Results 1 
7 of
7
Analysis and Geometry on Configuration Spaces
, 1997
"... 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 \Gamma X over a Riemannian manifold X ..."
Abstract

Cited by 72 (10 self)
 Add to MetaCart
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 \Gamma X over a Riemannian manifold X. This geometry is "nonflat" even if X = IR d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient r \Gamma , divergence div \Gamma , and LaplaceBeltrami operator H \Gamma = \Gammadiv \Gamma r \Gamma are constructed. The associated volume elements, i.e., all measures ¯ on \Gamma X w.r.t. which r \Gamma and div \Gamma become dual operators on L 2 (\Gamma X ; ¯), 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 \Gamma X . The corresponding Dirichlet...
Diffeomorphism Groups And Current Algebras: configuration space analysis in quantum theory
 PREPRINT 97073, SFB 343
, 1997
"... The constuction of models of nonrelativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space \Gamma of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on \Gamma, being the fr ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
The constuction of models of nonrelativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space \Gamma of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on \Gamma, being the free Hamiltonian. The case with interactions is also discussed together with its relation to the problem of unitary representations of the diffeomorphism group on R^d.
Canonical Dirichlet operator and distorted Brownian motion on Poisson spaces
, 1996
"... We derive the precise relation of the internal differential geometry of Poisson spaces developed in [1] to the corresponding external geometry given by the associated Fock structure. In particular, we identify the (internal) LaplaceBeltrami operator H \Gamma ß oe on Poisson space as the second qu ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We derive the precise relation of the internal differential geometry of Poisson spaces developed in [1] to the corresponding external geometry given by the associated Fock structure. In particular, we identify the (internal) LaplaceBeltrami operator H \Gamma ß oe on Poisson space as the second quantization of the Laplacian on IR d . Furthermore, the diffusion process generated by H \Gamma ß oe , which is a kind of distorted Brownian motion on the configuration space, is described explicitly. All this generalizes to the case where the underlying Euclidean space IR d is replaced by an (infinitedimensional) manifold.
The heat semigroup on configuration spaces
, 1992
"... In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural “Riemannianlike ” structure of the configuration space ΓX over a complete, connected, oriented, and stochastically complete Riemannian manifold X of infinite volume, the heat semigroup (e−tHΓ)t ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural “Riemannianlike ” structure of the configuration space ΓX over a complete, connected, oriented, and stochastically complete Riemannian manifold X of infinite volume, the heat semigroup (e−tHΓ)t∈R+ was introduced and studied in [J. Func. Anal. 154 (1998), 444–500]. Here, H Γ is the Dirichlet operator of the Dirichlet form E Γ over the space L 2 (ΓX,πm), where πm is the Poisson measure on ΓX with intensity m—the volume measure on X. We construct a metric space Γ ∞ that is continuously embedded into ΓX. Under some conditions on the manifold X, we prove that Γ ∞ is a set of full πm measure and derive an explicit formula for the heat semigroup: (e−tHΓF)(γ) = ∫ Γ ∞ F(ξ)Pt,γ(dξ), where Pt,γ is a probability measure on Γ ∞ for all t> 0, γ ∈ Γ∞. The central results of the paper are two types of Feller properties
ON THE STRUCTURE OF SPATIAL BRANCHING PROCESSES
"... Abstract. The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes (Φk)k∈Z by almost sure properties of their realizations witho ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes (Φk)k∈Z by almost sure properties of their realizations without making any assumptions about stationarity or existence of moments. This enables us to classify the clans of (Φk) into the regular part and the completely nonregular part. It turns out that the completely nonregular branching processes are built up from singleline processes, whereas the regular ones are mixtures of lefttail trivial processes with a Poisson family structure. 1. Introduction. The