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Infinite-dimensional degree theory and stochastic analysis

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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, 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.

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah, 51452, Saudi Arabia

    A. Al-Hussein

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    K. D. Elworthy

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  1. A. Al-Hussein
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Correspondence to K. D. Elworthy.

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Dedicated to Steve Smale on his 80th birthday, with thanks for inspiration and encouragement

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Al-Hussein, A., Elworthy, K.D. Infinite-dimensional degree theory and stochastic analysis. J. Fixed Point Theory Appl. 7, 33–65 (2010). https://doi.org/10.1007/s11784-010-0009-9

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