Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35) (London Mathematical Society Monographs) - Hardcover

Chen, Zhenqing; Fukushima, Masatoshi

Hardcover

ISBN 10: 069113605X ISBN 13: 9780691136059

Verlag: Princeton University Press, 2011

Inhaltsangabe

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

Über diesen Titel

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".

Von der hinteren Coverseite

"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

By Zhen-Qing Chen Masatoshi Fukushima

PRINCETON UNIVERSITY PRESS

Copyright © 2011 Princeton University Press
All right reserved.
ISBN: 978-0-691-13605-9

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

SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS

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 , v) is a bounded linear functional on the Hilbert space (D(E), E_a). By the Riesz representation theorem, there exists a unique element of D(E) denoted by G_af such that for every f [member of] H and v [member of] D(E),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.6)

{G_a; a > 0} so defined is a strongly continuous contraction resolvent on H, which in turn determines a strongly continuous contraction semigroup {T_t; t > 0} on H by (1.1.2). They are called the resolvent and semigroup generated by the closed symmetric form (E,D(E)), respectively.

The above-mentioned correspondences from {T_t; t > 0} to (E,D(E)) and from (E,D(E)) to {T_t; t > 0} are mutually reciprocal.

From now on, we shall take as H the space L²(E;m) on a s-finite measure space (E,B(E),m). In this book, we need to consider extensions of the domain D(E) of a closed symmetric form E on L²(E;m). For this purpose, we shall designate D(E) by F so that a closed symmetric form on L²(E;m) will be denoted by (E,F). We now proceed to introduce the notions of Dirichlet form and extended Dirichlet space.

Definition 1.1.1. For 1 = p = 8, a linear operator L on L^p(E;m) with domain of definition D(L) is called Markovian if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A real function ?, namely, a mapping from R to R, is said to be a normal contraction if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A function defined by ?(t) = (0 [??] t) [??] 1, t [member of] R, is a normal contraction which is called the unit contraction. For any e > 0, a real function ?_e satisfying the next condition is a normal contraction:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 1.1.2. A symmetric form (E,D(E)) on L²(E;m) is called Markovian if, for any e > 0, there exists a real function ?_e satisfying (1.1.7) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.8)

A closed symmetric form (E,F) on L²(E;m) is called a Dirichlet form if it is Markovian. In this case, the domain F is said to be a Dirichlet space.

Theorem 1.1.3. Let (E,F) be a closed symmetric form on L²(E;m) and {T_t}_t>0, {G_a}_a>0 be the strongly continuous contraction semigroup and resolvent on L²(E;m) generated by (E,F), respectively. Then the following conditions are mutually equivalent:

(a) T_t is Markovian for each t > 0.

(b) aGa is Markovian for each a > 0.

(c) (E,F) is a Dirichlet form on L²(E;m).

(d) The unit contraction operates on (E,F):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(e) Every normal contraction operates on (E,F): for any normal contraction ?

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. The implications (a) [??] (b) and (b) [??] (a) follow from (1.1.1) and (1.1.2), respectively. The implication (e) [??] (d) [??] (c) is obvious.

(c) [??] (b): We fix an a > 0 and a function f [member of] L²(E;m) with 0 = f = 1 [m], and introduce a quadratic form on F by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from (1.1.6) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

namely, G_af is a unique element of F minimizing f(v). Suppose (E,F) is a Dirichlet form on L²(E;m). There exists then for any e > 0 a real function ?_e satisfying (1.1.7) and (1.1.8). We let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every s [member of] [0, 1/a] and t [member of] R, we have |u(x) - f (x)/a| = |G_af (x) - f (x)/a| [m] and (u - f /a, u - f/a) = (G_af/aG_af(x) - f/a). Therefore, f(u) = (G_af) and consequently u = G_af [m], which means that -e = G_af = 1/a + e [m]. Letting e -> 0, we get (b).

It remains to prove the implication (a) [??] (e), which will follow from a more general theorem formulated below.

In what follows, we occasionally use for a symmetric form (E,F) on L²(E;m) the notations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 1.1.4. Let (E,F) be a closed symmetric form on L²(E;m). Denote by F_e the totality of m-equivalence classes of all m-measurable functions f on E such that |f| < 8[m] and there exists an E-Cauchy sequence {f_n, n = 1} [??] F such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on E. {f_n} [??] F in the above is called an approximating sequence of f [member of] F_e. We call the space F_e the extended space attached to (E,F). When the latter is a Dirichlet form on L²(E;m), the space F_e will be called its extended Dirichlet space.

Theorem 1.1.5. Let (E,F) be a closed symmetric form on L²(E;m) and F_e be the extended space attached to it. If the semigroup {T_t; t > 0} generated by (E,F) is Markovian, then the following are true:

(i) For any f [member of] F_e and for any approximating sequence {f_n} [??] F of f, the limit E(f , f) = limn_n->8 E(f_n, f_n) exists independently of the choice of an approximating sequence {f_n} of f.

(ii) Every normal contraction operates on (Fe, E): for any normal contraction ?

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) F = F_e [intersection] L₂(E;m). In particular, (E,F) is a Dirichlet form on L²(E;m).

Assertion (ii) of this theorem implies the implication (a) [??] (e) in Theorem 1.1.3, completing the proof of Theorem 1.1.3.

For f, g [member of] F_e, clearly both f + g and f - g are in F_e. Define E(f , g) = 1/4

(E(f + g, f + g) - E(f - g, f - g)), which is a symmetric bilinear form over F_e. (E,F_e) is called the extended Dirichlet form of (E,F).

If a given closed symmetric form (E,F) on L²(E;m) is a Dirichlet form, then the corresponding semigroup {T_t; t > 0} is Markovian by virtue of the already proven implication (c) [??] (a) of Theorem 1.1.3. So the extended Dirichlet space F_e satisfies all properties mentioned in Theorem 1.1.5.

Before giving the proof of Theorem 1.1.5, we shall fix a Markovian contractive symmetric linear operator T on L²(E;m) and make some preliminary observations on T.

By the linearity and the Markovian property of T on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Continues...)

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Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2011
• Sprache: Englisch
• ISBN 10: 069113605X
• ISBN 13: 9780691136059
• Einband: Tapa dura
• Anzahl der Seiten: 512
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar