Overview

The interplay between functional and stochastic analysis has wide implications for problems in partial differential equations, noncommutative or ""free"" probability, and Riemannian geometry. Written by active researchers, each of the six independent chapters in this volume is devoted to a particular application of functional analytic methods in stochastic analysis, ranging from work in hypoelliptic operators to quantum field theory. Every chapter contains substantial new results as well as a clear, unified account of the existing theory; relevant references and numerous open problems are also included. Self-contained, well-motivated, and replete with suggestions for further investigation, this book will be especially valuable as a seminar text for dissertation-level graduate students. Research mathematicians and physicists will also find it a useful and stimulating reference.

Full Product Details

Author:   M. M. Rao ,  D.R. Bell ,  B.K. Driver ,  S. Gudder
Publisher:   Birkhauser Boston Inc
Imprint:   Birkhauser Boston Inc
Edition:   2004 ed.
Dimensions:   Width: 15.50cm , Height: 2.30cm , Length: 23.50cm
Weight:   0.760kg
ISBN:  

9780817643324

ISBN 10:   081764332
Pages:   406
Publication Date:   23 September 2004
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Awaiting stock  
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Table of Contents

and Outline.- References.- Stochastic Differential Equations and Hypoelliptic Operators.- 1 Introduction.- 2 Integration by parts and the regularity of induced measures.- 3 A Hörmander theorem for infinitely degenerate operators.- 4 A study of a class of degenerate functional stochastic differential equations.- 5 Some open problems.- References.- Curved Wiener Space Analysis.- 1 Introduction.- 2 Manifold primer.- 3 Riemannian geometry primer.- 4 Flows and Cartan's development map.- 5 Stochastic calculus on manifolds.- 6 Heat kernel derivative formula.- 7 Calculus on W(M).- 8 Malliavin's methods for hypoelliptic operators.- 9 Appendix: Martingale and SDE estimates.- References.- Noncommutative Probability and Applications.- 1 Introduction.- 2 Traditional probability theory.- 3 Unsharp traditional probability theory.- 4 Sharp quantum probability.- 5 Unsharp quantum probability.- 6 Effects and observables.- 7 Statistical maps.- 8 Sequential products on Hilbert space.- 9 Quantum operations.- 10 Completely positive maps.- 11 Sequential effect algebras.- 12 Further SEA results.- References.- Advances and Applications of the Feynman Integral.- 1 Introduction.- 2 The operator valued Feynman integral.- 3 Evolution processes.- 4 The Feynman-Kac formula.- 5 Boundedness of processes.- 6 Path integrals on finite sets.- 7 The Dirac equation in one space dimension.- 8 Integration with respect to unbounded set functions.- 9 The Feynman integral with singular potentials.- 10 Quantum field theory.- References.- Stochastic Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphisms.- 1 Stochastic integrals for sernimartingales.- 2 Stochastic analysis of Lévy processes.- 3 Stochastic differential equation and stochastic flow.- 4 Appendix. Kolmogorov'scriterion for the continuity of random fields and the uniform convergence of random fields.- References.- Convolutions of Vector Fields-III: Amenability and Spectral Properties.- 1 Introduction.- 2 Elementary Aspects of Random Walks.- 3 Role of the Spectrum of Convolution Operators.- 4 Amenable Function Algebras and Groups.- 5 Spectra of Convolution Operators and Amenability.- 6 Beurling and Segal Algebras for Amenability.- References.

Reviews

Although most of the papers contain new results and/or proofs, their authors tried to make them reasonably self-contained and provide a wide background to the problems studied, so the papers should be accessible to advanced graduate students and serve as an introduction to some topics which have been paid considerable attention recently. a Applications of Mathematics

Although most of the papers contain new results and/or proofs, their authors tried to make them reasonably self-contained and provide a wide background to the problems studied, so the papers should be accessible to advanced graduate students and serve as an introduction to some topics which have been paid considerable attention recently. <p>a Applications of Mathematics

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