Geometric stochastic analysis on path spaces

TL;DR

This paper develops a geometric stochastic analysis framework on path spaces of Riemannian manifolds, using Brownian motion measures and stochastic differential equations to define calculus and Laplacian operators.

Contribution

It introduces a novel approach to analysis on manifold path spaces via Itô maps and stochastic differential equations, addressing fundamental open problems in calculus and Hodge theory.

Findings

Established a method for exterior differentiation on path spaces.

Constructed a Hodge-Kodaira operator and decomposition for one and two forms.

Extended the approach to more general diffusion measures.

Abstract

An approach to analysis on path spaces of Riemannian manifolds is described. The spaces are furnished with `Brownian motion' measure which lies on continuous paths, though differentiation is restricted to directions given by tangent paths of finite energy. An introduction describes the background for paths on ${\mathbb R}^m$ and Malliavin calculus. For manifold valued paths the approach is to use `It\^o' maps of suitable stochastic differential equations as charts . `Suitability' involves the connection determined by the stochastic differential equation. Some fundamental open problems concerning the calculus and the resulting `Laplacian' are described. A theory for more general diffusion measures is also briefly indicated. The same method is applied as an approach to getting over the fundamental difficulty of defining exterior differentiation as a closed operator, with success for one \& two forms leading to a Hodge -Kodaira operator and decomposition for such forms. Finally there is a brief description of some related results for loop spaces.