Stochastic Flows on Non-compact Manifolds

TL;DR

This thesis systematically studies stochastic differential equations on non-compact manifolds, solving the open problem of strong completeness and exploring their geometric and topological implications.

Contribution

It proves strong completeness under non-explosion assumptions on non-compact manifolds, extending results beyond compact spaces and linear cases, and analyzes the associated stochastic flows and differential operators.

Findings

Proved strong completeness of SDEs on non-compact manifolds.

Established existence of global smooth solution flows on ^n with unbounded derivatives.

Linked stochastic flow properties to topological and geometric features of manifolds.

Abstract

I was asked to make my, by now quite old PhD thesis, available on the arxiv, for parts of it was never submitted for publication. The thesis offers a systematic study of stochastic differential equations (SDEs) on non-compact spaces. In particular we solve the open problem on strong completeness. An SDE is strongly complete if its solution can be chosen to depend continuously in space and in time for all time. The question is whether non-explosion, with possibly additional assumptions, implies strong completeness. Strong completeness of an SDE implies that its solution depends continuously on the initial condition, opening up possibility for numerical solutions, and the existence of a perfect Cocycle (a basic assumption on random dynamical systems). This was known only for compact manifolds and for linear state spaces, methods for either are not applicable to a general space. We also obtain existence of the global smooth solution flow of SDEs on $R^n$ (sometimes allowing substantial growth of the derivative of the coefficients, removing the global Lipschitz continuity condition). Non-explosion, the $C_0$-property, and the derivative flow are studied. We showed Bismut-Witten Laplacians are essentially self-adjojnt, paving the way for studying theirs semigroups acting on functions and on differential forms. We relate the Markovian semi-group on differential one forms with the semi-group $P_t$ on functions (inter-twining), find a method for proving path integration formulas for $dP_tf$, path integration formula for semi-group on differential forms, moment bounds for the derivative flows. Relation are obtained between intrinsic topological and geometrical properties of the manifold and that of SDEs. Information on the homotopy and cohomology of the manifolds are obtained from moment stability of the stochastic flows.