Real and Stochastic Analysis

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.

Self-contained and written by active researchers, each chapter is devoted to a particular application of functional analytic methods in stochastic analysis Every chapter contains substantial new results as well as a clear, unified account of the existing theory, relevant references and numerous open problems Especially valuable as a seminar text for dissertation-level graduate students and as a reference

Inhalt
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.