Stochastic differential equations on Banach manifolds

Abstract

The theory of Stratonovich stochastic differential equations on
separable Banach manifolds modelled on M-type 2 Banach spaces is described.
Properties of the Sobolev-Slobodetskii spaces $W^{\theta,p}$ are given and they are
shown to be M-type 2. Moreover it is shown that the Nemytski maps are
locally Lipschitz on the Sobolev-Slobodetskii spaces
in certain ranges of their parameters.
Besov-Slobodetskii  spaces of loops
on Riemannian manifolds are constructed, with their manifold structure,
of such a class that the natural Brownian motion induced measures of stochastic
analysis are supported  on them. Some special stochastic differential equations are
constructed on these loop spaces including those which would give the
Brownian motions on loop groups described by Malliavin and Malliavin.
Conditions are given for non-explosion, and a semigroup on
differential forms is observed.