CHAPTER VIII - REGULARITY

Summary

THE INDUCED PROCESS ON THE DIFFEOMORPHISM GROUPS

(A) We shall use some results about the diffeomorphism groups of compact manifolds to obtain information about the flow F_t of an S.O.S. and the uniform convergence of piecewise li.near approximations. This was described, for compact manifolds, i.n (Elworthy 1978). The method mimics that used by Ebin & Marsden (1970) for ordinary differential equations.

Instead of using diffeomorphism groups of compact manifolds some readers might prefer to consider groups of diffeomorphisms of Rⁿ which are the identity outside the unit ball. Many of the properties are easier to see for these groups and they can be used in the compact manifold case by embedding the manifold in some open unit ball and then extending the S.O.S. so that it is the identity in a neighbourhood of the boundary of the ball. This is described in detail in (Carverhill & Elworthy 1982) where the method is used to give a unified treatment of many recent results, e.g. the generalized Itô formula (Bismut 1981a, b) and the criterion for the flow to be a diffeomorphism (Kunita 1981, 1982). A similar technique, using nuclear spaces is employed in (Ustunel).