Real and Stochastic Analysis: New Perspectives(Trends in Mathematics)

As in the case of the two previous volumes published in 1986 and 1997, the purpose of this monograph is to focus the interplay between real (functional) analysis and stochastic analysis show their mutual benefits and advance the subjects. The presentation of each article, given as a chapter, is in a research-expository style covering the respective topics in depth. In fact, most of the details are included so that each work is essentially self contained and thus will be of use both for advanced graduate students and other researchers interested in the areas considered. Moreover, numerous new problems for future research are suggested in each chapter. The presented articles contain a substantial number of new results as well as unified and simplified accounts of previously known ones. A large part of the material cov­ ered is on stochastic differential equations on various structures, together with some applications. Although Brownian motion plays a key role, (semi-) martingale theory is important for a considerable extent. Moreover, noncommutative analysis and probabil­ ity have a prominent role in some chapters, with new ideas and results. A more detailed outline of each of the articles appears in the introduction and outline to assist readers in selecting and starting their work. All chapters have been reviewed.

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.