Inhalt

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   Preface
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   0 Introduction
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  1 Preliminaries
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   1.1 Some Basic Notations
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   1.2 Absolute Continuity of Gaussian Measures
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   1.3 Wiener Processes and Stochastic Integrals
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  1.4 Lévy Processes
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   1.4.1 Lévy Processes and Infinite Divisible Distributions
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   1.4.2 The Lévy-Itô Decomposition and Stochastic Integrals
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   1.4.3 Stochastic integral with respect to Lévy noise
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   1.4.4 Symmetric alpha-Stable Processes
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  2 Absolute Continuity of Lévy Processes
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   2.1 Introduction
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   2.2 Girsanov's Theorem for Lévy Processes
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   2.3 Hellinger-Kakutani Theory
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   2.4 Conditions for Absolute Continuity of Lévy Processes
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   2.5 Gaussian Case
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   2.6 Proof of Theorem 2.4.1
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   2.7 Density of Lévy Processes
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   2.8 Appendix: The range of R is different with the Cameron-Martin Space in infinite dimension
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  3 Gluing and Coupling
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   3.1 Gluing Lemma
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   3.2 Proof of the Gluing Lemma
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   3.3 Coupling
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   4 Harnack Inequalities for Stochastic Differential Equations
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   4.1 Introduction to Harnack Inequalities
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   4.2 Harnack Inequalities: Known Results
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   4.3 Harnack Inequality: Framework I
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   4.4 Harnack Inequality: Framework II
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   4.5 Global Monotonicity Condition
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   4.6 Linear Growth Condition
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   4.7 Heat Kernel Estimates
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   4.8 Some Problems in Applying Girsanov's Theorem
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   5 Harnack Inequalities for Ornstein-Uhlenbeck Processes Driven by Wiener Processes
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   5.1 Ornstein-Uhlenbeck Processes
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  5.2 Harnack Inequalities
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   5.2.1 Main Theorem
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   5.2.2 Estimates of ||Gamma_T||
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   5.2.3 Estimates of Harnack Inequality
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   5.3 Properties Equivalent to Harnack Inequalities
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  5.4 Examples of Harnack Inequalities
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   5.4.1 Simple Cases
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   5.4.2 Diagonal Ornstein-Uhlenbeck Processes
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  5.5 Perturbations
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   5.5.1 Lipschitz Perturbation
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   5.5.2 Gradient Systems
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  5.6 Appendix
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   5.6.1 Finite Dimensional Approximation
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   5.6.2 Representations of Ornstein-Uhlenbeck Semigroups
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  6 Harnack Inequalities for Ornstein-Uhlenbeck Semigroups: Two Other Gaussian Cases
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   6.1 Harnack Inequalities for Fractional Orntein-Uhlenbeck Processes
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   6.1.1 Fractional Brownian Motions and Stochastic Integrals
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   6.1.2 Fractional Ornstein-Uhlenbeck Processes and Harnack Inequalities
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   6.2 Ornstein-Uhlenbeck Semigroups on Gaussian Probability Spaces
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   6.2.1 Gaussian Probability Spaces and Numerical Models
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   6.2.2 Ornstein-Uhlenbeck Semigroups
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   6.2.3 Harnack Inequalities and Examples
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   7 Harnack Inequalities for Ornstein-Uhlenbeck Processes Driven by Lévy Processes
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   7.1 Ornstein-Uhlenbeck Processes Driven by Lévy Processes
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   7.2 Semigroup Calculus Approach
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  7.3 Approach by Using Measure Transformation on State Spaces
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   7.3.1 Main Theorem for Harnack Inequality
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   7.3.2 alpha-Stable Ornstein-Uhlenbeck Processes
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   7.3.3 Harnack Inequalities for Markov Chains
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  7.4 Method of Coupling and Girsanov's Transformation
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   7.4.1 Harnack Inequalities: Using a Control Drift
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   7.4.2 Harnack Inequalities: Optimization Over All Drifts
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   7.4.3 Estimates of the Harnack Inequalities
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   7.4.4 Examples
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  7.5 Applications of the Harnack Inequalities
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   7.5.1 Regularizing Property
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   7.5.2 Heat Kernel Bounds
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   7.5.3 Hyperboundedness
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   8 Harnack Inequalities for Multivalued Stochastic Equations
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   8.1 Multivalued Maximal Monotone Operator
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   8.2 Harnack Inequalities for Multivalued Stochastic Differential Equations
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   8.3 Multivalued Stochastic Evolution Equations
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   8.4 Concentration of Invariant Measures
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   8.5 Harnack Inequalities
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   8.6 Applications of Harnack Inequalities
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  9 Functional Inequalities for Ornstein-Uhlenbeck Processes
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   9.1 Entropy Cost and HWI Inequalities
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   9.2 Proof of Entropy Cost Inequality
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   9.3 Proof of HWI Inequality
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   A Controllability of Infinite Dimensional Linear System
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   Bibliography
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   Index
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   Notation