj ) k, j # Zd with elements R kj = |||{ k a j ||| 1 .

(125)

The condition (123) implies that sup k # Zd

: R kj <.

(126)

j # Zd

This, see [LR], is enough for the existence of a sequence p # l 1+ such that : R kj p k
(127)

k # Zd

for some constant C and any j # Z d. By Schur's test the matrix R generates a bounded operator in l 2, p(Z d Ä R 1 ) with norm less than C (see [LR] and, e.g., [H]). It is easy to see that for any x # M p the matrix {a(x)= ({ k a j (x)) k, j # Zd generates then a bounded operator T x M p Ä T x M p with norm bounded by C uniformly in x. Thus we have proved that a # C 1(M p Ä TM p ). Let us now remark that for two matrixes R 1 , R 2 under condition (126) we can always find p such that (127) is satisfied for both of them and therefore both R 1 , R 2 generate bounded operators in l 2, p(Z d Ä R 1 ). n-times differentiability of a can be now proved by iterating the arguments above. K Remark 7.3. Given a finite number of vector fields of the classes Vect n(M) we can always choose the weight sequence p such that they all are n-times p-differentiable. 7.2. Existence and Uniqueness of Solutions for Infinite Systems of SDE Let us consider SDE d!(t)=a^(!(t)) dt+A (!(t)) b dw(t) on O M. Let us introduce the space S([0, T] Ä O M), T # R + , of random functions !(t, |), t # [0, T], with values in O M, equipped with the uniform metric R p(!, ')=sup E\ O p (!(t), '(t)).

(128)

t

Here \ O is the distance in O M generated by the Riemannian structure in M. We set S z ([0, T] Ä O M)=[! # S([0, T] Ä O M) : !(0, |)=z], z # O M. Given ! # S z ([0, T] Ä O M) we define the process ' k (t) # OM with the stochastic differential d' k (t)=B k (' k(t)), !(t)) dt+B k(' k(t), ! k (t)) B k(?( y k ), ?(z))=r( y k ) A k(z k ),

A k(z k )=r(z k )

&1

A k(?(z k )),

b k(?( y k ), ?(z))=r( y k ) a~ k(z k ), a~ k(z k )=r(z k ) &1 [a k(?(z k ))+ 12 Tr({A k(z k ) A k(z k )]

(129)

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

311

and initial point z k . Such a process can be constructed using a smooth embedding OM Ä R n. Given such an embedding we set ' k(t)=z k +

|

t

b k(' k(s), !(s)) ds+ 0

|

t

B k(' k(s), !(s)) b dw(s),

(130)

0

in R n, where b k resp. B k is considered as R n-valued vector resp. operator field. The Ito formula easily implies that ' k (t) # M provided x k # M. We introduce the mapping J za, A : S z ([0, T] Ä OM) Ä S z ([0, T] Ä O M)

(131)

J za, A(!) k =' k .

(132)

setting

Theorem 7.1. If a # C 1b(M p Ä T M p ) and A is as above, the mapping J defines a continuous mapping of the space (S z ([0, T] Ä O M), R p ) into itself; there exists m # Z + such that the m-composition power (J za, A ) m of J za, A is a contractive mapping of the space (S z ([0, T] Ä O M), R p ) into itself. z a, A

Proof. Let us first remark that the operator field B can be considered as a twice-differentiable mapping B: OM p Ä HS(K, Xp ), Xp :=l z, p(Z d Ä R n ). Then the mapping J za, A will have the form J za, A(!)(t)='(t)=

|

t

b('(s), !(s)) d(s) 0

+

|

t

B('(s), !(s)) dw(s)+ 0

|

t

c('(s), !(s)) ds,

(133)

0

where the third term is the Ito integral and the mapping c is defined by the expression c(z, y)= 12 Tr(A( y)* A( y))

(134)

It follows from the general theory of stochastic integrals that E

"|

t

B('$(s), !$(s)) dw(s)&

0

=E

|

t 0

|

t

"

2

B('"(s), !"(s)) dw(s) 0

Xp

&B('$(s), !$(s))&B('"(s), !"(s))& 2HS(K, Xp ) ds.

(135)

312

ALBEVERIO, DALETSKII, AND KONDRATIEV

Let us remark that c # C 1(M p Ä Xp ). We introduce the mapping X=b+c # C 1(O M p Ä Xp ). Because of formula (119), \ p('$(t), '"(t)) 2 c 1 &'$(t)&'"(t)& 2Xp c 1 +

\| "|

t 0

"

X('$({), !$({))&X('"({), !"({))

t

B('$(s), !$(s)) dw(s)& 0

|

"

2

d{ Xp 2

t

B('"(s), !"(s)) dw(s)

0

" +.

(136)

Xp

Formula (120) and Gronwall's inequality imply that R p(J za, A(!), J za, A('))MT R p(!, '),

(137)

for some constant M, which implies the continuity of J za, A , and R p((J za, A ) m (!), (J za, A ) m ('))

(MT ) m R p(!, '), m!

which implies the contractivity of (J za, A ) m for m big enough.

(138) K

Proof of Theorem 5.1. Existence and uniqueness of solutions on O M follows from the fixed point theorem applied to the contractive mapping (J za, A ) m in a similar way as in the case of SDE on Hilbert spaces; see, e.g., [DaF]. A standard application of Ito formula and Gronwall inequality to the process ! (considered as a process in Xp ) shows that for any z, z$ # O M E &! z (t)&! z$(t)& 2Xp e :t &z&z$& 2Xp

(139)

for some constant :, which implies that 2 ct O 2 E\ O p (! z (t), ! z$(t)) e \ p (z, z$)

(140)

(cf. (119)). It is now enough to remark that the projection of ! z to M gives a solution to SDE (60). Vice versa, each solution to (60) can be uniquely lifted to a solution of the corresponding SDE on O M. K Proof of Proposition 5.1. It is well known that the derivative of the solution ! x (t) to SDE (60) with respect to the initial data x can be described (at least heuristically) by the linear SDE with coefficients given by the derivatives of the coefficients of the initial equation. The solvability of it implies then the differentiability of ! x in the mean square sense. In order to avoid geometrical difficulties and to obtain some estimations for the

STOCHASTIC ANALYSIS ON PRODUCT MANIFOLDS

313

derivative !$x (t), we use the embedding M p /Xp . We assume that the embedding M/R n and therefore M p /Xp is isomeric. Then we have the isometric embeddings T M p /M p Ä Xp ,

(141)

T x M p /Xp . Any operator O # L(T x M p ) can be extended to the operator O # L(Xp ) by = means the orthogonal complement in Xp ). setting Oh=0 for h # T x M = p ( Obviously &O& L(Tx Mp ) =&O& L(Xp ) . Let h # FC 1(M p Ä TM p ) and consider the linear equation in Xp , t t d'(t)=Y(!(t)) '(t) dt+ B(!(t)) '(t) dw(t),

(142)

where Y(x) h(x)=( {X(x), h(x)), B(x) h(x)=( {(A )(x), h(x)), which is uniquely solvable for any initial data and by the general theory of SDE in Hilbert spaces defines a multiplicative functional processes !(t) (see, e.g., [DaF]). Let us consider the sequence of processes ! 1 #z, ! 2 =(J za, A ) m (! 1 ), ..., ! s = (J za, A ) m (! s&1 ), ... # S([0, T] Ä M p ), cf. Theorem 7.1. It is easy to see that for each s=1, 2, ... the process ?(! s(t)) is differentiable w.r.t. x in the sense of Definition 5.1. Let ' sh(x)(t) # T !s(t) M p /Xp be its derivative in the direction h. It can be checked directly by the standard arguments (see, e.g., [BDa]), that ' sh(x)(t) approximates in Xp the solution ' h(t) of Eq. (142) with initial data '(0)=h. This implies that ' h(t)=!$x (t) h(x).

(143)

Standard application of the Ito formula and Gronwall inequality shows that for h # Xp E &' h(t)& Xp e tC &h& Xp ,

(144)

for some constant C, which implies (66). K

ACKNOWLEDGMENTS We are greatly indebted to the late Yuri Daletskii for the steady inspiration his work gave to us. We deeply mourn his departure on December 12, 1997. We are glad to thank K. D. Elworthy, M. Rockner, and A. Thalmaier for useful and stimulating discussions. The financial support by SFB-256, DFG research Project AL 9149-3, and INTAS Project N 378 is gratefully acknowledged.

314

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