15386007
The Heat Semigroup on Configuration Spaces
In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural "Riemannian-like" structure of the configuration space X over a complete, connected, oriented, and stochastically complete Rie- mannian manifold X of infinite volume, the heat semigroup (e −tH )t∈R+ was introduced and studied in (J. Func. Anal. 154 (1998), 444-500). Here, H is the Dirichlet operator of the Dirichlet form E over the space L 2 ( X,�m), wherem is the Poisson measure on X with intensity m—the volume mea- sure on X. We construct a metric space ∞ that is continuously embedded into X. Under some conditions on the manifold X, we prove that ∞ is a set of fullm measure and derive an explicit formula for the heat semigroup: (e −tH F)() = R ∞ F(�) Pt,(d�), where Pt, is a probability measure on ∞ for all t > 0, ∈ ∞. The central results of the paper are two types of Feller prop- erties for the heat semigroup. The first one is a kind of strong Feller property with respect to the metric on the space ∞. The second one, obtained in the case X = R d , is the Feller property with respect to the intrinsic metric of the Dirichlet form E. Next, we give a direct construction of the independent infi- nite particle process on the manifold X, which is a realization of the Brownian motion on the configuration space. The main point here is that we prove that this process can start in every ∈ ∞, will never leave ∞, and has continuous sample path in ∞, provided dimX ≥ 2. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the Pt,(� ) above. Furthermore, we discuss the necessary changes to be done for constructing the process in the case dimX = 1. Finally, as an easy consequence we get a "path-wise" construction of the independent particle process on ∞ from the underlying Brownian motion.
16873301
Analysis and Geometry on Configuration Spaces
Abstract In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space Γ X over a Riemannian manifold X . This geometry is “non-flat” even if X = R d . It is obtained as a natural lifting of the Riemannian structure on X . In particular, a corresponding gradient ∇ Γ , divergence div Γ , and Laplace–Beltrami operator H Γ =−div Gamma;  ∇ Γ are constructed. The associated volume elements, i.e., all measures μ on Γ X w.r.t. which ∇ Γ and div Gamma; become dual operators on L 2 ( Γ X ;  μ ), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X . In particular, all these measures obey an integration by parts formula w.r.t. vector fields on Γ X . The corresponding Dirichlet forms E Γ μ on L 2 ( Γ X ;  μ ) are, therefore, defined. Each is shown to be associated with a diffusion process which is thus the Brownian motion on Γ X and which is sub- sequently identified as the usual independent infinite particle process on X . The associated heat semigroup ( T Γ μ ( t )) t >0 is calculated explicitly. It is also proved that the diffusion process, when started with μ , is time-ergodic (or equivalently E Gamma; μ is irreducible or equivalently ( T Γ μ ( t )) t >0 is ergodic) if and only if μ is Poisson measure π z dx with intensity z dx for some z ⩾0. Furthermore, it is shown that the Laplace–Beltrami operator H Γ =−div Gamma; ∇ Gamma; on L 2 ( Γ X ; π z dx ) is unitary equivalent to the second quantization of the Laplacian − Δ X on X on the corresponding Fock space ⊗ n ⩾0 L 2 ( X ;  z dx ) ⊗  n . As another direct consequence of our results we obtain a representation of the Lie-algebra of compactly supported vector fields on X on Poisson space. Finally, generalizations to the case where dx is replaced by an absolutely continuous measure and also to interacting particle systems on X are described, in particular, the case where the mixed Poisson measures μ are replaced by Gibbs measures of Ruelle-type on Γ X .
15622803
Analysis and Geometry on Configuration Spaces: The Gibbsian Case☆
Using a natural “Riemannian geometry-like” structure on the configuration spaceΓover Rd, we prove that for a large class of potentialsφthe corresponding canonical Gibbs measures on 1 can be completely characterized by an integration by parts formula. That is, if ∇Γ is the gradient of the Riemannian structure onΓone can define a corresponding divergence divΓφsuch that the canonical Gibbs measures are exactly those measuresμfor which ∇Γ and divΓφare dual operators onL2(Γ, μ). One consequence is that for suchμthe corresponding Dirichlet forms EΓμare defined. In addition, each of them is shown to be associated with a conservative diffusion process onΓwith invariant measureμ. The corresponding generators are extensions of the operatorΔΓφ :=divΓφ ∇Γ. The diffusions can be characterized in terms of a martingale problem and they can be considered as a Brownian motion onΓperturbed by a singular drift. Another main result of this paper is the following: Ifμis a canonical Gibbs measure, then it is extreme (or a “pure phase”) if and only if the corresponding weak Sobolev spaceW1, 2(Γ, μ) onΓis irreducible. As a consequence we prove that for extreme canonical Gibbs measures the above mentioned diffusions are time-ergodic. In particular, this holds for tempered grand canonical Gibbs measures (“Ruelle measures”) provided that the activity constant is small enough. We also include a complete discussion of the free case (i.e.,φ≡0) where the underlying space Rdis even replaced by a Riemannian manifoldX.
10834840
Stochastic analysis on configuration spaces: basic ideas and recent results
The purpose of this paper is to provide a both comprehensive and summarizing account on recent results about analysis and geometry on configuration spaces ΓX over Riemannian manifolds X. Particular emphasis is given to a complete description of the so–called “lifting– procedure”, Markov resp. strong resp. L1–uniqueness results, the non–conservative case, the interpretation of the constructed diffusions as solutions of the respective classical “heuristic” stochastic differential equations, and a self–contained presentation of a general closability result for the corresponding pre–Dirichlet forms. The latter is presented in the general case of arbitrary (not necessarily pair) potentials describing the singular interactions. A support property for the diffusions, the intrinsic metric, and a Rademacher theorem on ΓX , recently proved, are also discussed. AMS Subject Classification Primary: 60G60, 82C22 Secondary: 31C25, 82C31, 60H07, 60J60, 53B99, 82C21
16299239
Rademacher's Theorem on Configuration Spaces and Applications☆☆☆
Abstract We consider an L 2 -Wasserstein type distance ρ on the configuration space Γ X over a Riemannian manifold X , and we prove that ρ -Lipschitz functions are contained in a Dirichlet space associated with a measure on Γ X satisfying certain natural assumptions. These assumptions are in particular fulfilled by the classical Poisson measures and by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of ρ for a set to be exceptional. This result immediately implies, for instance, a quasi-sure version of the spatial ergodic theorem. We also show that ρ is optimal in the sense that it is the intrinsic metric of our Dirichlet form.
13919235
A Support Property for Innnite Dimensional Interacting Diiusion Processes. Une Propri Et E De Support Pour Des Processus De Diiusion En Dimension Innnie Avec Interaction
The Dirichlet form associated with the intrinsic gradient on Poisson space is known to be quasi-regular on the complete metric space ? = fZ +-valued Radon measures on IR d g. We show that under mild conditions, the set ? n ? is E-exceptional, where ? is the space of locally nite conngurations in IR d , that is, measures 2 ? satisfying sup x2IR d (fxg) 1. Thus, the associated diiusion lives on the smaller space ?. This result also holds for Gibbs measures with superstable interactions. Abstract Il est connu que la forme de Dirichlet associ ee au gradient intrins eque sur l'espace de Poisson est quasi-r eguli ere sur l'espace m etrique complet
15801431
Laplace operators and diffusions in tangent bundles over Poisson spaces
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on 1-forms and associated semigroups are considered. Their probabilistic interpretation is given.
17724280
LAPLACE OPERATORS ON DIFFERENTIAL FORMS OVER CONFIGURATION SPACES
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given.
118843692
On a martingale method for symmetric diffusion processes and its applications
Let X be a locally compact separable metric space and in be a Radon measure on X whose support is the whole space X. A symetric Markov process associated with a Dirichlet space on L 2 (X,m) is considered. Using a formula of Lyons-Zheng for an additive functional this process is studied by means of the martingale theory. A conservativeness criterion, a tightness criterion and some sample path properties for symmetric diffusion processes are found, with extensions of the method to non-symmetric situations
122000289
Heat kernels and spectral theory
Preface 1. Introductory concepts 2. Logarithmic Sobolev inequalities 3. Gaussian bounds on heat kernels 4. Boundary behaviour 5. Riemannian manifolds References Notation index Index.
122832312
HARMONIC ANALYSIS ON CONFIGURATION SPACE I: GENERAL THEORY
We develop a combinatorial version of harmonic analysis on configuration spaces over Riemannian manifolds. Our constructions are based on the use of a lifting operator which can be considered as a kind of (combinatorial) Fourier transform in the configuration space analysis. The latter operator gives us a natural lifting of the geometry from the underlying manifold onto the configuration space. Properties of correlation measures for given states (i.e. probability measures) on configuration spaces are studied including a characterization theorem for correlation measures.
121004187
Correlation functions and the uniqueness of the state in classical statistical mechanics
AbstractA general criterion is derived which assures the uniqueness of the state of a classical statistical mechanical system in terms of a given system of correlation functions. The criterion is
$$\sum\limits_k {(m_{k + j}^A )} ^{ - 1/k} = \infty$$
 for allj and all bounded setsA, where
$$m_k^A = (k!)^{ - 1} \int\limits_A \cdots \int\limits_A {\varrho _k } (x_1 , \ldots ,x_k )dx_1 \ldots dx_1 .$$

121329419
States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures
This paper is devoted to the solution of the problem of characterizing correlation measures arising naturally in classical statistical mechanics of point particles. A correlation measure ρ must be related to a (not necessarily unique) probability measure μ over an infinite particle configuration space X by the formula μ(H)=∝NH(ξ)dμ where {NH} is a certain family of integer valued random variables. We prove that there are three conditions, namely (S) symmetry, (P) positivity, and (N) normalization, which together are sufficient as well as necessary for ρ to be a correlation measure. The main operative condition is (P), which says that ξφ(x)dρ≧0 must hold for every function φ for which Sφ(ξ)≧0 identically for ξ ε X, where φ → Sφ is a certain linear operator whose properties we study. Condition (P) gives rise to a large class of inequalities satisfied by the ρ-measures of certain sets. The theory is also generalized to the case when there is a group of translations acting in the one-particle space, the concern then being with measures ρ as well as μ that are invariant with respect to the group.
119402685
THE COINCIDENCE APPROACH TO STOCHASTIC POINT PROCESSES
The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for 'completely' regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the 'fermion' process that is suitable whenever the points are repulsive.
117640453
On a spectral representation for correlation measures in configuration space analysis
The paper is devoted to the study of configuration space analysis by using the projective spectral theorem. For a manifold $X$, let $\Gamma_X$, resp.\ $\Gamma_{X,0}$ denote the space of all, resp. finite configurations in $X$. The so-called $K$-transform, introduced by A. Lenard, maps functions on $\Gamma_{X,0}$ into functions on $\Gamma_{X}$ and its adjoint $K^*$ maps probability measures on $\Gamma_X$ into $\sigma$-finite measures on $\Gamma_{X,0}$. For a probability measure $\mu$ on $\Gamma_X$, $\rho_\mu:=K^*\mu$ is called the correlation measure of $\mu$. We consider the inverse problem of existence of a probability measure $\mu$ whose correlation measure $\rho_\mu$ is equal to a given measure $\rho$. We introduce an operation of $\star$-convolution of two functions on $\Gamma_{X,0}$ and suppose that the measure $\rho$ is $\star$-positive definite, which enables us to introduce the Hilbert space ${\cal H}_\rho$ of functions on $\Gamma_{X,0}$ with the scalar product $(G^{(1)},G^{(2)})_{{\cal H}_{\rho}}= \int_{\Gamma_{X,0}}(G^{(1)}\star\bar G{}^{(2)})(\eta) \rho(d\eta)$. Under a condition on the growth of the measure $\rho$ on the $n$-point configuration spaces, we construct the Fourier transform in generalized joint eigenvectors of some special family $A=(A_\phi)_{\phi\in\D}$, $\D:=C_0^\infty(X)$, of commuting selfadjoint operators in ${\cal H}_\rho$. We show that this Fourier transform is a unitary between ${\cal H}_{\rho}$ and the $L^2$-space $L^2(\Gamma_X,d\mu)$, where $\mu$ is the spectral measure of $A$. Moreover, this unitary coincides with the $K$-transform, while the measure $\rho$ is the correlation measure of $\mu$.
54689496
Heat kernel upper bounds on a complete non-compact manifold.
Let M be a smooth connected non-compact geodesically complete Riemannian manifold, ? denote the Laplace operator associated with the Riemannian metric, n = 2 be the dimension of M. Consider the heat equation on the manifold

ut - ?u = 0,

where u = u(x,t), x I M, t > 0. The heat kernel p(x,y,t) is by definition the smallest positive fundamental solution to the heat equation which exists on any manifold (see [Ch], [D]). The purpose of the present work is to obtain uniform upper bounds of p(x,y,t) which would clarify the behaviour of the heat kernel as t ? +8 and r = dist(x,y) ? +8.
61265778
Probability Metrics and the Stability of Stochastic Models
General topics in the theory of probability metrics relations between compound, simple and primary distances applications of minimal p. distances ideal metrics.