K. David Elworthy

Generalized Ito formulae and space-time Lebesgue-Stieltjes integrals of local times

Generalized Ito formulae are proved for time dependent functions of continuous real valued semi-martingales. The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded two-dimensional variation. In particular a class of functions with discontinuous first derivative is included. An estimate of Krylov allows further weakening of these conditions when the semi-martingale is a diffusion.

Bisimut type formulae for differential forms

Abstract Formulae are given dPt, d*Pt, and ΔPtAP,c/> for Pt the heat semigroup acting on a q-form . The formulae are Brownian motion expectations ofcomposed with random translations determined by Weitzenbock curvature terms. Derivatives of the curvature are not involved.

Equivariant Diffusions on Principal Bundles

Let M be a smooth finite dimensional manifold and P(M,G) a principal fibre bundle over M with structure group G a Lie group

The Space of Stochastic Differential Equations

One of the main tools arising from Ito’s calculus is the theory of stochastic differential equations, now with applications to many areas of science, economics and finance. This article is a remark on some aspects of the geometry and topology of certain spaces of stochastic differential equations, making no claims to relevance to the actual theory or its applications. It is based on work with Yves LeJan & Xue-Mei Li reported in [ELL99],[ELJL04] and in preparation in [ELJL]. It was stimulated by contacts with Steve Rosenberg and his article with Sylvie Paycha, [PR04]. However the topological constructions and remarks, in all except 2.4 (which is taken from [ELJL]), are essentially well known and any novelty arises from their interpretation in terms of stochastic differential equations and flows.

Intertwining and the Markov Uniqueness Problem on Path Spaces

There are two open problem on the analysis of continuous paths on a Riemannian manifold, the Markov uniqueness and the independence of the closure of the differential operator

Filtering with non-Markovian Observations

d

Example: Riemannian Submersions and Symmetric Spaces

on its initial domain. The operator

The Commutation Property

d

Decomposition of Diffusion Operators

acts naturally on

Example: Stochastic Flows

BC^1