##
**Dirichlet forms on loop space.**
*(English)*
Zbl 0683.31002

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.

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