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

By Zhenqing Chen and Masatoshi Fukushima

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.

• Language: English
• Publisher: Princeton University Press
• Release date: Oct 31, 2011
• ISBN: 9781400840564

Book preview

∞.

Preface

The seminal paper of A. Beurling and J. Deny on the theory of Dirichlet spaces appeared in 1959, just half a century ago. Since then the theory of Dirichlet spaces has been growing in close relationship with the theory of Markov processes, especially symmetric Markov processes. The scope of the original Dirichlet space theory has since been greatly expanded both in theory and applications. Books bearing the term Dirichlet forms or symmetric Markov processes in the title continue to be published. For instance, the following volumes, listed in the chronological order, are devoted to this theory: [138, 63, 140, 64, 15, 119, 47, 73, 100, 141].

In 1960’s and 1970’s, the study of the Beurling-Deny theory was motivated by a desire to comprehend and develop the boundary theory for Markov processes as is evident in the papers by M. Fukushima, H. Kunita, M. L. Silverstein, and Y. LeJan and in the two books of Silverstein. Indeed, the concept of the reflected Dirichlet space and the space of functions with finite Douglas integrals involved there were the outgrowth of the idea in the preceding works by W. Feller in 1957 and J. L. Doob in 1962 reinterpreted in terms of Dirichlet forms. But the study in this direction was left halfway until Z.-Q. Chen gave an appropriate reformulation of the reflected Dirichlet space in 1992.

Over the last 30 years, time changes of symmetric Markov processes have been extensively studied. It is well understood now that time change of a symmetric Markov process by means of a positive continuous additive functional (PCAF in abbreviation) with full support corresponds precisely to the replacement of the symmetrizing measure while keeping the extended Dirichlet space invariant. The relevant stochastic calculus is also well developed accompanied by basic decomposition theorems of (not necessarily positive) additive functionals. However, the intrinsic Beurling-Deny decomposition of the trace Dirichlet form or characterization of the time-changed process by a non-fully supported PCAF in terms of a (generalized) Douglas integral has been obtained only quite recently; this is the topic of Chapter 5 of this book.

The notion of the quasi-regular Dirichlet form due to S. Albeverio, Z.-M. Ma, and M. Rockner and a related result of P. J. Fitzsimmons have enabled us to reduce the study of a general symmetric (not necessarily Borel) right process to the study of a symmetric Borel special standard process. It is established by Z.-Q. Chen, Z.-M. Ma, and M. Rockner that a Dirichlet form is quasi-regular if and only if it is quasi-homeomorphic to a regular Dirichlet form on a locally compact separable metric space. This quasi-homeomorphism allows one to transfer problems concerning those quasi-regular Dirichlet forms and symmetric right processes on (possibly infinite dimensional) general Hausdorff topological spaces to problems for regular Dirichlet forms and symmetric Hunt processes on locally compact separable metric spaces.

The development of the Dirichlet form theory benefits from its interaction with other areas of probability theory such as the theory of general Markov processes, martingale theory, and stochastic analysis, and with analytic potential theory, harmonic analysis, Riemannian geometry, theory of function spaces, partial differential equations and pseudo-differential operators, and mathematical physics. On the other hand, Dirichlet form theory has wide range of applications to these fields. For example, it is an effective tool in studying various probabilistic models as well as analytic problems with non-smooth data or in non-smooth media, such as reflecting Brownian motion on non-smooth domains, Brownian motion with random obstacles, diffusions and analysis on fractals, Markov processes and analysis on metric measure spaces, and diffusion processes and differential analysis on path spaces or loop spaces over a compact Riemannian manifold. It is a powerful machinery in studying various stochastic differential equations in infinite dimensional spaces, stochastic partial differential equations, various models in statistical physics such as quantum field theory and interacting particle systems. It also provides a probabilistic means to study various problems in partial differential equations with singular coefficients, analytic potential theory, and theory of function spaces.

The aim of this book is twofold. First, it gives a systematic introduction to the essential ingredients of both the probabilistic part and the analytic part of the theory of quasi-regular Dirichlet form. This is done in the first four and a half chapters of the book, where the theory of quasi-regular Dirichlet form and that of regular Dirichlet form are developed in a unified way. Second, it presents some recent developments of the theory in the last two and a half chapters. Its aim is, along with the characterization of the trace Dirichlet form by the Douglas integral, to give a comprehensive account of the reflected Dirichlet space and then to show the important role they play in the recent development of the boundary theory of symmetric Markov processes.

We strived to make the contents of this book self-contained so that it, especially its first four and a half chapters, can be used as a textbook for advanced graduate students. Chapters 2, 3, 5, 6, and 7 contain many examples illustrating the theory presented. Exercises given throughout the book are an integral part of the book. The solutions to these exercises are given in Appendix B.

The rest of the book is organized as follows. In Chapter 1, we introduce the concepts of Dirichlet form on L2(E, B;m) and its extended Dirichlet space, where (E, B) is a measurable space without any topological assumption imposed on E. In the remaining sections of Chapter 1, we give a quick introduction to the basic theory of quasi-regular Dirichlet forms, where E is assumed to be a topological Hausdorff space with the Borel σ -field B(E) being generated by the continuous functions on E. After the concept of a quasi-regular Dirichlet form is introduced, it is shown that every quasi-regular Dirichlet form is quasi-homeomorphic to a regular Dirichlet space on a locally compact separable metric space.

In Chapter 2, we investigate the transience and recurrence of the semigroups associated with general Dirichlet forms. Analytic potential theory for regular Dirichlet forms, such as capacity, smooth measures, and their potentials, are studied in Section 2.3. Various equivalent characterizations of the local property for a quasi-regular Dirichlet form are given in Section 2.4. Some basic examples of Dirichlet forms corresponding to symmetric Markov processes are presented in Section 2.2, including symmetric pure jump step processes, symmetric Levy processes, one-dimensional diffusions, multidimensional Brownian motions, and Brownian motions on manifolds. An example of a quasi-regular but not regular Dirichlet form on Rn is given in Example 5.1.11.

Probabilistic potential theory of symmetric Markov processes and its relationship to analytic potential theory of the associated Dirichlet forms are presented in Chapter 3.

In Chapter 4, additive functionals of symmetric Markov processes are studied. In particular, the one-to-one correspondence between positive continuous additive functionals and the smooth measures is established. Fukushima decomposition, which serves as a counterpart of Ito’s formula for symmetric Markov processes, is presented in Section 4.2. It plays an important role in analyzing the sample path properties of the processes as well as in other areas of the Dirichlet form theory. Beurling-Deny decomposition of a regular Dirichlet form is derived in Section 4.3 by utilizing martingale additive functionals.

The first half of Chapter 5 is devoted to a study of time changes of symmetric Markov processes, their Dirichlet form characterization, and applications. In the second half, Feller measures are introduced. They are used to characterize trace Dirichlet forms and to identify jump and killing measures of the time-changed process.

The reflected Dirichlet space of a Dirichlet form is introduced and investigated in Chapter 6. It is first introduced under the regular Dirichlet form setting. The transient and recurrent cases are treated separately. It is then extended to the quasi-regular Dirichlet form setting by using quasi-homeomorphism. Concrete examples of reflected Dirichlet spaces are exhibited for a number of Dirichlet forms including most of those appearing in Section 2.2. The important role that reflected Dirichlet spaces would play in the boundary theory for symmetric Markov processes is indicated by the fact that the active reflected Dirichlet form is the maximal Silverstein extension of the Dirichlet form.

In Chapter 7, we present some recent developments of boundary theory of symmetric Markov processes, emphasizing the role of reflected Dirichlet spaces and the function spaces of finite Douglas integrals. In the second half of Chapter 7, we develop the theory when the boundary is countable and give many concrete illustrative examples.

In Appendix A, we present basic materials on (not necessarily symmetric) right processes that are utilized in the text.

For readers’ convenience, an index of some useful results is provided in the Catalogue of Some Useful Theorems at the end of the book.

The material starting from Section 5.5 to the end of Chapter 7 appears here for the first time in a book. Except for a few sections, most results in other parts of the book are not new. However, their proofs and presentations can be found to be novel in many places. As compared to the book by M. Fukushima, Y. Oshima, and M. Takeda published in 1994 [73], the approach presented in this book is more probabilistic and the framework of a quasi-regular Dirichlet form is employed for the first time in parallel with a regular one. On the other hand, [73] contains more detailed expositions of analytic properties of a regular Dirichlet form as well as some other relevant topics including a construction of an associated Hunt process and a Girsanov-type transformation. We refer readers to the book by Z. M. Ma and M. Rockner [119] published in 1992 for further readings on quasi-regular (non-symmetric) Dirichlet forms, in particular for basic concrete examples of quasi-regular Dirichlet forms in infinite dimensions that are not touched upon in the present volume.

The materials in Sections 3.5 and 5.4 and Appendix A owe a lot to the work by P. J. Fitzsimmons, R. K. Getoor, P. A. Meyer, and M. J. Sharpe on right processes, additive functionals, and energy functionals.

The notes at the end of this book provide information on other closely related books, sources of materials, and related literature.

In the bibliography, we list only literature that is directly linked to the topics of this book. But we admit with apology that it is still far from being complete partly due to a great diversity and vast literature of related areas.

In August 2003, the second-named author gave an invited lecture series of London Mathematical Society at the University of Wales Swansea, kindly arranged by N. Jacob. The lecture notes contained a time-change theory as well as a preliminary account of the Douglas integrals for diffusions. They eventually grew into the present book, in collaboration with the first author. We are indebted to N. Jacob for creating an opportunity to write the preliminary version of the book.

Thanks are due to M. Takeda for allowing us to use several ingredients from the recently published Japanese book by Fukushima and Takeda [74].

We are grateful to K. Burdzy, W. T. Fan, M. Hino, N. Kajino, P. Kim, K. Kuwae, S. Lou, and Y. Oshima for reading parts of the manuscript and providing us with helpful comments and lists of typos. We thank the reviewers of this book for their helpful comments. We also thank Vickie Kearn and Stefani Wexler at Princeton University Press for their truly kind assistance during the preparation of this manuscript.

The research of the authors was supported in part by NSF grants in the United States and a Grant-in-Aid for Scientific Research in Japan.

Zhen-Qing Chen

Seattle

Masatoshi Fukushima

Osaka

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 [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 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 σ-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 [39] 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 σ-finite measure on it. Let Bm(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 ε E: f (x) ≠ g(x)}) = 0. For p ≥ 1 and a numerical function f ε Bm(E), we put

The family of all m-equivalence classes of f ε Bm(Ef p < ∞ is denoted by Lp(E; mp, namely, a complete normed linear space. We denote by L∞(E; m) the family of all m-equivalence classes of f ε Bm(E) which are bounded m-a.e. on E. L∞(E; m) is a Banach space with norm

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

For a moment, let us consider an abstract real Hilbert space H for f ε H 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 (ε, D(ε)), the family of all strongly continuous contraction semigroups {Tt; t ≥ 0}, the family of all strongly continuous contraction resolvents {Rα; α > 0}, and the family of all non-positive definite self-adjoint operators A. Here we mention the correspondences among the first three objects only.

ε or (ε, D(εE)) 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 ∈ D(E) and a, b

ε(f, g) = ε(g, f), ε(f, f) ≥ 0, and

ε(af + bg, h) = aε(f, h) + bε(g, h).

For α > 0, we define

εα(f, g) = ε(f, g) + α(f, g), f, g ε D(E).

We call a symmetric form (ε, D(ε)) on H closed if D(E. D(E) is then a real Hilbert space with inner product Eα for each α > 0.

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

We call a family of symmetric linear operators {Gα; α > 0} on H a strongly continuous contraction resolvent if for every α, β > 0 and f ∈ H,

The semigroup {Tt; t ≥ 0} and the resolvent {Gα; α > 0} as above correspond to each other by the next two equations:

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

{Gα; α > 0} determined by (1.1.1) from {Tt; t > 0} is called the resolvent of {Tt; t ≥ 0}.

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

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

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

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

which means that Φ(v) = (f, v) is a bounded linear functional on the Hilbert space (D(E), Eα). By the Riesz representation theorem, there exists a unique element of D(E) denoted by Gαf such that for every f ε H and v ∈ D(E),

{Gα; α > 0} so defined is a strongly continuous contraction resolvent on H, which in turn determines a strongly continuous contraction semigroup {Tt; 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 {Tt; t > 0} to (E, D(E)) and from (E, D(E)) to {Tt; t > 0} are mutually reciprocal.

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

DEFINITION 1.1.1. For 1 ≤ p ≤ ∞, a linear operator L on Lp(E; m) with domain of definition D(L) is called Markovian if

, is said to be a normal contraction if

A function defined by ϕ(tt1, t , is a normal contraction which is called the unit contraction. For any ε > 0, a real function ϕε satisfying the next condition is a normal contraction:

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

A closed symmetric form (ε, 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 (ε, F) be a closed symmetric form on L2(E; m) and {Tt}t>0, {Gα}α>0 be the strongly continuous contraction semigroup and resolvent on L2(E; m) generated by (ε, F), respectively. Then the following conditions are mutually equivalent:

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

(b) αGα is Markovian for each α > 0.

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

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

(e) Every normal contraction operates on (ε, F): for any normal contraction ϕ

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 α > 0 and a function f L2(E; m) with 0 ≤ f ≤ 1 [m], and introduce a quadratic form on F by

It follows from (1.1.6) that

namely, Gαf is a unique element of F minimizing Φ(v). Suppose (ε, F) is a Dirichlet form on L2(E;mto obtain

and t , we have

. Letting ε → 0, we get (b).

It remains to prove the implication (a) ⇒ (e

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

DEFINITION 1.1.4. Let (ε, F) be a closed symmetric form on L2(E; m). Denote by Fe the totality of m-equivalence classes of all m-measurable functions f on E such that |f | < ∞ [msuch that limn→∞ fn = f m-a.e on E. {fn} ⊂ F in the above is called an approximating sequence of f Fe. We call the space Fe the extended space , F). When the latter is a Dirichlet form on L2(E; m), the space Fe will be called its extended Dirichlet space.

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

(i) For any f Fe and for any approximating sequence {fn} ⊂ F of f, the limit (f, f) = limn(fn, fn) exists independently of the choice of an approximating sequence {fn} of f.

(ii) Every normal contraction operates on (Fe): for any normal contraction ϕ

(iii) F = Fe ∩ L2(E; m). In particular, , 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 Fe, clearly both f + g and f – g are in Fe(f, g, which is a symmetric bilinear form over Fe, Fe) is called the extended Dirichlet form , F).

, F) on L2(E; m) is a Dirichlet form, then the corresponding semigroup {Tt; t > 0} is Markovian by virtue of the already proven implication (c) ⇒ (a) of Theorem 1.1.3. So the extended Dirichlet space Fe satisfies all properties mentioned in Theorem 1.1.5.

Before giving the proof of Theorem 1.1.5, we shall fix a Markovian con-tractive 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 L2(E; m) ∩ L∞(E; m),

Due to the σ-finiteness of m, we can construct a Borel function η ∈ L1(E; m) which is strictly positive on E. If we put ηη(x) = (nη(x1, then 0 < ηn ≤ 1, ηn ↑ 1, n → ∞. Hence we can define an extension of T from L2(E; m) ∩ L∞(E; m) to L∞(E; m) as follows:

By the symmetry of T, (g, T(f · ηn)) = (Tg, f · ηn) for g ∈ bL1(E; m). Letting n → ∞, we see that the function Tf, f ∈ L∞(E; m), defined by (1.1.9), satisfies the identity

denotes the integral ∫E Consequently, Tf is uniquely determined up to the m-equivalence for f ∈ L∞(E; m). T becomes a Markovian linear operator on L∞(E; m) and satisfies

Further, if a sequence {fn} ⊂ L∞(E; m) is uniformly bounded and converges to f m-a.e. as n → ∞,

LEMMA 1.1.6. (i) For any g ∈ L∞(E; m),

(ii) For any g ∈ L∞(E;m), define

It holds for g ∈ L2(E; m) ∩ L∞(E; m) that

(iii) For any g ∈ L∞(E; m) and for any normal contraction ϕ,

(iv) For any f, g ∈ L∞(E; m),

Proof. (i) For a simple function on E expressed by

where ai , Bi ∈ B(E) with Bi ∩ Bj = Ø for i ≠ j , we have

≥ = 1} of this type such that

(ii) Recall that {ηn, n ≥ 1} is an increasing sequence of positive functions that is defined preceding (1.1.9). For g ∈ L2 ∩ L∞, we have (Tg2, ηn) = (g2, Tηn) by the symmetry of T. By letting n → ∞, we get ∫ETg2dm = ∫Eg2T1dm < ∞ and (g – Tg, g)=

.

(iii) For g ∈ L∞(E; m) and k = 1, 2, . . ., we put

When g is a simple function of the type (1.1.17),

Here

.

For any normal contraction ϕ, it holds that

Thus for a simple function s. For any g ∈ L∞(E; m), we can take uniformly bounded simple functions s →∞ s = g [m→ ∞ and then k → ∞, we have by (1.1.12) that (1.1.15) holds.

for each fixed k instead of AT, the bilinear form defined by

satisfies the Schwarz inequality

.

> 0} be a specific family of normal contractions defined by

For any m-measurable function g on E with |g| < ∞ [m], ATo g+1 o go g and Lemma 1.1.6(iii). We can then extend AT(g) to g by letting

LEMMA 1.1.7. (i) For g ∈ L2(E; m), AT(g) = (g – Tg, g).

(ii) (Fatou’s property) For any m-measurable functions gn, g on E with |gn| < ∞, |g| < ∞ [m], limn→∞ gn = g [m],

(iii) For any m-measurable function g on E with |g| < ∞ [m] and for any normal contraction ϕ, AT(ϕ o g) ≤ AT(g).

(iv) The triangular inequality (1.1.16) holds for every m-measurable functions f and g that are finite m-a.e. on E.

Proof. (i) follows from Lemma 1.1.6(ii) and the contraction property of T on L2(E; m).

(ii) We first give a proof when |gn| ≤ M, |g| ≤ M for some M and limn→∞gn = g [m]. From the linearity, the Markovian property of T on L∞(E; m), and (1.1.11), we have for b

Since the identity (1.1.18) holds when s is a simple function like (1.1.17), we get from the above inequality

On the other hand,

hence

Taking a sequence of simple functions s such that s → g [m],

Integrating both sides with respect to m and taking the defining formula (1.1.13) into account, we arrive at the desired (1.1.21) using the Fatou’s lemma in the Lebesgue integration theory.

When gn and g are not necessarily uniformly bounded, we can use the results obtained above to get

→ ∞, we still have the inequality (1.1.21).

(iii) It holds for f = ϕ o g, f o g →∞f = f. Equations (1.1.15) and (1.1.21) then lead us to

(iv) If we let fn := ϕn o f and gn := ϕn o g, then limn→∞ (fn + gn) = f + g[m] so that (1.1.16) and (1.1.21) yield

Proof of Theorem 1.1.5. (i) For any f ∈ Fe, take its approximating sequence {fn} ⊂ F. fn . Let us prove that

(f, f) does not depend the choice of the approximating sequence.

Since f – f ∈ Fe and {fn – f } ⊂ F is its approximating sequence, we have from Lemma 1.1.7 and (1.1.4)

increases as t , we can get from the triangular inequality and the inequality obtained above that

→.8. The proof of (1.1.22) is complete. (ii) For any f ∈ Fe and any normal contraction ϕ, we are led from Lemma 1.1.7(iii) and (1.1.22) to

Hence it suffices to show ϕ o f ∈ Fe. For an approximating sequence {fn} ⊂ F of f, we obtain by Lemma 1.1.7 and (1.1.4)

Thus ϕ º fn ∈ F . This means that ϕ o fn are elements of F with uniformly bounded ε-norm. Therefore, the Cesàro mean gk is an ε-Cauchy sequence by Theorem A.4.1. Since limk→∞ gk = ϕ o f [m], we arrive at ϕ o f ∈ Fe.

(iii) The first identity follows from (1.1.4), Lemma 1.1.7, and (1.1.22). Since every normal contraction operates on (ε, F) by (ii), (ε, F

Remark 1.1.8. Property (1.1.22) in particular implies that if {fk, k ≥ 1} ⊂ F is an ε-Cauchy sequence and fk → 0 [m], then ∈(fk, fk

COROLLARY 1.1.9. (Fatou’s lemma) Suppose {fk, k ≥ 1} ⊂ Fe and f ∈ Fe. If fk → f [m], then

Proof. It follows from (1.1.21) and (1.1.22) that

In the remainder of this section, (ε, F) is a Dirichlet form on L2(E; m).

Exercise 1.1.10. Show that for f, g ∈ Fe ∩ L∞(E; m), f · g ∈ Fe

We state two lemmas for later use.

LEMMA 1.1.11. (i) Let ≥1 be a sequence of normal contractions satisfying (t) = t for every t . Then (f) – f ε1 = 0 for any f ∈ F.

(ii) Suppose {fn} ⊂ F is ε1-convergent to f ∈ F. Then, for any normal contraction ϕ, {ϕ(fn)} is ε1-weakly convergent to ϕ(f). If further ϕ(f) = f, then the convergence is ε1-strong.

Proof. (f) = f , then f ∈ F f f ε1. Since G1(L2) is ε1-dense in F by (1.1.6) and ε1(f , G1g) = (f , g) → (f, g) = ε1(f, G1g) for every g ∈ L2, we can conclude that f → ∞ to f weakly in (F, ε1means that the convergence is strong as well.

(ii) ε1-norm of ϕ(fn) is uniformly bounded and, for any g ∈ L2(E; m), ε1(G1g, ϕ(fn) – ϕ(f)) = (g, ϕ(fn) – ϕ(f)) → 0, n → ∞. Hence the first assertion follows. If ϕ(f) = f, then as n → ∞,

LEMMA 1.1.12.Let f be an m-measurable function on E with |f| < ∞ [m]. If, for the contractions of (1.1.19), f o f ∈ Fe for every ≥ 1, and f ε < ∞, then f ∈ Fe.

Proof. Without loss of generality, we assume that f is non-negative. For each, choose an approximating sequence f, k ∈ F for f . We put

and it converges to f m-a.e. as k . Furthermore,

Take a strictly positive m-measurable function g with ∫E gdm as k → ∞ and the latter converges to f converges to f as well as m-a.e. on E if we choose a suitable subsequence {k } of {kvl, k ε obtained above, we can conclude that f ∈ Fe admits the Cesàro mean of a subsequence of w ∈ F

A numerical function K(x, B) of two variables x ∈ E, B ∈ B(E), is said to be a kernel on the measurable space (E, B(E)) if, for each fixed x ∈ E, it is a (positive) measure in B and, for each fixed B ∈ B(E), it is a B(E)-measurable function in x. We then put

Kf ∈ B+(E) for f ∈ B+(E) because the latter is an increasing limit of simple functions. A kernel K is called Markovian if K(x, E) ≤ 1 for every x ∈ E. A Markovian kernel K defines a linear operator on the space of bounded B(E)-measurable functions by (1.1.23). A Markovian kernel K on E is said to be conservative or a probability kernel if K(x, ·) is a probability measure for each x ∈ E.

We call a kernel K(x, ·) (or an operator K) on (E, B(E)) m-symmetric if

Let K be an m-symmetric Markovian kernel on (E, B(E)) and f ∈ bB(E) ∩ L2(E; m). We then have from (1.1.23) (Kf)2(x) ≥ (Kf 2)(x), which yields by integrating with respect to m and using (1.1.24) the contraction property

This means that K can be regarded as a bounded linear operator on the space of m-essentially bounded m-measurable functions in L2(E; m), which is dense in L2(E; m). Hence K is uniquely extended to a linear contraction symmetric operator on L2(E; m).

So far we have assumed that (E, B(E)) is only a measurable space. In the rest of this section, we assume that E is a Hausdorff topological space. In this case, we shall use the notation B(E) exclusively for the Borel field, namely, the σ-field of subsets of E generated by open sets. The space of B(E)-measurable real-valued functions will be denoted by B(E). We sometimes need to consider a larger σ-field B*(E, where P(E) denotes the family of all probability measures on E and Bµ(E) is the completion of B(E) with respect to µ ∈ P(E).

DEFINITION 1.1.13. (i) A family {Pt; t ≥ 0} is called a transition function on (E, B(E)) (resp. (E, B*(E))) if Pt is a Markovian kernel on (E, B(E))(resp. (E, B*(E))) for each t ≥ 0 and the following four conditions are satisfied:

(t.1) PsPtf = Ps+tf for s, t ≥ 0 and f ∈ bB(E) (resp. f ∈ bB*(E)). Here Ptf (x) := ∫Ef(y)Pt(x, dy).

(t.2) For each B ∈ B(E), Pt(x, B) is B([0, ∞)) × B(E)-measurable (resp. B([0, ∞)) × B*(E)-measurable) in two variables (t, x) ∈ [0, ∞) × E.

(t.3) For each x ∈ E, P0(x, ·) = δx(·), where δx denotes the unit mass concentrated at the one-point set {x}.

(t.4) limt↓0 Ptf (x) = f (x) for any f ∈ bC(E) and x ∈ E.

A transition function {Pt; t ≥ 0} is called a transition probability if Pt is conservative for every t > 0.

(ii) A family {Rα; α > 0} is called a resolvent kernel on (E, B(E)) (resp. (E, B*(E))) if, for each α > 0, αRα is a Markovian kernel on (E, B(E)) (resp. (E, B*(E))) and

Property (t.1) is called the semigroup property or Chapman-Kolmogorov equation. Identity (1.1.25) is called the resolvent equation. For a transition function {Pt; t ≥ 0} on (E, B(E)) (resp. (E, B*(E))), it is easy to verify that

determines uniquely a resolvent on (E, B(E)), (resp. (E, B*(E))), which is called the resolvent kernel of the transition function {Pt; t ≥ 0}.

A topological space E is called a Lusin space (resp. Radon space) if it is homeomorphic to a Borel (resp. universally measurable) subset of a compact metric space F. For a topological space E, a measure m on (E, B(E)) is said to be regular if, for any B ∈ B(E), m(B) = inf{m(U): B ⊂ U, U open} = sup{m(K): K ⊂ B, K compact}. Any Radon measure on a locally compact separable metric space is regular. Any finite measure on a Lusin space or on a Radon space is regular.

LEMMA 1.1.14. Let {Pt; t ≥ 0} be a family of Markovian kernels on a Lusin space E equipped with the Borel field B(E) or on a Radon space equipped with the σ-field B*(E) of its universally measurable subsets.

(i) Suppose {Pt; t ≥ 0} satisfies (t.1), (t.3)and

(t.4)' For every f ∈ bC(E), Ptf (x) is right continuous in t ∈ [0, ∞) for each x ∈ E.

Then {Pt; t ≥ 0} is a transition function.

(ii) Suppose {Pt; t ≥ 0} satisfies (t.1), (t.4) and, for a σ-finite measure m on E, {Pt; t ≥ 0} is m-symmetric in the sense that Pt is m-symmetric for each t > 0. Let Tt be the symmetric linear operator on L2(E; m) uniquely determined by Pt. Then {Tt; t ≥ 0} is a strongly continuous contraction semigroup on L2(E; m).

Proof. We give a proof for a family {Pt; t ≥ 0} of Markovian kernels on a Lusin space (E, B(E)). The proof for a Radon space (E, B*(E)) is the same.

(i) It suffices to establish (t.2). Let H be the collection of functions in bB(E) such that Ptf (x) is measurable in two variables (t, x). H is then a linear space closed under the operation of taking uniformly bounded increasing limits. By (t.4)', it holds that bC(E) ⊂ H. Hence (t.2) follows from Proposition A.1.3.

(ii) We may assume that E is a Borel subset of a compact metric space (F, d) and identify L2(E; m) with L2(F; m) by setting m(F \ E) = 0. We first show that bC(F) ∩ L2(F; m) is dense in L2(F; m). Since m is σ-finite, it suffices to assume that m is a finite measure and that the indicator function of a set B ∞ B(F) can be L2-approximated. For any ε, there exist a compact set K and an open set U such that K ⊂ B ⊂ U, m(U \ K) < ε. If we let g(x) = d(x, Uc)/(d(x, Uc) + d(x, K)), x ∈ F, then g ∈ bC(F) ∩ L2(F; mg – 1B .

For any f ∈ L2(E; m) and ε > 0, take a function g ∈ bC(F) ∩ L2(E; mf – g 2 < ε. Because of the contraction property of {Tt; t Ttf – f Ptg – g 2 + 2ε. Further,

which tends to 0 as t ↓ 0 by (t.4)

By virtue of Lemma 1.1.14, any m-symmetric transition function {Pt; t ≥ 0} on a Lusin space (E, B(E)) or a Radon space (E, B*(E)) determines a unique strongly continuous contraction semigroup {Tt; t ≥ 0} on L2(E; m), which in turn decides a Dirichlet form (ε, F) on L2(E; m) according to Theorem 1.1.3. (ε, F) is called the Dirichlet form of the transition function {Pt; t ≥ 0}. In this case, the resolvent {Gα; α > 0} of {Tt; t ≥ 0} is the unique extension of the resolvent kernel {Rα; α > 0} of {Pt; t ≥ 0} from bB(EL2(E; m) to L2(E; m). Moreover, we have from (1.1.6) that for f ≥ bB(E) ∩ L2(E; m),

Conversely, if the resolvent kernel {Rα; α > 0} of a transition function {Pt; t ≥ 0} satisfies (1.1.28) for a Dirichlet form (ε, F) on L2(E; m), then {Pt; t ≥ 0} is m-symmetric and its Dirichlet form coincides with (ε, F).

In the rest of this chapter, we give a quick introduction to the basic theory of quasi-regular Dirichlet forms. The importance of a quasi-regular Dirichlet form is that they are in one-to-one correspondence with symmetric Markov processes having some nice properties. We will show that any quasi-regular Dirichlet form is quasi-homeomorphic to a regular Dirichlet form on a locally compact separable metric space. Thus the study of quasi-regular Dirichlet forms can be reduced to that of regular Dirichlet forms.

1.2. EXCESSIVE FUNCTIONS AND CAPACITIES

In this section, let E be a Hausdorff topological space with the Borel σ-field B(E) being assumed to be generated by the continuous functions on E and m be a σ-finite measure with supp[m] = E. Here for a measure ν on E, its support supp[ν] is by definition the smallest closed set outside which ν vanishes. Let (ε, F) be a symmetric Dirichlet form on L2(E; m), and {Tt; t ≥ 0} and {Gα; α > 0} be its associated semigroup and resolvents on L2(E; m).

DEFINITION 1.2.1. For α > 0, u ∈ L2(E; m) is called α-excessive if e–αtTtu ≤ u m-a.e. for every t > 0.

Remark 1.2.2. (i) If u is α-excessive, then u ≥ 0. This is because

and so u ≥ limt→∞ eααtTtu = 0.

(ii) The constant function 1 is α-excessive if m(E) < ∞. For f ∈ L2+(E; m), Gαf is α-excessive.

(iii) If u1 ≥ 0, u2 ≥ 0 are α-excessive functions, then so are uu2 and u1.

LEMMA 1.2.3. Let u ∈ L2+(E; m) be α-excessive for α > 0. Assume there is v ∈ F such that u ≤ v. Then u ∈ F and Eα(u, u) ≤ Eα (v, v).

Proof. By the symmetry and contraction property of Tt in L2(E; m), for each t > 0, (f, g – e–αtTtg) is a non-negative symmetric bilinear form on L2(E; m). So it satisfies the following Cauchy-Schwarz inequality:

Thus we have by the α-excessiveness of u,

and so

It follows then that

We conclude from (1.1.4)–(1.1.5) that u ∈ F with εα(u, u) ≤ εα(v, v

LEMMA 1.2.4. The following statements are equivalent for u ∈ F and α > 0:

(i) u is α-excessive.

(ii) εα(u, v) ≥ 0 for every non-negative v ∈ F.

Proof. (i) ⇒ (ii): It follow from (1.1.5) that

as t ↓ 0.

(ii) ⇒ (i): For v ∈ L2+(E; m) and t > 0, since

we have

This implies that u – e–αt

For a closed subset F of E, define

THEOREM 1.2.5. Let α > 0 and f be a non-negative function defined on E. For an open set D, denote LD, f = {u ∈ F : u ≥ f m-a.e. on D}. Suppose LD, f ≠ Ø. Then

(i) there is a unique fD ∈ LD, f such that

(ii) fD is the unique function in LD, f such that

(iii) εα(fD, v) ≥ 0 for every v ε F with v ≥ 0 m-a.e. on D. In particular, fD is α-excessive and εα(fD, v) = 0 for every v .

(iv) fD ≤ f if and only if fD f is an α-excessive function. In this case, fD = f m-a.e. on D. fD is the minimum element among a-excessive functions in LD, f in the sense that, if u ∈ LD, f is a-excessive, then fD ≤ u.

(v) If open sets D1 ⊂ D2 and and

(vi) For open sets D1 ⊂ D2, if f fD2is an α-excessive function, then (fD2)D1= fD1. If further f fD1 is α-excessive, then

(vii) For open sets D1 ⊂ D2, (fD1)D2 = fD1.

Proof. (i) Because LD, f is a closed convex set in the Hilbert space (F, Eα), it has a unique minimizer fD.

(ii) For every u ∈ LD, f and 0 < ε < 1, fD + ε(u – fD) = (1 – ε)fD + εu ∈ LD, f and so Eα(fD + ε(u – fD), fD + ε(u – fD)) ≥ Eα(fD, fD). This implies that εα(fD, u – fD) ≥ 0. Now suppose v ∈ LD, f is another function such that for every u ∈ LD, f, εα(v, u – v) ≥ 0. As fD ∈ LD, f, εα(v, fD – v) ≥ 0. But with εα(fD, v – fD) ≥ 0, we have εα(fD – v, fD – v) ≤ 0. Therefore, v = fD.

(iii) For any v ε F with v ≥ 0 m-a.e. on D, fD + εv ∈ LD, f for every ε > 0. One immediately deduces from εα(fD + εv, fD + εv) ≥ εα(fD, fD) that εα(fD, v) ≥ 0.

(iv) This follows immediately from (iii) and Lemma 1.2.3.

(v) The first part follows from (iv). The second part follows from (i).

(vi) By (iv), f = fD2 on D2 and hence by definition, (fD2)D1 = fD1. The second assertion follows from (iii) and (iv).

by (iii). We therefore have by (ii) that fD1 = (fD1)D

The function fD is called the α-reduced function of f on D.

Remark 1.2.6. (i) If f is α-excessive in F, then fD is the εα-orthogonal projection of f . This is because f = (f – fD) + fDby Theorem 1.2.5(iv) and fD by Theorem 1.2.5(iii).

(ii) By (iii) and (iv) of Theorem 1.2.5, if g ∈ F is α-excessive, then εα(fD, g) = εα(fD, gD

DEFINITION 1.2.7. ((h, α)-capacity) Fix α > 0. Let h ≥ 0 be a function on E satisfying one of the following two conditions:

(i) h ∈ F and h is α-excessive;

(ii) h hD is a α-excessive function for every open set D ⊂ E with LD, h ≠ ∅. (This is equivalent to, by Theorem 1.2.5(iv), that h ≥ hD for every open set D ⊂ E with LD, h ≠ ∅). Define for open subset D ⊂ E, and for an arbitrary subset A ⊂ E,

and for an arbitrary subset A ⊂ E,

Remark 1.2.8. (i) Important cases are h = 1 and h = Gαϕ for some strictly positive ϕ ∈ L2(E; m).

(ii) Under either of conditions (i) and (ii), h = hD [m] on D whenever LD, h ≠ ∅.

(iii) When h > 0 [m] on E, then Caph,α(A) = 0 implies that m(A) = 0.

for any open set D by Lemma 1.2.3.

(v) We shall use the following comparison in α > 0 for the capacity: if h1 is 1-excessive, h2 is 2-excessive, and h2 ≤ h1, then

In fact, we have for an open set D,

In the remainder of this section h ≥ 0 is a non-trivial function on E satisfying