Über die Autorin bzw. den Autor

Zhen-Qing Chen is professor of mathematics at the University of Washington. Masatoshi Fukushima is professor emeritus at Osaka University in Japan. His books include Dirichlet Forms and Symmetric Markov Processes.

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"This is an excellent book that provides a systematic treatment of one of the most fundamental concepts in modern probability theory. It will certainly find lots of interest among all mathematicians who work at the interplay of stochastics and analysis."--Karl-Theodor Sturm, University of Bonn

"Modern theory of Dirichlet forms is widely considered to be one of the main achievements in the analysis of stochastic process, and the authors of this book are among the world's leading experts in the field."--Michael Röckner, Bielefeld University

Aus dem Klappentext

"This is an excellent book that provides a systematic treatment of one of the most fundamental concepts in modern probability theory. It will certainly find lots of interest among all mathematicians who work at the interplay of stochastics and analysis."--Karl-Theodor Sturm, University of Bonn

"Modern theory of Dirichlet forms is widely considered to be one of the main achievements in the analysis of stochastic process, and the authors of this book are among the world's leading experts in the field."--Michael Röckner, Bielefeld University

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SYMMETRIC MARKOV PROCESSES, TIME CHANGE, AND BOUNDARY THEORY

PRINCETON UNIVERSITY PRESS

Contents

Notation............................................................................ixPreface.............................................................................xiChapter 1. SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS.......................1Chapter 2. BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS.........................37Chapter 3. SYMMETRIC HUNT PROCESSES AND REGULAR DIRICHLET FORMS.....................92Chapter 4. ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES.......................130Chapter 5. TIME CHANGES OF SYMMETRIC MARKOV PROCESSES...............................166Chapter 6. REFLECTED DIRICHLET SPACES...............................................240Chapter 7. BOUNDARY THEORY FOR SYMMETRIC MARKOV PROCESSES...........................300Appendix A. ESSENTIALS OF MARKOV PROCESSES..........................................391Appendix B. SOLUTIONS TO EXERCISES..................................................443Notes...............................................................................451Bibliography........................................................................457Catalogue of Some Useful Theorems...................................................467Index...............................................................................473

Chapter One

1.1. DIRICHLET FORMS AND EXTENDED DIRICHLET SPACES

The concepts of Dirichlet form and Dirichlet space were introduced in 1959 by A. Beurling and J. Deny and the concept of the extended Dirichlet space was given in 1974 by M. L. Silverstein. They all assumed that the underlying state space E is 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 s-finite measure space (E,B(E),m) without any topological assumption on E and establish the correspondence of the above-mentioned notions to the semigroups of symmetric Markovian linear operators. The present arguments are a little longer than the usual ones under the topological assumption found in and [73, §1.4] but they are quite elementary in nature.

Only at the end of this section, we shall assume that E is a Hausdorff topological space and consider the semigroups and Dirichlet forms generated by symmetric Markovian transition kernels on E.

Let (E,B(E)) be a measurable space and m a s-finite measure on it. Let B^m(E) be the completion of B(E) with respect to m. Numerical functions f , g on E are said to be m-equivalent (f = g [m] in notation) if m({x [member of] E: f (x) [not equal to] g(x)}) = 0. For p = 1 and a numerical function f [member of] B^m(E), we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The family of all m-equivalence classes of f [member of] B^m(E) with ||f||_p < 8 is denoted by L^p(E;m), which is a Banach space with norm ||·||_p, namely, a complete normed linear space. We denote by L₈(E;m) the family of all m-equivalence classes of f [member of] B^m(E) which are bounded m-a.e. on E.L⁸(E;m) is a Banach space with norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that L²(E;m) is a real Hilbert space with inner product

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For a moment, let us consider an abstract real Hilbert space H with inner product [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for ||f||_H is denoted by _f _H. As is summarized in Section A.4, there are mutual one-to-one correspondences among four objects on the Hilbert space H: the family of all closed symmetric forms (E,D(E)), the family of all strongly continuous contraction semigroups {T_t; t = 0}, the family of all strongly continuous contraction resolvents {R_a; a > 0}, and the family of all non-positive definite self-adjoint operators A. Here we mention the correspondences among the first three objects only.

E or (E,D(E)) is said to be a symmetric form on H if D(E) is a dense linear subspace of H and E is a non-negative definite symmetric bilinear form defined on D(E) × D(E) in the sense that for every f, g, h [member of] D(E) and a, b [member of] R

E(f , g) = E(g, f ), E(f , f) = 0, and

E(af + bg, h) = aE(f , h) + bE(g, h).

For a > 0, we define

E_af , g) = E(f , g) + a(f , g), f , g [member of] D(E).

We call a symmetric form (E,D(E)) on H closed if D(E) is complete with norm [square root of E₁(f , f)]. D(E) is then a real Hilbert space with inner product Ea for each a > 0.

A family of symmetric linear operators {T_t; t > 0} on H is called a strongly continuous contraction semigroup if, for any f [member of] H,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We call a family of symmetric linear operators {G_a; a > 0} on H a strongly continuous contraction resolvent if for every a, ß > 0 and f [member of] H,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The semigroup {T_t; t = 0} and the resolvent {G_a; a > 0} as above correspond to each other by the next two equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.1)

the integral on the right hand side being defined in Bochner's sense, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.2)

{G_a; a > 0} determined by (1.1.1) from {T_t; t > 0} is called the resolvent of {T_t; t = 0}.

Given a strongly continuous contraction symmetric semigroup {T_t; t > 0} on H, for each t > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.3)

defines a symmetric form E^(t) on H with domain H. For each f [member of] H, E^(t)(f , f) is non-negative and increasing as t > 0 decreases (this can be shown, for example, by using spectral representation of {T_t; t > 0}). We may then set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.5)

which becomes a closed symmetric form on H called the closed symmetric form of the semigroup {T_t; t > 0}. We call E^(t) of (1.1.3) the approximating form of E.

Conversely, suppose that we are given a closed symmetric form (E,D(E)) on H. For each a > 0, f [member of] H and v [member of] D(E), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which means that [Phi](v) = (f ,...