Dowell, Richard Malcolm (1980) Differentiable approximations to Brownian motion on manifolds. PhD thesis, University of Warwick.

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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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Ornstein-Uhlenbeck process, Banach spaces, Riemannian manifolds
Official Date:	March 1980
Dates:	Date Event March 1980 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Elworthy, K. D.
Sponsors:	Science Research Council (Great Britain) (SRC)
Extent:	iv, 104 p.
Language:	eng

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