Vector fields on mapping spaces and related Dirichlet forms and diffusions (Q1378321)

The authors introduce the notion of a Caratheodory-Finsler manifold (C-F manifold), i.e. a Finsler manifold \(M\) such that the Caratheodory metric is complete and generates the topology on \(M\). It is well-known that for a compact metric space \(S\) the space \(E:= C(S;M)\) of all continuous maps from \(S\) to a Finsler manifold \(M\) is again a Finsler manifold in a natural way. The authors show that if \(M\) is, in addition, a C-F manifold, so is \(C(S;M)\), providing a large number of examples of C-F manifolds such as path and loop spaces. Then for a probability measure \(\mu\) on a C-F manifold \(E\), they study a class of so-called \({\mathcal D}\)-admissible resp. strongly \({\mathcal D}\)-admissible vector fields on \(M\) which act on a linear space \({\mathcal D} \subset C^1_b(E)\) and which lead to closable pre-Dirichlet forms on \(L^2 (E;\mu)\) resp. (which is stronger) satisfy an integration by parts formula w.r.t. \(\mu\). Assuming one is given countably many such vector field so that the corresponding pre-Dirichlet forms with domain \({\mathcal D}\) are summable, hence the sum is closable, the authors then prove, employing a method due to Byron Schmuland, that the closure is a quasi-regular Dirichlet form or equivalently generates a diffusion process on \(E\). A number of examples are discussed.