Stochastic differential equations on Banach manifolds - Brzeźniak, Z.; Elworthy, K. D. (Methods of Functional Analysis and Topology, 2025)
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^θ,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.