# Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

Zhen-Qing Chen
Masatoshi Fukushima
Pages: 512
https://www.jstor.org/stable/j.ctt7s6w6

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1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Notation
(pp. ix-x)
4. Preface
(pp. xi-xvi)
Zhen-Qing Chen and Masatoshi Fukushima
5. Chapter One SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS
(pp. 1-36)

The concepts of Dirichlet form and Dirichlet space were introduced in 1959 by A. Beurling and J. Deny [8] and the concept of the extended Dirichlet space was given in 1974 by M. L. Silverstein [138]. They all assumed that the underlying state spaceEis a locally compact separable metric space. Concrete examples of Dirichlet forms (bilinear form, weak solution formulations) have appeared frequently in the theory of partial differential equations and Riemannian geometry. However, the theory of Dirichlet forms goes far beyond these.

In this section, we work with a σ-finite measure space (E, 𝓑(E),m) without any...

6. Chapter Two BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS
(pp. 37-91)

In this section, we introduce the concepts of the transience, recurrence, and irreducibility of the semigroup for general Markovian symmetric operators and present their characterizations by means of the associated Dirichlet form as well as the associated extended Dirichlet space. These notions are invariant under the time changes of the associated Markov process. We shall also examine them in the concrete examples of the next section.

As in Section 1.1, we let (E, 𝓑(E),m) be a σ-finite measure space and consider a strongly continuous contraction semigroup$\{ T_t ;t > 0\}$of symmetric Markovian operators and a Dirichlet form (𝓔,𝓕) on$L^2 (E;m)$,which...

7. Chapter Three SYMMETRIC HUNT PROCESSES AND REGULAR DIRICHLET FORMS
(pp. 92-129)

As is clearly embodied by Theorem 1.4.3, three theorems of Section 1.5, and Theorem 3.1.13 below, the study of general symmetric Markov processes can be essentially reduced to the study of a symmetric Hunt process associated with a regular Dirichlet form. So without loss of generality, we assume throughout this chapter, except for the last parts of Sections 3.1 and 3.5, thatEis a locally compact separable metric space,mis a positive Radon measure onEwith supp[m] =E, and$X = (X_t ,{\mathbf{P}}_x )$is anm-symmetric Hunt process on (E, 𝓑(E)) whose Dirichlet form (𝓔,𝓕) is regular on$L^2 (E;m)$....

8. Chapter Four ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES
(pp. 130-165)

This chapter is devoted to the study of additive functionals of symmetric Markov processes under the same setting as in the preceding chapter, namely, we letEbe a locally compact separable metric space, 𝓑(E) be the family of all Borel sets ofE, andmbe a positive Radon measure onEwith supp[m] =E, and we consider anm-symmetric Hunt process$X = (\Omega ,\mathcal{M},{X_t},\zeta ,{{\mathbf{P}}_x})$on (E, 𝓑(E)) whose Dirichlet form (𝓔, 𝓕) on$L^2 (E;m)$is regular on$L^2 (E;m)$.The transition function and the resolvent ofXare denoted by$\{ P_t ;t \geq 0\} ,\{ R_{\alpha ,} \alpha > 0\}$,respectively. 𝓑*(E) will denote the family of all universally...

9. Chapter Five TIME CHANGES OF SYMMETRIC MARKOV PROCESSES
(pp. 166-239)

Time change is one of the most basic and very useful transformations for Markov processes, which has been studied by many authors. The following is a prototype of the problem that will be studied in this chapter. SupposeXis a Lévy process in$\mathbb{R}^n$that is the sum of a Brownian motion in$\mathbb{R}^n$and an independent rotationally symmetric α-stable process in$\mathbb{R}^n$,where n ≥ 1 and α ∈ (0, 2). Denote byB(x,r) the open ball in$\mathbb{R}^n$centered at$x \in \mathbb{R}^n$with radiusr. Its Euclidean closure is denoted by$\overline{B(x,r)}$.Let$F = \overline{B(0,1)} \cup \partial B(x_0 ,1)$,where$x_0 \in \mathbb{R}^n$with...

10. Chapter Six REFLECTED DIRICHLET SPACES
(pp. 240-299)

Reflected Dirichlet space was introduced for a regular Dirichlet form by M. L. Silverstein in 1974 in [138, 139] and further investigated by the firstnamed author in [16]. Reflected Dirichlet space plays an important role for the boundary theory of symmetric Markov processes. While it is possible to introduce reflected Dirichlet form for a quasi-regular Dirichlet form directly, we choose to do it first in the regular Dirichlet form setting as this allows us to define the reflected Dirichlet space via terminal random variables and harmonic functions of finite energy. This approach sheds more insight into the probabilistic meaning and...

11. Chapter Seven BOUNDARY THEORY FOR SYMMETRIC MARKOV PROCESSES
(pp. 300-390)

Let$X^0$be an$m_0$-symmetric right process on a state space$E_0$.The boundary theory of symmetric Markov processes concerns all possible symmetric extensions of$X^0$to some state space Ẽ containing$E_0$as an intrinsic open subset so that they admit no-sojourn (that is, spend zero Lebesgue amount of time) at$\tilde E\ \backslash \ E_0$.The no-sojorn condition ensures that the extension process is$m_0$-symmetric after the measure$m_0$is extended to Ẽ by setting$m_0 (\tilde E\backslash E_0 ) = 0$.The initial “minimal” process$X^0$can be taken as the part process on$E_0$of a symmetric Hunt processXonEkilled upon...

12. Appendix A. ESSENTIALS OF MARKOV PROCESSES
(pp. 391-442)
13. Appendix B. SOLUTIONS TO EXERCISES
(pp. 443-450)
14. Notes
(pp. 451-456)
15. Bibliography
(pp. 457-466)
16. Catalogue of Some Useful Theorems
(pp. 467-472)
17. Index
(pp. 473-479)