  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.
