Infinite-dimensional degree theory and stochastic analysis

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, and Üstünel & 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.