Something about the author Dr. Shizan Fang(bom in 1963 ). Professor of University of Burgundy(Dijon. FranceS, obtained his PhD degree at University of Paris VI in February 1990 and then worked there as "maitre de Conferences" during 1990-1996. His main interests of research are in the field of "Analysis. Geometry and Probability". He has published some first rate results on the subjects "Geometric Analysis on the Wiener Space". "Geometric Stochastic Analysis on Riemannian Path Spaces and Loop Gr

书摘
The theory of Malliavin Calculus was initiated by P. Malliavin in 1976, to study the Hrmander hypoellipticity problem. Since then the research on infinite dimensional analysis became very active. The theory was well developed during 1980s in the framework of Grosss Abstract Wiener space. The application to the study of the loop space over a compact Lie group or more generally over a compact Riemannian manifold started at the beginning of 1990.
The functionals defined by It stochastic calculus are the typical examples of smooth functions in Malliavin calculus. In this note, we shall pay much attention to classical Wiener spaces. Not only it does give more direct connections with Analysis, Geometry and others through It stochastic differential equations, but also it has a particular structure: It filtration. In this framework, Girsanov theorem plays a central role.
The existence of divergence operator allows to do the integration by parts on the Wiener space. In L2 case, it is in connection with the fundamental geometric formula: Weitzenbck formula. The establishment in Lp needs the tool of Riesz transform. The proof given in this note is heavily dependent on the linear structure of the Wiener space. The problem for Riemannian path spaces is open.
This note was based on a course given at Beijing Normal University during JuneJuly 2002, some materials were talked at the seminar “Dynamic system and Probability” at Mathematical Institute of Burgundy, Dijon. The author is grateful to Professor Fengyu Wang for giving him the opportunity to visit and work at “Stochastic Center” in Beijing Normal University. The presence and the interest of Professor Mufa Chen in the course was an encouragement to the author. The author would like also to express many thanks to Professor Bernard Schmitt for inviting him to give a series of talks at his seminar on the subject at Dijon.

1 Brownian motions and Wiener spaces
1.1 Gaussian family
1.2 Brownian motion
1.3 Classical Wiener spaces
1.4 Abstract Wiener spaces
2 Quasiinvariance of the Wiener measure
2.1 Convergence theorem for L2martingales
2.2 CameronMartin theorem
2.3 Girsanov theorem
3 Sobolev spaces over the Wiener space
3.1 Definitions and examples
3.2 Integration by parts
3.3 Sobolev spaces Dp1(W)
3.4 High order Sobolev spaces
4 OrnsteinUhlenbeck operator
4.1 Definitions
4.2 The spectrum of L
4.3 Vector valued OrnsteinUhlenbeck operator
5 Existence of divergence: L2case
5.1 Energy identity
5.2 Weitzenbck formula
5.3 Γ2 criterion
6 OrnsteinUhlenbeck semigroup
6.1 Mehler formula
6.2 Hypercontractivity of Pt
6.3 Some other properties of Pt
7 Riesz transform on the Wiener space
7.1 Hilbert transform on the circle S1
7.2 Riesz transform on the Wiener space
7.3 Meyer inequalities
8 Existence of divergence: Lpcase
8.1 Meyer multipliers
8.2 Commutation formulae
8.3 Smoothness for δ(Z)
9 Malliavins density theorem
9.1 Non\|degenerated functionals
9.2 Examples
10 Tangent processes and its applications
10.1 Tangent processes
10.2 Path space over a compact Lie group
10.3 Path space over a unimodular Lie group
Appendix: Stochastic differential equations
General notation
Notes and Comments
Bibliography
Index

书评
Malliavin Calculus is the theory of infinite dimensional differential calculus, which is suitable for functionals involved in diffusion theory, stochastic control, finanial market models, etc. It also provides infinite dimensional examples in Dirichlet forms theory, in Functional Inequalities Analysis, etc.
The main purpose of this book is to give a foundation of Malliavin Calculus, as well as some insights toward further researches in the field of path and loop spaces.