Hyperfinite Dirichlet Forms and Stochastic Processes
Description
This monograph treats the theory of Dirichlet forms from a comprehensive point of view, using "nonstandard analysis." Thus, it is close in spirit to the discrete classical formulation of Dirichlet space theory by Beurling and Deny (1958). The discrete infinitesimal setup makes it possible to study the diffusion and the jump part using essentially the same methods. This setting has the advantage of being independent of special topological properties of the state space and in this sense is a natural one, valid for both finite- and infinite-dimensional spaces.
The present monograph provides a thorough treatment of the symmetric as well as the non-symmetric case, surveys the theory of hyperfinite Lévy processes, and summarizes in an epilogue the model-theoretic genericity of hyperfinite stochastic processes theory.
Reviews / Votes
From the reviews:
"This book deals with the relationship between Dirichlet forms and stochastic processes from the perspective of nonstandard analysis. . The book has a very extensive bibliography and gives an interesting and valuable survey of the development and the applications of the theory of Dirichlet forms. . the book will serve as a starting point for new explorations." (Tom L. Lindström, Mathematical Reviews, Issue 2012 k)
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Content
- Intro
- Hyperfinite Dirichlet Formsand Stochastic Processes
- Preface
- Contents
- Chapter 1: Hyperfinite Dirichlet Forms
- 1.1 Hyperfinite Quadratic Forms
- 1.2 Domain of the Symmetric Part
- 1.3 Resolvent of the Symmetric Part
- 1.4 Weak Coercive Quadratic Forms
- 1.5 Hyperfinite Dirichlet Forms
- 1.6 Hyperfinite Representations
- 1.7 Weak Coercive Quadratic Forms, Revisited
- Chapter 2: Potential Theory of Hyperfinite Dirichlet Forms
- 2.1 Exceptional Sets
- 2.1.1 Exceptional Sets
- 2.1.2 Co-Exceptional Sets
- 2.2 Excessive Functions and Equilibrium Potentials
- 2.3 Capacity Theory
- 2.4 Relation of Exceptionality and Capacity Theory
- 2.5 Measures of Hyperfinite Energy Integrals
- 2.6 Internal Additive Functionals and Associated Measures
- 2.7 Fukushima's Decomposition Theorem
- 2.7.1 Decomposition Under the Individual Probability Measures Pi
- 2.7.2 Decomposition Under the Whole Measure P
- 2.8 Internal Multiplicative Functionals
- 2.8.1 Internal multiplicative functionals
- 2.8.2 Subordinate Semigroups
- 2.8.3 Subprocesses
- 2.8.4 Feynman-Kac Formulae
- 2.9 Alternative Expression of Hyperfinite Dirichlet Forms
- 2.10 Transformations of Symmetric Dirichlet Forms
- Chapter 3: Standard Representation Theory
- 3.1 Standard Parts of Hyperfinite Markov Chains
- 3.1.1 Inner Standard Part of Sets
- 3.1.2 Strong Markov Processes and ModifiedStandard Parts
- 3.2 Hyperfinite Dirichlet Forms and Markov Processes
- 3.2.1 Separation of Points
- 3.2.2 Nearstandardly Concentrated Forms
- 3.2.3 Quasi-Continuity
- 3.2.4 Construction of Strong Markov Processes
- Chapter 4: Construction of Markov Processes Associated With Quasi-Regular Dirichlet Forms
- 4.1 Main Result
- 4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms
- 4.3 Relation with Capacities
- 4.4 Path Regularity of Hyperfinite Markov Chains
- 4.5 Quasi-Continuity and Nearstandard Concentration
- 4.6 Construction of Strong Markov Processes
- 4.7 Necessity for Existence of Dual Tight Markov Processes
- Chapter 5: Hyperfinite Lévy Processes
- 5.1 Standard Lévy Processes
- 5.2 Hyperfinite Lévy Processes: Definitions and Characterizations
- 5.3 Standard Parts of Hyperfinite Lévy Processes
- 5.4 Lindstrøm's Hyperfinite Lévy-Khintchine Formula
- 5.5 Lindstrøm's Hyperfinite Representation Theorem for Lévy Processes
- 5.6 Hyperfinite Lévy Processes: Extensions and Applications
- Chapter 6: Epilogue: Genericity of Hyperfinite Loeb Path Spaces
- 6.1 Adapted Probability Logic
- 6.2 Universality, Saturation, Homogeneity
- 6.3 Hyperfinite Adapted Spaces
- 6.4 Probability Logic and Markov Processes
- Appendix
- A.1 General Topology
- A.2 Structure of * R
- A.3 Internal Sets and Saturation
- A.4 Loeb Measure
- A.5 Linear Spaces
- Historical Notes, Bibliographical Complements
- References
- Notation Index
- Index
