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Hyperfinite Dirichlet Forms and Stochastic Processes

Authors: Sergio Albeverio, Ruzong Fan, Frederik Herzberg

Publisher: Springer Berlin Heidelberg

Book Series : Lecture Notes of the Unione Matematica Italiana

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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.

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Table of Contents

Frontmatter

Chapter 1. Hyperfinite Dirichlet Forms

Abstract

The interplay between methods from functional analysis and the theory of stochastic processes is one of the most important and exciting aspects of mathematical physics today. It is a highly technical and sophisticated theory based on decades of research in both areas. Numerous papers have been writ- ten on the standard theory of Dirichlet forms. Apart from the articles and monographs cited below, other notable contributions to the area include: Albeverio and Bernabei [4], Albeverio et al. [31], Bliedtner [85], Bouleau [89], Bouleau and Hirsch [90], Chen et al. [103], Fabes, Fukushima, Gross, Kenig, RÖckner and Stroock [144], Fitzsimmons and Kuwae [163], Fukushima [168– 170], Fukushima and Tanaka [176], Fukushima and Ying [179,180], Grothaus et al. [189], Hesse et al. [198], Jacob [208–210], Jacob and Moroz [211], Jacob and Schilling [212], Jost et al. [214], Kassmann [221], Kim et al. [229], Kumagai and Sturm [237], Ma and RÖckner [261,262], Ma et al. [263], Oshima [283], RÖckner and Wang [304], RÖckner and Zhang [306], Schmuland and Sun [316], Shiozawa and Takeda [318], da Silva et al. [319], Stannat [325], and Takeda [333, 334].

Sergio Albeverio, Ruzong Fan, Frederik Herzberg

Chapter 2. Potential Theory of Hyperfinite Dirichlet Forms

Abstract

Probabilistic potential theory has been a very important component in the study of hyperfinite Dirichlet space theory. It provides a probabilistic inter- pretation of potential theory; and, more generally, it establishes a beautiful bridge between functional analysis and the theory of Markov processes. There are many applications of this theory, especially in the area of infi- nite dimensional stochastic analysis and mathematical physics. Our purpose in this chapter is to develop the probabilistic potential theory associated with hyperfinite Dirichlet forms and the related Markov chains. The motivation is twofold. On the one hand, we want to establish a relationship between the standard Dirichlet space theory and the hyperfinite counterpart. On the other hand, we want to provide new methods for the theory of hyperfinite Dirichlet forms itself. Infinite dimensional stochastic analysis has been devel- oped extensively in the last decades. We hope to convince the reader that nonstandard analysis can provide a new tool to deal with problems in this exciting area, see particularly Chap. 4, for example.

Sergio Albeverio, Ruzong Fan, Frederik Herzberg

Chapter 3. Standard Representation Theory

Abstract

The purpose of this chapter is to study the standard projection of hyperfinite Markov chains, and the standard projection of hyperfinite Markov chains associated with hyperfinite quadratic forms. At first, we shall introduce in Sect. 3.1 a concept of irregularity; and then we shall prove that if a hyperfinite Markov chain X(·, t) has a set of irregularities, its standard part x(·, t) is a strong Markov process. In Sect. 3.2, we shall find conditions on hyperfinite quadratic forms which guarantee that the modified standard parts of associated hyperfinite Markov chains are strong Markov processes.

Sergio Albeverio, Ruzong Fan, Frederik Herzberg

Chapter 4. Construction of Markov Processes Associated With Quasi-Regular Dirichlet Forms

Abstract

In this chapter, we consider the construction of tight strong Markov processes associated with standard quasi-regular Dirichlet forms by using the language of nonstandard analysis. In Fan [154], the construction of strong Markov pro- cesses associated with standard symmetric Dirichlet forms was considered. In this chapter, we consider nonsymmetric Dirichlet forms. We consider a standard Dirichlet form (F(·, ·),D(F)) whose state space is a regular Hausdorff space Y. We impose conditions directly on the Dirichlet form (F(·, ·), D(F)) to obtain associated tight strong Markov processes. The motivation comes from the desire to find the relationship between the family of Dirichlet forms and the family of strong Markov processes.

Sergio Albeverio, Ruzong Fan, Frederik Herzberg

Chapter 5. Hyperfinite Lévy Processes

Abstract

So far, we have concentrated in this book on studying arbitrary Markov pro- cesses which admit a modification whose paths are right-continuous with left limits (i.e. càdlàg Markov or Feller processes, cf. e.g. [294, Chap. 3, Theorem 2.7]), and their relations to energy forms and Hamiltonians from the perspective of nonstandard analysis.

Sergio Albeverio, Ruzong Fan, Frederik Herzberg

Chapter 6. Epilogue: Genericity of Hyperfinite Loeb Path Spaces

Abstract

An important part of this book has been concerned with the investigation of hyperfinite Markov chains \( {X\,=\,(X_t)_{t\in T}}\) on an internal probability space Ω. If we confine ourselves to a finite time horizon – say, 1 – and if we assume \({\Delta {t}}\, \)to be a reciprocal hypernatural number, then \( {H} \,=\,{1/{\Delta{t}}}\in\,{^*}\mathbb{N}\,\backslash\mathbb{N}\, \)will be the number of possible transitions occurring between \( {0\,\in\,T}\,\)and \( {1\,\in\,T\,.}\,\)Because X was assumed to be a hyperfinite Markov chain starting at a given point x₀ in the state space, the number of possible paths between 0 and 1 will be hyperfinite. Thus, provided we are only interested in studying events that lie in the filtration generated by X, we may assume Ω to be hyperfinite.

Sergio Albeverio, Ruzong Fan, Frederik Herzberg

Backmatter

Title: Hyperfinite Dirichlet Forms and Stochastic Processes
Authors: Sergio Albeverio
Ruzong Fan
Frederik Herzberg
Copyright Year: 2011
Publisher: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-19659-1
Print ISBN: 978-3-642-19658-4
DOI: https://doi.org/10.1007/978-3-642-19659-1

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