Vector fields on mapping spaces and related Dirichlet forms and diffusions

Abstract

A. Let μ be a Radon measure on an infinite dimensional smooth manifold E. Associated to μ there are various additional structures on E. This is seen from the example of Gaussian where E is a separable Banach space inducing an abstract Wiener space structure on E, from the example of path and loop on finite dimensional Riemannian manifolds with measures induced by Brownian motions and Brownian bridges which are usefully analyzed using special tangent spaces [18], from the notions of of measures leading to classes of admissible vector fields describing the directions in which μ can be differentiated [4], and from very general considerations [11]. Here we describe a class of vector fields determined by μ and the differential structure of E which also have a claim to be called admissible but are defined in terms of Dirichlet form theory rather than differentiability of μ. Finite or suitably bounded countable families of such vector fields are shown to give rise to quasi-regular Dirichlet forms on E with their associated diffusion processes, Markovian semigroups, and infinitesimal generators. The ideas are valid for general separable metrizable manifolds but an adequately rich class of differentiable test functions is needed. Such would be assured if E were modeled on a space admitting smooth partitions of unity with bounded derivatives. However this is not so for of continuous paths (such as classical Wiener space) and for such mapping it is often convenient to use cylindrical functions. On the other hand we wish to include such cases as iterated path (paths on path spaces) and other examples of of maps into infinite dimensional manifolds. To do this we introduce in Section 2 the notion of a Caratheodory-Finsler (C-F) manifold: a class of Finsler manifolds possessing a rich enough family of test functions. Closed submanifolds of separable Banach spaces, with induced Finsler structure are C-F manifolds, as is the space of continuous maps of a compact metric space into a C-F manifold. In this way we are able to give a unified treatment which covers and extends the existing results on path and loop spaces.