Malliavin隨機變分引論

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 Groups". "Stochastic Differential Equations and Flow of Homeomorphism". Abstract of the book Malliavin Calculus is the theory of infinite dimensional differential calculus, which is suitable for functionals involved in diffusion theory, stochastic control, financial 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. ^[1] 

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