Analysis and Geometry on Configuration Spaces

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 spaceΓXover a Riemannian manifoldX. This geometry is “non-flat” even ifX=Rd. It is obtained as a natural lifting of the Riemannian structure onX. In particular, a corresponding gradient ∇Γ, divergence divΓ, and Laplace–Beltrami operatorHΓ=−divGamma; ∇Γare constructed. The associated volume elements, i.e., all measuresμonΓXw.r.t. which ∇Γand divGamma;become dual operators onL2(ΓX; μ), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume elementdxonX. In particular, all these measures obey an integration by parts formula w.r.t. vector fields onΓX. The corresponding Dirichlet forms EΓμonL2(ΓX; μ) are, therefore, defined. Each is shown to be associated with a diffusion process which is thus the Brownian motion onΓXand which is sub- sequently identified as the usual independent infinite particle process onX. The associated heat semigroup (TΓμ(t))t>0is calculated explicitly. It is also proved that the diffusion process, when started withμ, is time-ergodic (or equivalently EGamma;μis irreducible or equivalently (TΓμ(t))t>0is ergodic) if and only ifμis Poisson measureπz dxwith intensityz dxfor somez⩾0. Furthermore, it is shown that the Laplace–Beltrami operatorHΓ=−divGamma;∇Gamma;onL2(ΓX;πz dx) is unitary equivalent to the second quantization of the Laplacian −ΔXonXon the corresponding Fock space ⊗n⩾0 L2(X; z dx)⊗ n. As another direct consequence of our results we obtain a representation of the Lie-algebra of compactly supported vector fields onXon Poisson space. Finally, generalizations to the case wheredxis replaced by an absolutely continuous measure and also to interacting particle systems onXare described, in particular, the case where the mixed Poisson measuresμare replaced by Gibbs measures of Ruelle-type onΓX.