Preliminaries 1
1 The Brownian Motion and the Wiener Measure . . . . . . . . . . 1
Gross-Sobolev Derivative, Divergence and Ornstein-Uhlenbeck
Operator 9
1 The Construction of V and its properties . . . . . . . . . . . . . 10
1.1 Relations with the stochastic integration . . . . . . . . . . 12
2 The Ornstein-Uhlenbeck Operator . . . . . . . . . . . . . . . . . 16
II Meyer Inequalities 19
1 Some Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 ~(I + f~)-1/2 as the Riesz Transform . . . . . . . . . . . . . . . 21
III Hypercontractivity 27
IV LP-Multipliers Theorem, Meyer Inequalities and Distributions 31
1 LP-Multipliers Theorem . . . . . . . . . . . . . . . . . . . . . . . 31
V Some applications of the d i s t r i b u t i o n s 41
1 Extension of the Ito-Clark formula . . . . . . . . . . . . . . . . . 42
2 Lifting of,_q'(~d) with random variables . . . . . . . . . . . . . . 45
VI Positive d i s t r i b u t i o n s and applications 53
1 Capacities and positive Wiener functionals . . . . . . . . . . . . . 55
VII Characterization of independence of some Wiener functionals 61
1 Independence of Wiener functionals . . . . . . . . . . . . . . . . 61
VIII
1
2
3
Moment i n e q u a l i t i e s for Wiener functionals 69
Exponential tightness . . . . . . . . . . . . . . . . . . . . . . . . 60
Coupling inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 73
An interpolation inequality . . . . . . . . . . . . . . . . . . . . . 77
IX Introduction to the theorem of Ramer 81
Bibliography
Subject Index
Notations
91
94
95