Stochastic analysis on configuration spaces: basic ideas and recent results

TL;DR

This paper reviews recent advances in the analysis and geometry of configuration spaces over Riemannian manifolds, focusing on lifting procedures, uniqueness results, diffusions, and Dirichlet forms, with applications to stochastic differential equations.

Contribution

It provides a comprehensive overview of recent results, including a general closability theorem for Dirichlet forms with singular interactions and detailed analysis of diffusions on configuration spaces.

Findings

Complete description of the lifting procedure

Results on Markov and strong uniqueness

A general closability theorem for Dirichlet forms

Abstract

The purpose of this paper is to provide a both comprehensive and summarizing account on recent results about analysis and geometry on configuration spaces $\Gamma_X$ over Riemannian manifolds $X$. Particular emphasis is given to a complete description of the so--called ``lifting--procedure'', Markov resp. strong resp. $L^1$--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 $\Gamma_X$, recently proved, are also discussed.