Dirichlet forms on loop space. (English) Zbl 0683.31002

Getzler, Ezra

Bull. Sci. Math., II. Sér. 113, No. 2, 151-174 (1989).

Pre-Dirichlet forms on a Wiener space (B,H,\(\mu)\) of the type \[ {\mathcal E}_{\pi}(f,f)=\int_{B}| d_{\pi}f|^ 2_ Hd\mu,\quad f\in W^{\infty}(B), \] are studied. Here \(W^{\infty}(B)\) denotes the set of Malliavin test functions on B, \(\pi\) is a map from B to some Riemannian manifold M satisfying Malliavin’s regularity condition \[ \det ((d\pi)(d\pi)^*)^{-1}\in L^ p(B;\mu)\quad for\quad all\quad p<\infty \] and \(d_{\pi}\) denotes “differentiation along the fibres of \(\pi\) ”. It is indicated how to construct an associated diffusion process and a formula for the Ricci curvature Ric of \({\mathcal E}_{\pi}\) in the sense of D. Bakry and M. Emery [C. R. Acad. Sci. Paris, Sér. I, T. 299, 1984, pp. 999-1000] is derived. Finally, the operator Ric is calculated in the case where M is equal to a compact Lie group G and \(\pi\) comes from the so-called path-ordered exponential in which case the fibre of \(\pi\) (when restricted to H) over the identity e of G can be identified with the space of based loops in G.
Despite the value of the paper as a very stimulating piece of work there are some points in it which are not dealt with in a sufficiently precise way. For example, the fundamental question whether \({\mathcal E}_{\pi}\) with domain \(W^{\infty}(B)\) is closable as a quadratic form on \(L^ 2(B;\mu)\), hence leads, indeed, to a Dirichlet form (which is necessary for the existence of an associated diffusion on process), is not discussed.

Reviewer: M.Röckner

MSC:

31C25	Dirichlet forms
60H07	Stochastic calculus of variations and the Malliavin calculus
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J65	Diffusion processes and stochastic analysis on manifolds
22E99	Lie groups

Keywords:

Pre-Dirichlet forms; Wiener space; associated diffusion process; Ricci curvature