Table of Contents
Preface vii
1 Introduction 1
2 Garsia-Rodemich-Rumsey Inequality 7
2.1 Metric Garsia-Rodemich-Rumsey Inequality 7
2.2 Volume Metric Garsia-Rodemich-Rumsey Inequality 11
2.3 Sample Path Holder Continuity of Random Fields 16
3 Analysis With Respect to Gaussian Measure in Rd 19
3.1 Gaussian Measure in Rd 19
3.2 Some Inequalities Related to Gaussian Measure 21
3.3 Brunn-Minkowski Inequality 27
3.4 Hermite Polynomials 35
3.5 Spectral Gap and Logarithmic Sobolev Inequalities 43
3.6 Variance Identity and Inequality 47
3.7 Correlation Inequality 48
3.8 Hypercontractivity 51
3.9 Hermite Polynomials in Physics and Hermite Functions 54
3.10 Segal-Bargmanu Space and Complex Hermite Polynomials 58
3.11 Segal-Bargmann Transform 63
4 Gaussian Measures on Banach Space 67
4.1 Random Variables in Banach Space 67
1.1 Abstract Wiener Space 80
4.1 Canonical Wiener Space 88
4.4 Right Tail Estimate 90
4.5 Small Ball (Left Tail) Estimate 96
5 Nonlinear Functional on Abstract Wiener Space 103
5.1 Fock Space and Chaos Expansion 103
5.2 Polarization 112
5.3 Multiple Wiener-Ito Integrals 117
5.4 Multiple Stratonovich Integrals 126
5.5 Right Tail Estimate for Homogeneous Chaos 133
5.6 Chaos Expansion of Exit Time and Skeleton of Wiener Functional 139
6 Analysis of Nonlinear Wiener Functionals 153
6.1 Gross-Sobolev Derivatives 153
6.2 Divergence Operator 163
6.3 Regularity of Density of Wiener Functional 169
6.4 Girsanov Transformation: Finite Dimension 177
6.5 Girsanov-Ramer Theorem in Abstract Wiener Space 182
6.6 Wick Product 187
6.7 Wick Renormalizatiou 194
6.8 (Noncommutative) Composition of Wiener Functional 201
6.9 Stop Brownian Motion at Anticipative Exit Time 211
7 Some Inequalities 219
7.1 Complex Hypercontractivity 219
7.2 Meyer's inequality 231
7.3 Multiplier Theorem 236
7.4 Littlewood-Paley-Stein-Meyer Theory 241
7.5 Meyer's Inequalities Revisited 257
7.6 Interpolation Inequality 265
7.7 Grothendieck Inequality 268
8 Convergence in Density 273
8.1 General Nonlinear Wiener Functional 273
8.2 Weak Convergence 276
8.3 Representation of the Derivatives of the Density 280
8.4 Random Variables in the q-th Wiener Chaos 290
8.5 Uniform Estimation of Difference of Derivatives of Densities 292
8.6 Density Convergence of Higher Bank Hermite Polynomials 297
9 Local Time and (Self-) Intersection Local Time 311
9.1 Local Time of Brownian Motion 311
9.2 Chaos Expansion of Self-inter sec lion Local Time 314
9.3 Exponential Integrability 320
9.4 Renormalizatiou When d ≠ 3 324
9.5 L2-Modulus of Continuity of Local Time of Brownian Motion 329
10 Stochastic Differential Equation 341
10.1 Existence, Uniqueness and Non-explosion 341
10.2 Hörmander Theorem 347
10.3 Exponential Integrability 364
10.4 Itô Wiener Chaos Expansion 371
10.5 FKG Inequality and Variance Inequality 373
10.6 Hypercontractivity, Spectral and Logarithmic Sobolev Inequality 376
10.7 Convergence to Density for Eigodie Diffusion 384
11 Numerical Approximation of Stochastic Differential Eequation 395
11.1 LP Convcrgence Rate 395
11.2 Convergence in Dβ,p and Convergence in Density 403
11.3 Weak Convergence Rate 417
11.4 Wong-Zakai Approximation 421
Appendix A Appendix 427
A.1 Some Elementary Results from Analysis 427
A.2 Martingales 439
Bibliography 453
Index 467