Real and Stochastic Analysis

• 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.

"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." -Applications of Mathematics