Derivative Flows of Stochastic Differential Equations: Moment Exponents and Geometric Properties

this paper have been submitted for publication elsewhere. it is shown that the behaviour of the linearization is intimately connected with the topology of the underlying state space. Some of these results and their geometric consequences are described in xx1,2,4 below, while in x3 we describe criteria in terms of the coefficients of the equation which give the relevant properties, and even existence in the non-compact case, of the flows. The derivative flow arises naturally from differentiation under the expectation sign in the Feyman-Kac formula. This leads to useful, and simple, formulae exhibiting the smoothing properties of non-degenerate Markov semigroups [EL94] extending Bismut's well known formula ( which uses the Ricci curvature of the associated metric). There are also analogous formulae for the heat equation for q-forms on a Riemannian manifold, and these are described in x5