Analysis and geometry on R+-marked configuration spaces

Original · EN

We carry out analysis and geometry on a marked configuration space Ωₓʳ⁺ over a Riemannian manifold X with marks from the space R+ as a natural generalization of the work [J. Func. Anal. 154 (1998), 444--500]. As a transformation group G on this space, we take the ``lifting'' to Ωₓʳ⁺ of the action on X× R+ of the semidirect product of the group Diff of diffeomorphisms on X with compact support and the group R+ˣ of smooth currents, i.e., all C∞ mappings of X into R+ which are equal to one outside a compact set. The marked Poisson measure π on Ωₓʳ⁺ with Lévy measure σ is proven to be quasiinvariant under the action of G. Then, we derive a geometry on Ωₓʳ⁺ by a natural ``lifting'' of the corresponding geometry on X× R+. In particular, we construct a gradient ∇Ω and divergence divΩ. The associated volume elements, i.e., all probability measures μ on Ωₓʳ⁺ with respect to which ∇Ω and divΩ become dual operators on L²(Ωₓʳ⁺,μ) are identified as the mixed Poisson measures with mean measure equal to a multiple of σ. As a direct consequence of our results, we obtain marked Poisson space representations of the group G and its Lie algebra g. We investigate also Dirichlet forms and Dirichlet operators connected with (mixed) marked Poisson measures. In particular, we obtain conditions of ergodicity of the semigroups generated by the Dirichlet operators. A possible generalization of the results of the paper to the case where the marks belong to a homogeneous space of a Lie group is noted.