Brownian Motion and Index Formulas for the de Rham Complex (Mathematical Research) - Softcover

Taira, Kazuaki

Softcover

ISBN 10: 3527401393 ISBN 13: 9783527401390

Publisher: Wiley VCH, 1998

Synopsis

This book is an easy–to–read reference providing a link between partial differential equations (pde), stochastic analysis, and index theory. Most mathematicians working in pde are only vaguely familiar with the powerful ideas of stochastic analysis. On the other hand, the additional intuition which Taira´s book conveys might provide better insight and be helpful for their work.
In addition, the book provides a nice compendium for a large variety of facts from differential geometry, functional analysis, pseudodifferential operators, and Markov processes – for quickly looking up a theorem.

About this title

From the Back Cover

The purpose of this monograph is to give an analytic proof of an index formula for the relative de Rham cohomology groups which may be considered as a generalization of the celebrated Hodge–Kodaira theory for the absolute de Rham cohomology groups. More precisely, let X be a compact oriented smooth Riemannian manifold without boundary, and Y a submanifold of X. The purpose is to find an operator D such that ind D = _X(X) – _X(Y) where _X(X) and _X(Y) are the Euler–Poincaré characteristics of X and Y, respectively. The crucial point is how to introduce spaces of currents on X and Y in which the index formula for D holds. In deriving this index formula, the theory of harmonic forms satisfying an interior boundary condition plays a fundamental role. The approach here has a great advantage of intuitive interpretation of the index formula in terms of Brownian motion from the point of view of probability theory, and the result may be stated as follows: Brownian motion describes the topology of a compact Riemannian manifold through its Euler–Poincaré characteristic.

Edition details

• Publisher: Wiley VCH
• Publication date: 1998
• Language: English
• ISBN 10: 3527401393
• ISBN 13: 9783527401390
• Binding: Paperback
• Edition number: 1
• Number of pages: 216