Contents

Preface vii

Chapter 1. The derivative operator 1

1.1. Hermite polynomials and chaos expansions 1

1.2. Derivative operator: Definition and properties 3

1.3. Derivative in the white noise case 9

Chapter 2. The divergence operator 13

2.1. Properties of the divergence 13

2.2. The divergence in the white noise case 15

2.3. Clark-Ocone formula 17

Chapter 3. The Ornstein-Uhlenbeck semigroup 19

3.1. Mehler’s formula 19

3.2. Hypercontractivity 20

3.3. The generator of the Ornstein-Uhlenbeck semigroup 23

Chapter 4. Sobolev spaces and equivalence of norms 27

4.1. Meyer inequalities 27

4.2. A continuous family of seminorms and Sobolev spaces 29

4.3. Continuity of the divergence 30

Chapter 5. Regularity of probability laws 33

5.1. Existence and formulas for probability densities 33

5.2. Regularity of the density 35

Chapter 6. Support properties. Density of the maximum 39

6.1. Properties of the support of the law 39

6.2. Regularity of the law of the maximum of continuous processes 40

Chapter 7. Application of Malliavin calculus to diffusion processes 45

7.1. Differentiability of the solution 46

7.2. Existence of densities under ellipticity conditions 48

7.3. Regularity of the density under H¨ ormander’s conditions 49

Chapter 8. The divergence operator as a stochastic integral 55

8.1. Skorohod integral 55

8.2. Stochastic calculus for fractional Brownian motion 59

Chapter 9. Central limit theorems and Malliavin calculus 67

9.1. Central limit theorems via chaos expansions 67

9.2. Stein’s method and Malliavin calculus 72

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