Infinite-dimensional degree theory and stochastic analysis

Abstract

The main aim of this paper is to describe how stochastic analysis is applied to infinite-dimensional degree theory for measurable maps of Banach spaces and Fredholm maps between Banach manifolds. It is based on work of Getzler, Kusuoka, andÜstünel & Zakai.Topics include the following: measure-theoretic versions of Sard's theorem and inequality, pull-backs of measures by Fredholm maps, integral formulae for the degree, infinite-dimensional area formulae, generalised McKean-Singer formulae for Euler characteristics, and generalised Rice formulae. Introductory material on Gaussian measures and stochastic analysis is included.Mathematics Subject Classification (2010). 47H11, 58B15, 58D20, 28C20, 58J65, 58C30.