Differentiable approximations to Brownian motion on manifolds

Abstract

The main part of this thesis is devoted to generalised Ornstein-Uhlenbeck processes. We show how to construct such processes on 2-uniformly smooth Banach spaces. We give two methods of constructing Ornstein-Uhlenbeck type processes on manifolds with sufficient structure, including on finite dimensional Riemannian manifold where we actually construct a process on the orthonormal bundle 0(M) and project down to M to obtain the required process. We show that in the simplest case on a finite dimensional Riemannian manifold the two constructions give rise to the same process. We construct the infinitesimal generator of this process. We show that, given a Hilbert space and a Banach space E with W a Brownian motion on E whose index set includes [0,R], and X: H->L(E;H), V:H->H satisfying sufficient boundedness and Lipschitz conditions, the solutions of the family of o.d.e.'s dxβ=X(xβ)vβdt+V(xβ)dt (where vβ is an O-U velocity process on E), indexed by weΩ where Ω is the probability space over which W is defined, converges in L2-norm to a solution of dx=X(x)dW+V(x)dt, both solutions having the same starting point. We show that the convergence is uniform over [O, R] in probability, and include a proof of Elworthy, from 'Stochastic Differential Equations on Manifolds' (Warwick University preprint, 1978) to show that convergence still occurs when the processes are constructed on suitable manifolds (Elworthy's proof is for piecewise linear approximations). We extend our results to include 0-U processes in 'force-fields'. We follow the method of Elworthy to show the uniform convergence of the flows of the constructed processes. Finally we prove similar convergence theorems for piecewise-linear approximations, following the proofs of Elworthy