| 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. |
| 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 |
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