Functional Inequalities Markov Semigroups and Spectral Theory

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By Fengyu Wang, School of Mathematical Sciences, Beijing Normal University

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Included in series The Science Series of the Contemporary Elite Youth, Audience mathematicians, libraries, departments of mathematics Contents Chapter 0: Preliminaries Chapter 1 Poincare Inequality and Spectral Gap Chapter 2 Diffusion Processes on Manifolds and Applications Chapter 3 Functional Inequalities and Essential Spectrum Chapter 4: Weak Poincare Inequalities and Convergence of Semigroups Chapter 5 Log-Sobolev Inequalities and Semigroup Properties Chapter 6 Interpolations of Poincare and Log-Sobolev Inequalities Chapter 7 Some Infinite Dimensional Models Bibliography Index

Bibliographic & ordering Information Hardbound, publication date: FEB-2006 ISBN-13: 978-0-08-044942-5 ISBN-10: 0-08-044942-5 Imprint: ELSEVIER Price: Order form USD 98.95 EUR 79.95 GBP 54.99 Books and book related electronic products are priced in US dollars (USD), euro (EUR), and Great Britain Pounds (GBP). USD prices apply to the Americas and Asia Pacific. EUR prices apply in Europe and the Middle East. GBP prices apply to the UK and all other countries. See also information about conditions of sale & ordering procedures, and links to our regional sales offices. 050/501 Last update: 17 Mar 2008

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Preface Since in a standard situation (e.g. in the symmetric case), any C0-contraction semigroup (and hence its generator) on a Hilbert space is uniquely determined by the associated quadratic form, it is reasonable to describe the properties of the semigroup and its generator by using functional inequalities of the quadratic form. In particular, if the associated form is a Dirichlet form, then the corresponding semigroup is (sub-) Markovian. The purpose of this book is to present a systematic account of functional inequalities for Dirichlet forms and applications to Markov semigroups (or Markov processes in a regular case). The functional inequalities considered here only involve in the Dirichlet form and one or two norms of functions, and can be easily illustrated in many cases. On the other hand, these inequalities imply plentiful analytic properties of Markov semigroups and generators, which are related to various behaviors of the corresponding Markov processes. For instance, the Poincar6 inequality is equivalent to the exponential convergence of the semigroup and the existence of the spectral gap. Moreover, the Gross log-Sobolev inequality is equivalent to Nelson's hypercontractivity of the semigroup and is strictly stronger than the Poincard inequality. So, it is natural for us to ask for more spectral information and semigroup properties from more general functional inequalities. This is the starting point of the book. In this book, we introduce functional inequalities to describe: (i) the spectrum of the generator: the essential and discrete spectrums, high order eigenvalues, the principal eigenvalue, and the spectral gap; (ii) the semigroup properties: the uniform integrability, the compactness, the convergence rate, and the existence of density; (iii) the reference measure and the intrinsic metric: the concentration, the isoperimetric inequality, and the transportation cost inequality. For reader's convenience and for the completeness of the account, we summarize some necessary preliminaries in Chapter 0. Corresponding to various levels of spectral and semigroup properties, Chapters 1, 3, 4, 5 and 6 focus on several different functional inequalities respectively: Chapter 1 and Chapter 5 introduce the above mentioned Poincard and log-Sobolev inequalities respectively, Chapter 6 the interpolations of these two inequalities, Chapter 3 the super Poincard inequality, and Chapter 4 the weak Poincard inequality. Each of these chapters presents a correspondence between the underlying

ii

Preface

functional inequality and the properties of the semigroup and its generator, as well as sufficient and necessary conditions for the functional inequality to hold. Moreover, the general results are illustrated by concrete examples, in particular, examples of diffusion processes on manifolds and countable Markov chains. These chapters are relatively (although not absolutely) independent, so that one may read in one's own order without much trouble. Chapter 2 is devoted to diffusion processes on Riemannian manifolds and applications to geometry analysis. In particular, the estimation of the first eigenvalue is related to the Poincar~ inequality, while the results concerning gradient estimates, the Harnack inequality and the isoperimetric inequality will be used in the sequel to illustrate other functional inequalities. The results included in w concerning the first eigenvalue have been introduced in a recent monograph [47] by Professor Mu-Fa Chen. Chen's monograph emphasizes the main idea of the study which is crucial for understanding the machinery of the work, while the present book provides the technical details which are useful for further study. Finally, in Chapter 7 we establish functional inequalities for three infinite-dimensional models which have been studying intensively in stochastic analysis and mathematical physics. At the end of each chapter (except Chapter 0), some historical notes and open questions for further studies are addressed. The notes are not intended to summarize the principal results of each paper cited but merely to indicate the connection to the main contents of each chapter in question, while the open problems are listed mainly based on my own interests. Thus, these notes are far from complete in the strict sense. At the end of the book, a list of publications and an index of main notations and key words are presented for reader's reference. These references are presented not for completeness but for a usable guide to the literature. I regret that there might be a lot of related publications which have not been mentioned in the book. Due to the limitation of knowledge and the experience of writing, I would like to apologize in advance for possible mistakes and incomplete accounts appeared in this book, and to appreciate criticisms and corrections in any sense. I would like to express my deep gratitude to my advisors Professor ShiJian Yan and Professor Mu-Fa Chen for earnest teachings and constant helps. Professor Chen guided me to the cross research field of probability theory and Riemannian manifold, and emphasized probabilistic approaches in research, in particular, the coupling methods which he had worked on intensively. Our fruitful cooperations in this direction considerably stimulated other work included in this book. During the past decade I also greatly benefited from col-

Preface

iii

laborations and communications with Professors M. RSckner, A. Thalmaier, V. I. Bogachev, F.-Z. Gong, K. D. Elworthy, M. Cranston and X.-M. Li. In particular, the work concerning the weak Poincar6 inequality and applications is due to effective cooperations with Professor M. RSckner. At different stages I received helpful suggestions and encouragements from many other mathematicians, in particular, Professors S. Aida, S. Albeverio, D. Barkry, D. Chaifi, D.-Y. Chen, T. Couhlon, S. Fang, M. Fukushima, G.-L. Gong, L. Gross, E. Hsu, C.-R. Hwang, W.S. Kendall, R. Leandre, M. Ledoux, Z.-H. Li, Z.-M. Ma, P. Malliavin, Y.-H. Mao, S.-G. Peng, E. Priola, M.-P. Qian, I. Shigekawa, D. Stroock, K.-T. Sturm, Y.-L. Sun, J.-L. Wu, L. Wu, J.-A. Yan, T.-S. Zhang, Y.-H. Zhang and X.-L. Zhao. I would also like to thank Professor Yu-Hui Zhang, Mr Wei Liu and graduate students in our group for reading the draft and checking errors. Finally, it is a pleasure to acknowledge the generous support of this work by the National Natural Science Foundation of China (1002510, 10121101), the Teaching and Research Awarded Project for Outstanding Young Teachers, and the 973-Project.

Feng-Yu Wang Beijing, June 2004

Chapter 0 Preliminaries In this chapter we briefly recall some necessary preliminaries of the book from Dirichlet forms, Markov processes, spectral theory and Riemannian geometry. Results included in this part are well-known and fundamental in these fields. w and w are mainly summarized from [137], most results in w can be

found in [225] and [155], and w

0.1

is mainly selected from [33] and [35].

Dirichlet forms, sub-Markov semigroups and generators

Let us start with some basic facts on semigroups, resolvents and generators. Let (1~, II"]I) be a real Banach space. A pair (L, ~(L)) is called a linear operator on ~ if ~(L) is a linear subspace of ~ and L: ~(L) -, ~ is a linear map. We sometimes simply denote the operator by L. The operator L is called closed if its graph {(f, Lf): f E ~(L)} is closed in ~ • ]~. A linear operator (L, ~(L)) is called closable if the closure of its graph is the graph of a linear operator (L, ~ ( L ) ) which is called the closure of (L, ~(L)).

Definition 0.1.1 A family {Pt}t>~0 of linear operators on ~ with ~(Pt) for all t ~> 0 is called a strongly continuous (or Co-) contraction semigroup on ]~, if (1) lim P t f - Pof - f, f c 1t~. t--~0

(2) IlPtll := sup{llPtfll: f E I~, [Ifll >.O. For a given C0-contraction semigroup {Pt}t>~o (simply denoted by Pt in the sequel), define ~(L) "- { f E ]~" lim -1(Ptf - f) exists in ]~} t-~O t

L f "= lim -1( P t f - f ) , t~O t

f E ~(L).

2

Chapter 0

Preliminaries

Then (L, ~(L)) is a linear operator on 1~, which is called the (infinitesimal) generator of Pt. The following well-known result provides a complete characterization for generators of C0-contraction semigroups (see e.g. [225]). T h e o r e m 0.1.1 (Hille-Yoshida Theorem) A linear operator (L, ~ ( L ) ) is the generator of a Co-contraction semigroup if and only if (1) L is densely defined, i.e. ~ ( L ) is dense in 1~. (2) For any )~ > O, ()~ - L) is invertible and I1()~- L) -111 < A-1. In this case the corresponding semigroup is uniquely determined by L and is denoted by Pt - e tL, and L is closed. Let (L, ~(L)) be the generator of a C0-contraction semigroup Pt. We have R )~f "= ()~- L ) - l f =

/0=

e-'Xs Ps f ds ,

f E I~, )~ > O.

We call {R.x : )~ > 0} the resolvent of L or Pt, see w for this notion of linear operators on complex Banach spaces. Now, let us consider I~ := H, a real Hilbert space with inner product (,). Then an operator (L, ~(L)) provides a bilinear map 8 : ~ ( L ) x ~ ( L ) ~ IR with 8 ( f , g ) := - ( L f , g) for f , g E ~(L). In general, (s is called a bilinear form on IE if ~ ( 5 ~) is a linear subspace of IE and 5~: ~ ( d ~) x ~ ( d ~) -~ H is a bilinear map. If moreover 8 ( f , f) /> 0 for f E ~ ( 8 ) , then (5~, ~ ( 8 ) ) i s called a positive definite form on ]HI. For a bilinear form (g~, ~ ( # ) ) , we define its symmetric part by 8-( f , g ) : - 1 (8(f, g) + 8(g, f)), f, g E ~ ( 8 ) . Moreover, let 8a(f, g) := a(f, g) + 8 ( f , g), f, g E ~ ( 8 ) , a ~> 0. D e f i n i t i o n 0.1.2

Let (8, ~ ( s

be a densely defined positive definite form

on ]HI. (1) (8, ~ ( 8 ) ) (2) (8, ~ ( 8 ) ) (3) (s ~ ( s exists a constant

is called symmetric if 5~(f, g) = 5~(g, f), f, g E ~ ( E ) . is called closed if ~ ( 8 ) is complete under the norm 81/2. is called a coercive closed form on ]HI if it is closed and there K > 0 such that

181(f, g)l < KS1 (f, f)1/281 (g, g)l/2,

f, g e ~(E).

(0.l.1)

Condition (0.1.1) is called the weak sector condition. The following result gives a correspondence between the coercive closed forms and the generators of C0-contraction semigroups. T h e o r e m 0.1.2

(1) Let (s ~ ( s

be a coercive closed form. Define

~ ( L ) "= { f c ~ ( 8 ) " the map 8 ( f , .)" ~ ( 8 ) - + R is continuous under I1" II},

0.1

Dirichlet forms, sub-Markov semigroups and generators

3

and for f E ~(L) define L f e H via - ( L f, g) - 8 ( f , g) for all g e ~(E). Then L is the generator of a Co-contraction semigroup Pt with resolvent {Ra},x>0 satisfying R~ (H) C ~ ( E ) and s

g) = (f, g),

f e H, g e ~(E), )~ > O.

(o.1.2) In particular, (L, ~(L)) satisfies the weak sector condition: there exists K > 0 such that f, g e ~(L).

I((1 - L)f,g}[ 0 satisfies (0.1.2). Finally, let us consider the Markovian setting. Let (E, ~ , #) be a measure space and let H := L2(#), the set of all measurable real functions which are square-integrable with respect to #, that is, letting ~ ( E ) be the set of all measurable real functions on E, we have

S d. We write f ~< g or f < g if the corresponding inequality holds #-a.e. D e f i n i t i o n 0.1.3 (1) A bounded linear operator P " L2(#) ~ L2(#) is called sub-Markovian if 0 ~< P f ~< 1 for all f c L2(#) with 0 ~< f ~o is called a sub-Markov (rasp. Markov) semigroup if each Pt is sub-Markovian (rasp. Markovian). (2) A closed densely defined linear operator (L, ~ ( L ) ) on L2(#) is called a Dirichlet operator if (L f, (f - 1) + ) ~< 0 for all f e ~(L). If moreover 1 E ~ ( L ) and L1 = 0 then L is called a conservative Dirichlet generator. (3) A coercive closed form (8, ~(g~)) on L2(#) is called a Dirichlet form if for any f e ~ ( 8 ) , one has f + A 1 e ~ ( E ) and E(f+f+Al,

f-f+A1)>70,

E(f-f+Al,

f + f + A1) >7 0.

(0.1.4)

A Dirichlet form (8, ~ ( E ) ) is called conservative if 1 e ~ ( E ) and E(f, 1) = d~ f ) = 0 for all f e _@(d~

4

Chapter 0

Preliminaries

Proposition 0.1.3 (1) If (s ~(5~)) is a Dirichlet form then so is its symmetric part (~, ~(s (2) A symmetric closed form (s ~(s is a Dirichlet form if and only if for any f ~ ~ ( ~ ) one has f+ A 1 ~ ~ ( ~ ) and

d~ + A 1, f+ A 1) ~< d~

f).

(3) A symmetric closed form (8, ~ ( 8 ) ) is a Dirichlet form if and only if for any T : I~ ---, I~ with T(O) = 0 and IT(x) - T(y)l 0 be the corresponding resolvent. Then the following are equivalent.

(1) i is a nirichlet operator ( resp. conservative Dirichlet operator). (2) {Pt}t>~o is sub-Markovian (resp. Markovian). (3) For each )~ > O, )~R~ is sub-Markovian (resp. Markovian). If (5,~(5)) satisfies the weak sector condition (0.1.3) and ( 8 , ~ ( 8 ) ) is the associated coercive closed form, then they are also equivalent. (4) For any f E ~ ( 8 ) one has f+A1 E ~ ( 8 ) and E ( f + f + A 1 , f - f + A 1 ) ~> 0 (resp. moreover 1 e ~ ( 8 ) with 8(1, f) = 1 for all f e ~ ( 8 ) ) .

Corollary 0.1.5 Let (s ~(8)) be a coercive closed form associated to the generator ( L , ~ ( / ) ) , the semigroup {Pt}t~o and the resolvent {R~}~>0. Let P~ (resp. R*~) be the adjoint operator (see Definition 0.3.2 below) of Pt (resp. R;~) on 52(#) for t >~0 (resp. A > 0), and let (L*, ~(L*)) be the corresponding generator. Then the following are equivalent. (1) (s ~(s is a Dirichlet form (resp. conservative nirichlet form). (2) L and L* are Dirichlet operators (resp. conservative Dirichlet operators). (3) {Pt}t>~o and {Pt*}t~>0 are sub-Markovian (resp. Markovian). (4))~R~ and AR*~ are sub-Markovian (resp. Markovian) for each )~ > O.

In applications, 8 is often explicitly defined on a smaller domain ~ ( 8 ) so that (8, 2(8)) is not closed. To determine a closed form, one needs to find a closed extension of (8, ~(8)). To this end, we introduce the notion of closability of the form. Definition 0.1.4 A positive definite bilinear form (E, :~(d~)) is called closable if it has a closed extension (5~', ~(8')), i.e. (8', ~(8')) is a closed form with .~(oz') D _~(8) and 8'l~(8 ) oz. -

-

0.1

Dirichlet forms, sub-Markov semigroups and generators

5

Proposition 0.1.6

Let (s ~ ( s be a positive definite bilinear form satisfying the weak sector condition (0.1.1). (1) (8, ~ ( 8 ) ) is closable if and only if for any 8-Cauchy sequence {fn} C ~ ( 8 ) ( f i.e. 8 ( f n - fro, fn - fro) ~ 0 as n, m ~ oc) with fn ~ O(n ~ c~) in L2(#), one has E(fn, fn) ---+O(n ---+c~). (2) (E, ~(E)) is closable if and only if so is its symmetric part (8, ~(8)). (3) If (8, ~ ( 8 ) ) is closable, then it extends uniquely to the completion of ~ ( E ) with respect to the norm 81/2, denoted by (~, ~ ( ~ ) ) . If moreover ~ ( 8 ) is dense in L2(#) then (~, ~(~)) is the smallest coercive closed form extending (8, ~ ( 8 ) ) , and is called the closure of (8, ~ ( 8 ) ) .

P r o p o s i t i o n 0.1.7

(1) Let (L,~(L)) be a negative definite operator on L2(p), satisfying the weak sector condition (0.1.3). Define E(f, g) := - ( L f, g),

f, g e ~(L).

Then (8, ~ ( L ) ) is closable on L 2(p). (2) Let ( 8 ( k ) , ~ ( f ( k ) ) ) , k e N, be closable (rasp. closed) positive definite symmetric forms on L2(p). Let

~ ( ~ ) "-

f C N ~(8(k))'EE(k)(f'f) k/>l

< oc ,

k=l

CO

8(f, g) "-- ~

E (k) (f, g),

f, g e ~(E).

k=l

Then (8, ~ ( ~ ) ) is closable (rasp. closed) on L2(#). (3) Let ( 8 , ~ ( 8 ) ) be a coerceive closed form and {fn} C ~ ( 8 ) such that {8(f~, fn)} is bounded and fn ~ f e L2(#) as n ~ c~, then f e ~ ( E ) and

lim 8 ( f n , f ) - E ( f , f ) < lim 8 ( f n , fn). n---,oc

n--~c~

Finally, the following result (see [91, Theorem 1.5.2]) enables us to extend the domain of a Dirichlet form. T h e o r e m 0.1.8

Let (8, ~ ( 8 ) ) be a symmetric Dirichlet form on L2(#). For any measurable function f , if there exists an 8-Cauchy sequence {fn} C ~ ( 8 ) such that f n ---* f #-a. e. , then the limit g~(f, f ) " - lim E ( f n , f n ) exists and n---,oc

does not depend on the choice of {fn}. If moreover f e L2(#) then f e ~ ( 8 ) .

According to Theorem 0.1.8 we may extend the Dirichlet form to the extended domain ~ ( E ) "- {f e ~ ( E ) "

f~ ~ f #-a.e. for some

6

Chapter 0

Preliminaries

8-Cauchy sequence {fn } C ~ ( E ) }, where ~ ( E ) is the set of all measurable real functions on E. Throughout the book, all real or complex functions are assumed to be finite.

0.2

D i r i c h l e t forms and M a r k o v p r o c e s s e s

In this section we introduce the correspondence between Dirichlet forms and Markov processes, i.e. to show how these two objects determine each other. We first recall the notion of Markov processes. Let E be a Hausdorff topological space with Borel a-field ~ , that is, the a-field induced by open sets. A stochastic process with state space E describes the behavior of a particle randomly moving on E. If the particle is allowed to move out from E, then we add a new point A to stand for the "died state" of the particle. Thus, the whole state space becomes EA := E U{A } equipped with the natural one-point compaction topology, i.e. a subset G of EA is open if it is either an open set in E or a set containing A with compact complement. If in particular E itself is compact, then A is isolated. Let ~ A be the corresponding Borel a-field. For any function f on E, we extend it to EL by letting f (A) = 0. For simplicity, we only consider the standard Markov process defined on the canonical path space over E. A map w. : [0, oc) ~ EA is called a canonical path if it is right continuous and has left limit at each point t > 0 with wt ~ A. Let J? denote the set of all canonical pathes over E such that wt = A for all t ~> ~(w) : - inf{t ~> 0: wt - A}, where ~ is called the lifetime. For each t 1> 0, let

xt :12 ~ E;

xt(w) :=wt,

wEs

and let ~ t := a(x~ : s 0 is called the natural filtration of the path process over E. To make the filtration right continuous, let ~ + " - N s > t ~ s , t ~> 0. In the sequel, whenever (12, ~ ) is equipped with a probability measure I~, the filtration under consider is automatically extended to its completion with respect to I~. D e f i n i t i o n 0.2.1 A family of probability measures {IPz : x E EA} on (s ~cr is called a Markov process on E, if (1) ~X(xo E A) = 5~(A) := 1A(X),X e E A , A E ~ A , where 1A is the indicator function of A. In particular, ~A(xt -- A, t/> 0) -- 1. (2) For any F E ~cr is ~A-measurable.

0.2

Dirichlet forms and Markov processes

7

(3) For any s, t >/0 and any x E E, A E ~/x,

(o.2.i)

IPx (xt+~ E A J ~ ) = Fx (xt+~ E AJx~),

where IpX(.]x~) is the conditional probability of IPx under the a-field induced by x~. The equation (0.2.1) is called the Markov property (with respect to the filtration { ~ t } ) . If for any x E E one has ] ? x ( ~ = c r 1, then we may drop A and call {FX:x E E} a nonexplosive (or conservative) Markov process on E. Given a Markov process {Fx : x E EA}, and given u E ~ ( E A ) , the set p

of all probability measures on EA, let IP" " - / _ JE

FXu(dx), which is called the A

distribution of the Markov process starting from u. In this book, we only consider the time-homogeneous Markov process for which the Markov property (0.2.1) can be written as

FX(xt+~ E A I ~ ) = I~x~ (xt E A),

I~X-a.s., x E EA, A E ~ A , s, t >/0. (0.2.2) For a time-homogenous Markov process {IPx : x E EA }, define

Ptf (x) "- EX f (xt) "- / ~ f (xt)dF x,

f E ~ + ( E ) , x E E,

where ~ + ( E ) is the set of nonnegative measurable functions on E. f ( A ) = 0 by convention, we have (x)

Since

x E E , t >~O.

-

It is easy to see from (0.2.2) that {Pt}t~o is a sub-Markov semigroup on :~b(E) := {f E ~ ( E ) : Ilfi] := sup If] < cr which is a Banach space and the norm is called the uniform norm. To introduce the definition of strong Markov processes, let us recall the notion of a stopping time. A mapping T: [2 ~ [0, cr is called a { ~ t }-stopping time if {T ~< t} E ~ t for all t ~> 0. Given a stopping time ~- we define the a-field

{r

r

.< t} e

t

0}

Definition 0.2.2 A time-homogenous Markov process {I~x : x E EA} is called a strong Markov process if for any { ~ t }-stopping time T, any u E ~ (EA) and any A E ~/x,

I~" (xt+r E A I ~ )

= ~xr (xt E A),

IP'-a.s.

on

{~- < cr

Now, let us connect Markov processes with Dirichlet forms.

(0.2.3)

8

Chapter 0.

Preliminaries

D e f i n i t i o n 0.2.3 Let # be a measure on (E, ~ ) . Let (d~, ~ ( 8 ) ) be a Dirichlet form on L2(#) with {Tt}t>>,o the associated sub-Markov C0-contraction semigroup. A Markov process {I?x 9 x E EA} is called associated with (d~, ~(d~)) if its semigroup Pt satisfies Ptf - Ttf #-a.e. for all t > 0 and all f e gYb(E) ~ L2(#).

Let {I?x : x E EA} be a Markov process with semigroup {Pt}t>~o. If {Pt}t>~o is strongly continuous on L2(#) with generator satisfying the weak sector condition, then Pt is associated with a unique coercive closed form ( 8 , ~ ( 8 ) ) such that for any f E ~ ( 8 ) one has f+ A 1 E ~ ( 8 ) and 8 ( f + f + A 1, f - f + A 1) /> 0. If moreover P~ is also sub-Markovian, it is P r o p o s i t i o n 0.2.1

the case when # is a Pt-supermedian measure, i.e. f E P t f d# ~o is strongly continuous if

9- { f e C b ( E ) N L 2 ( # ) "{Ptf}te[o,1)is uniformly integrable in L2(#)} is dense in L2(#). D e f i n i t i o n 0.2.4 Let (5~, ~ ( 8 ) ) be a Dirichlet form on L2(#). (1) For any g C E, let ~ ( 8 ) K := {f e ~ ( 8 ) : f[gc = 0 #-a.e.}. (2) An increasing sequence {Kn}n>~l of closed subsets of E is called an 8-nest if [.J @(8)K, is dense in @(8) with respect to ~1/2. n~l

(3) A subset N c E is called 8-exceptional if N c (7 K~ for some 8-nest n~l

{Kn}n~l. (4) A function f is called 8-quasi-continuous if there exists an 8-nest f[gn is continuous for each n I> 1. We now introduce the notion of quasi-regular Dirichlet form which is associated with the special standard process.

{Kn}n~l such that

D e f i n i t i o n 0.2.5 A Dirichlet form (8, ~ ( 8 ) ) on L2(#)is called quasi-regular if (1) There exists an 8-nest {Kn}n>~l of compact sets.

0.3

Spectral theory

9

(2) The set { f e ~ ( 8 ) : f has a d~-quasi-continuous #-version} is dense in ~ ( # ) with respect to ~1/2. (3) There exist an E-exceptional set N C E and {fn}n>~l C ~ ( E ) having 8-quasi-continuous p-versions {fn}~>l separating the points of N c. T h e o r e m 0.2.2 Assume that a ( C ( E ) ) = J~. A Dirichlet form (8, ~ ( 8 ) ) on L2(#) is quasi-regular if and only if it is associated with a p-tight special standard process {IPx : x E EA}, i.e. the process is strongly Markovian with respect to {~,~+} and satisfies (1) For some (and hence all) L, e ~ ( E / x ) equivalent to #, if T, Tn(n ~ 1) are J~+-stopping times such that ~-n ~ 7, then x ~ ~ x~(n ~ c)c) P'-a.s. on {T < ~}, and x~ is Vn~l ~,+Tn-measurable. As mentioned above the filtration is understood as its completion with respect to F'. (2) There exists an increasing sequence {Kn}n~l of compact metrizable sets in E such that IP'( lim aK~ < ~) -- 0, .where aK~ "-- inf{t >~ 0" xt ~ Kn}. n---+oo

In this case P t f is 8-quasi-continuous for all t > 0 and all f c ~ b ( E ) ~ L 2 (#), i.e. the process is properly associated with (E, ~ ( 8 ) ) . Let E be a locally compact separable metric space with # being a positive Radon measure of full support, i.e. # are finite on compact sets and strictly positive on non-empty open sets. Let Co(E) be the set of all continuous real functions with compact supports. Then we call a symmetric Dirichlet form (8, ~(d~)) on L2(p) regular if ~ ( E ) N C o ( E ) is a core, i.e. it is dense in ~ ( E ) under the norm E 1/2 and dense in Co(E) under the uniform norm. Moreover, a strong Markov process is called a Hunt process if it is strongly Markovian with respect to { ~ + } and quasi-left continuous, i.e. for a n y , e ~ ( E / x ) and any ~ + - s t o p p i n g times ~-, T~(n >~ 1) with ~-n ~ ~-, one has x ~ ~ x~(n ~ oc) P ' a.s. on {T < c~}. The following result is taken from [91, Theorem 7.2.1]. T h e o r e m 0.2.3 If (go,~(go)) is a symmetric regular Dirichlet form on L2(#), then it is associated with a #-symmetric Hunt process.

0.3

Spectral theory

Let (L, ~ ( L ) ) be a linear operator on the complex Banach space (]~, I1'1]). D e f i n i t i o n 0.3.1 Let p(L) be the set of all ~ c C such that the range ~ ( X - L ) of X - L is dense in ~ and )~-L has a bounded inverse R~ "- ( ) , - L ) -1. We call p(L) the resolvent set of L and R~ the resolvent of L at ~ for each E p(L). Moreover, a(L) := C \ p(L) is called the spectrum of i a n d is decomposed into the following three disjoint parts:

10

Chapter 0

Preliminaries

(a) The point spectrum ap(L) is the set of all A E C such that there exists f r 0 with L f = Af , i.e. A - L does not have any inverse. The element f is called an eigenvector of L corresponding to the eigenvalue A. (b) The continuous spectrum a~(L) is the set of all A E C such that A - L has an unbounded densely defined inverse. (c) The residual spectrum a~(L) is the set of all A E C such that A - L has an inverse whose domain is not dense. Finally, the above notions of a linear operator on a real Banach space are defined by those of the complexification of the operator. P r o p o s i t i o n 0.3.1 Let (L, ~ ( L ) ) be closed. Then for any A E p(L), R~ "( A - L) -1 is defined on the whole space, i.e. ~ ( A - L ) = I~. D e f i n i t i o n 0.3.2 Let (]HI,(,)) be a real or complex Silbert space, and let (L, ~ ( L ) ) be a densely defined linear operator on H. (1) Let ~(L*) be the set of all f E ]E such that there exists (and hence unique) g E ]HI satisfying (Lh, f) = (h,g), h E ~(L). Let L * f "= g for f E ~(L*). We call (L*, ~(L*)) the adjoint operator of (L, ~(L)). (2) (L, ~ ( L ) ) is called symmetric if L* D L, i.e. (L*, ~(L*)) is an extension of (L, @(L)). (3) If L* = L, then ( L , ~ ( L ) ) i s called self-adjoint. If L* is self-adjoint, i.e., L** = L*, then (L, ~ ( L ) ) is called essentially self-adjoint. P r o p o s i t i o n 0.3.2 Let (L, @(L)) be a densely defined operator on H. Then: (1) (L*, @(L*)) is closed, hence any self-adjoint operator is closed. (2) L is closable if and only if @(L*) is dense, in this case L - L** and L* = L*. T h e o r e m 0.3.3 Let (L, @(L)) be a densely defined negative definite symmetric operator on H. Then the quadric from 8 ( f , g ) "- - ( L f , g) defined on @(L) is closable and its closure is associated with a unique self-adjoint operator which is called the Friedrichs extension of (L,@(L)). (L,@(L)) is essentially self-adjoint if and only if its closure is self-adjoint, in this case the closure is the unique self-adjoint extension. We now consider the spectrum of a self-adjoint operator, which is determined by the resolution of identity (or the spectral family). D e f i n i t i o n 0.3.3 A family of projection operators {E~ 9 A E I~} on ]HI is called a resolution of the identity if (1) E),~E), 2 = EA1AA2, /~1, )~2 E ]I~. (2) E _ ~ f "- lira E),f = 0, E ~ f "- lira E),f - f, A~-cc

A---+eo

f E H.

0.3

Spectral theory

11

(3) E~+ = E~, i.e. lim E~, f = E~f,

f E H.

P r o p o s i t i o n 0.3.4 Let {E~ " )~ E I~} be a resolution of the identity. For any f, g e H, (E.f, g) is a boundedly variational function and hence d(E~f, g) is a signed measure on R. In particular, if ]]f]] - 1 then dllE~(f)]l 2 "= d(E~f, f} is a probability measure. T h e o r e m 0.3.5 (L, ~ ( L ) ) is a self-adjoint operator on H if and only if there exists a unique resolution of the identity {E~ : )~ ~ ~} such that Lf -

-

exists

/

in

,

w h e r e / _ ~ )~dE~f is defined as the Riemann integral on H. oo

Remark

0.3.1

Let F be a complex-valued Borel function on IR.

F()~)dExf exists if and only if

Then

IF()~)I2d]IExfll 2 < oo. In particu-

lar, if F is a real-valued continuous function, then (F(L), ~ ( F ( L ) ) ) defined below is a self-adjoint operator:

F ( n ) f "-

?

F()~)dExf,

~ ( F ( ~ ) ) "-

(:x:)

{

f"

}

F()~)2dl[Exfll 2 < oc . (2o

D e f i n i t i o n 0.3.4 The resolution of the identity {E~ 9 ~ c IR} satisfying (0.3.1) is called the spectral family of (L, ~ ( L ) ) . T h e o r e m 0.3.6 Let (L, ~ ( L ) ) be a self-adjoint operator on H. Then: (1) a(L) C I~ and at(L) = O. (2))~ C ap(L) if and only if E~ r E~_, and the eigenspace of )~, i.e. the space spanned by eigenvectors with eigenvalue )~, is ~ ( E ~ - E~,). (3) A E ac(L) /f and only if E~ = E~_ and E~ 1 r E~ 2 for any )~1 < )~ < )~2. In other words, ac(L) = a(L) \ ap(L). (4) For any real continuous function F, one has a(F(L)) = F(a(L)). If moreover F is strictly monotonic then ap(F(L)) - F(ap(L)). Theorem 0.3.6 (4) is called the spectral mapping theorem, which follows immediately from the definition of F(L). Indeed, for any ~0 E a(L), there exists {fn} C ~ ( L ) with ]]f~]] = 1 such that the support of dE~fn is contained in

[)~0 - -1, ~ 0

1] 9 Then

+ -

it is trivial to see that IIF(L)fn - F()~o)fnll 0, one has lim

sup #(IPflPl{.gfj>rjg,})-O,

where []. lip is the norm in L~(#). semicompactness.

(0.3.4)

This leads to the following notion of

(1) The following quantity is called the measure of nonsemicompactness of a set D C L~(#)"

D e f i n i t i o n 0.3.9

p(D)'=

inf

supllfl{ifl>lgl}llp,

geLS(#) feD

and p ( P ) " - p(PB(O, 1)) is called the measure of non-semicompactness of P. Moreover, D (rasp. P) is called semicompact if p(D) - 0 (rasp. p(P) - 0 ) . (2) D is called order bounded in n ~ ( # ) if there exists nonnegative u e LP(#) such that D C I-u, u] "- {f e L ~ ( # ) ' I I I < u}. An operator is called order bounded if it sends order bounded sets onto order bounded sets. (3) An operator P is called AM-compact if it sends semicompact sets onto relatively compact sets, or equivalently sends order bounded sets onto relatively compact sets. The following result is due to [183] (see also [71] for the context of Banach lattice).

If P is order bounded and AM-compact on L~(#) then ~(P) = p(P). Consequently, an order bounded operator is compact if and only if it is semicompact and AM-compact. T h e o r e m 0.3.11

In applications, it is convenient to rewrite the quantity p in the following simple way. We leave the proof as exercise.

For any strictly positive r E LP(#) (which exists since # is a-finite), one has p(D) - limccsupy~DIIfl{.fl>rr p. Consequently, P r o p o s i t i o n 0.3.12

p(P) is equal to the following nP-tail norm of P: IIP[Ip,T = Jimccsuply,,p~rr liP"

0.3

Spectral theory

15

We recall the following two interpolation theorems which are crucial in the analysis of semigroups. T h e o r e m 0.3.13 (1) Riesz-Thorin's interpolation theorem Let 1 0, let Pm= #(P >~ m). We have

ha(r~2) 2 0 such that

Therefore,

8pmo

1

4 - Cr 2 1 and fn ~ f in L2(#) as n ~ c~. Moreover, by Proposition 0.1.7(3) one has

lim ~(fn, f) - 8(f, f) ~< lim 8(fn, fn). n - - - + (:X:)

n - - + (:x:)

Since 8(fn, fn) ~< E(f, f), it follows that lim 8 ( f - f n ,

f - - f n ) -- 0. Therefore,

n---~ (x)

~ ( 8 ) N ~ b ( E ) is dense in ~ ( 8 ) with respect to ~1/2.

[--1

The quantity L s ( f ) defined below is a generalization of the Lipschitz constant considered above. D e f i n i t i o n 1.2.1 Let ( E , ~ , # ) be a probability space and ( E , ~ ( E ) ) a conservative symmetric Dirichlet form on n2(#). Let ~ ( L s ) " - {f E ~ ( E ) " fn "- ( f A n) V ( - n ) e ~ ( E ) , n >~ 1}. For f e ~ ( L s ) , the following quantity is called the 8-Lipschitz constant of f: L E ( f ) "-- lim sup

{ 8(fng, f n ) - ~18 ( f n2, g ) ' g

C ~ ( 8 ) , #([g[) ~ 1

}

1/2

n---+ oo

1 Since taking g - 1 one has 8(f~, f ~ ) - ~ E ( f n 2, 1) - E(fn, fn) ) 0, it follows from Proposition 0.1.7 (3) that L E ( f ) 2 >~ lim 8(f~, f~) ~> ~ 3~(f' f)

if f C @r

Chapter 1

28

Poincar~ Inequality and Spectral Gap

In particular, L r >1 0 and the equality holds if and only if E(f, f) - 0. Indeed, if 8 ( f , f) = 0, then by the symmetry 8(f~, g)2 < 8 ( f n , fn)8(g, g) -- 0 for any n ~> 1 and any g E ~ ( E ) . The following result shows that LE is a generalization of the usual Lipschitz constant on manifolds. P r o p o s i t i o n 1.2.2 Let E be a complete Riemannian manifold with # a probability measure equivalent to the volume measure dx. Let 8 ( f , g) "= #((V f, Vg)),

f, g e H 2'1(#).

Then L s ( f ) -IIVflloo = Lip(f) for any nipschitz continuous function f . Proof.

For any n ~> 1, we have 1 In " -- sup {oZ(Yng, f n ) - - - ~ e ( f 2 n , g ) ' g e - ~ ( ' 8 ) ,

-sup{#(glWAI2)

9g

#(IgI)

#(Igl) < 1} 1/2

1} 1/2.

Since ~ ( E ) is dense in L2(#), it follows that In - I I V / n l I ~ , Hence Ls(f) =

lim In = llV fll~ = Lip(f).

[~

n--.-+oo

Let us consider the case of jump processes. P r o p o s i t i o n 1.2.3 Let (E, ,,~, #) be a probability space. Consider the qmeasure q(x, dy) with q(., A) e dY(E) for each A e fir and q(x, .) is a measure on ~ absolutely continuous with respect to # for each x E E. Assume that q(.,E) e LI(#) and that q is #-reversible (i.e. #(dx)q(x, dy) is a symmetric measure on E • E). Define

E(f,g) "- ~

xE

( f (x) - f (y))(g(x) - g(y))#(dx)q(x, dy),

~ ( 8 ) "- ( f e L 2 ( # ) ' E ( f , f )

< oc}.

Then (E, ~ ( E ) ) is a symmetric Dirichlet form and ~ ( L s ) - dY(E). Moreover, for f E ~ ( L s ) one has

1

L s ( f ) 2 - ~ess, sup

/

If(x) - f(y)12q(x, dy).

Proof. It is easy to see that d~b(E) C ~ ( 8 ) which is dense in L2(#) since for any f e L2(#) one has ( f A n ) V ( - n ) ~ f i n L 2 ( # ) a s n - - - oc. Next, it is easy to see that for any Lipschitz continuous function 9 on I~ one has

1.2

Concentration of measures

29

oz(~ o f, ~ o f) ~ Lip( ~)2oz(f, f) for any f E ~ ( E ) . Then to prove that (8, ~ ( 8 ) ) is a Dirichlet form it suffices to show the closeness. Let {F(n)}n>~l be a Cauchy sequence in ~ ( oz) with respect to OZl/2. Then there exists a unique F E L2(#) such that F (n) ~ F in L2(#) as n --. oc. Without loss of generality, we assume also F (n) ---, F #-a.s. Since q(x, dy) is absolutely continuous with respect to #, one has F (n) (x) - F (n) (y) --~ F ( x ) - F(y),

# ( d x ) q ( x , dy)-a.e.

Then by Fatou lemma and noting that {F(n)}n~O is a Cauchy sequence with respect to 81/2, we obtain 8(F, F) ~< lim oP(F (n) , F (n)) < oo, n---~oo lim ~ ( F (n) - F, F (n) - F ) < lira lira oZ(F (n) - F (m) , F (n) - F (m)) - O. n--,oo m---,cc n--,c~

Thus, (oz, ~ (oz)) is closed. Since for any f e ~ ( E ) one has fn "- (f A n)V ( - n ) e ~ ( 8 ) , we have ~ ( L E ) - ~ ( E ) . For any f e ~ ( E ) and any g e ~(d~ 1 E ( f n g , A ) -- ~8(f~n, g) -- "41/ExE ( g ( x ) - ~ - g ( y ) ) ( f n ( X ) -

f n ( y ) ) 2 # ( d x ) q ( x , dy).

By the symmetry of # ( d x ) q ( x , dy), oZ(fng, fn) - -~

-~

g(x)

/.

}

(fn(X) - fn(y))2q(x, dy) #(dx).

Hence, LE(f) 2 = lim ~essz sup n---,cc

(fn(X) - fn(y))2q(x, dy)

x

/> ~ s s . sup

(fn (x) -- fn(Y))2q(x, dy)

z

= ~ess. sup

(I(x) - f ( y ) ) 2 q ( x , dy).

Z

Therefore, the proof is finished by noting that (fn(X)- fn(y)) 2 0, let fe with #(re) - 0 and #(f~) = 1 such that A1 + e ~> 8(re, fe). Next, let ce be the median of re, i.e. #(re < ce) V #(re > ce) < 1/2. Set ge := f e - ce. We have 1 A1 + ~ >~ 8(g~,g~) - -~

>i -21

•

(Ig+ (Y) - g+ (x)l + Ig~-(Y) - g~- (z)l

xE 1 9 + ( Y ) - g + ( z ) r 2 J ( d z ' d Y ) + 2

J ( d z , dy)

xE Ig2(Y)-g-[(x)12J(dx'dy)

>1ao(f~ > ~).(g+2) + ao(f~ < c~).(g2 2) ) #(g~)

inf

# (B) E (0,1/2]

)~o(B) ) #(f~)

This implies (1.3.4) by letting s ~ 0.

inf

# (B) E (0,1/2]

)~o(B) =

inf

# (B) E(0,1/2]

)~o(B).

1.3

Poincard inequalities for jump processes

35

Now, for any B with #(B) e (0, 1/21, consider the form 8 B ( f , f ) " -- -~

•

(fiB(x) (f(x)

2

-- f l B ( y ) ) 2 J ( d x ,

- f(y))2j(dx,

dy)

dy) + KB(f2),

xB

where

KB(f2) "-- /B /

(x)J(B c • dx).

Letting ko(B) 9 -

KB(A) + J ( ( B N A) x (B \ (B N A))) #(A N B) J ( ( B N A) • (B n A) ~) inf t,(ANB)>0 #(A n B)

inf tt(ANB)>0

we have k0 (B) >~ k~. Noting that J(A • B ) + K B ( A ) / 2

1. Then kl > 0 but ~ = 0. If let K ( A ) : = 1A(0), then ~0 = 0, but k0 > 0.

Proof.

Let A C Z+ with 0 < #(A) < 1. If 0 E A, then there exists i E A such that i + 1 ~ A. We have J(A • A c) >1 J(i, i + 1) = bi#i >1 1/Z. If 0 ~ A, then there exists i E A such that i - 1 ~ A, i i> 1. We have J ( A • A ~) ~> J(i, i - 1) = ai#i >i 1/Z. Thus, kl ~ 4/Z. On the other hand, let

fn(i) := i 5-11{i0.

n

E (fn(i + 1 ) - fn(i))2#ibi

8(A,A) n

#(fn2)- #(fn)2

E fn(i)2#i-

E fn(i)#i

i=1

i--1

Since f~(i) - f ~ ( i - 1) ~< ci ~-2 and #~b~ - 1/Z for all i t> 1, and #i = O(i -~) for big i, we have

E(fn, fn)

O(n 25-3)

~ ( f n 2) -- ~ ( f n ) 2

O(n 5-1) -- O(1og n) 2

for b i g n . Since 5 E (1, 2) implies 2 5 - 3 < 5 - 1, it follows that ~1 ~< lim

E(A, fn)

=0.

n-L, co ~ ( f n 2) -- # ( f n ) 2

Finally, for K(A) . - XA(O) and the above test function fn, the value of 8(fn, fn) does not change and hence one has )~0 - 0. But for any A E with #(A) > 0, if 0 ~ A then #(A) E (0, 1) and hence (1.3.6) holds. If 0 E A then J ( A • A ~) + K(A) >/K(A) = 1. Therefore, ko > O.

[::]

1.3

Poincar~ inequalities for jump processes

37

The above example suggests that k0 and kl are too big to provide nontrivial lower bounds of A0 and )~1 respectively. This leads us to define "smaller" quantities in place of k0 and kl. To this end, we define the new quantities by using "smaller" measures in place of J and K. Let us fix a nonnegative symmetric function 7 E ~ • ~ , and a nonnegative function S E ~ such that

fA xE 1{~>0) (x, y)JY)(dx, dy) + /A I { s > ~ V(x,

~< #(A),

A E ~,~. (1.3.7)

In particular, for the q-pair (q(x), q(x, dy)), one may simply choose V(x, y) "-- s-l(q(x,E)Vq(y,E)),

[0,1],

S(x) "- ( 1 - ~ ) - l ( q ( x ) - q ( x , E ) ) ,

1 0 where ~ "- c~ and ~ " - 0 . For a e (0, 1], let

j(a) (dx, dy)

"-

1{~>0) (x, y)?(x, y)-~J (dx, dy),

g (~) (dx) "- l{s>0)(x)S(x)-~g(dx), and define d~(~), k~a), k~~) and k~~)' as e, k0, ]gl, ~ by using J(~) and K (a) in place of J and K respectively. T h e o r e m 1.3.3

Assume (1.3.7). We have A~1) ~< k~1) ~< 1 and

k~1/2)2 A0 ~ 2-- Arl)~ ~

k~1/2) 2

1+ r

k~1)2'

where /~1) i8 defined as A0 with J and K replaced by j(1) and K (1).

Proof. By Theorem 1.3.1 and (1.3.7) we have )~1) ~ ~1) ~ 1. Let EA "E U { A } and extend a function f E ~ to EA by taking f(A) - 0. Define / J(a)(C) j(a) (C) . K(a) (A) 0

if C E o~ x o~", if C = A x {A} or {A} x A , A c ~,~, otherwise.

Then .I(~) is a symmetric measure on (EA x EA, ~/~ x ~A). Moreover,

12/E If(x)- f(y)lJ(~)(dx, dy)+/EIf(x)lK( xE

_ 1/E

AxEA

If (x) - f (y)I.I (~) (dx, dy).

)(dx) (1.3.8)

38

Chapter 1 Poincar6Inequality and Spectral Gap

Therefore, for f with #(f2) = 1, (1.3.3) implies

k(1)2

if(x)2 - f(y)2[J(1)(dx,

1

~< ~i o.g~(1)(f,f)/E ~ •

dy)}2

(f(x) + f(y))2j(1)(dx, dy)

=18(1)(f ' f) ~EAxE, a (2f(x) 2 + 2f(y) 2 - ( f ( x ) - f(y))2)j(1)(dx, dy) o'~:~

(f, f)(2 - og~ (f, f)).

This implies d~

f) ~> 1 - ~ / 1 - k~1)2 and hence

k~1)2.

A~1) ) 1 - 1 1 -

(1.3.9)

Next, by Schwartz inequality we have

k~1/2)2

If(x) 2 - f(y)2[j(1/2)(dx, dy)} 2

{1/E

, , f)

/z(B)E (0,1/2]

inf #(B)E(0,1/2]

R e m a r k 1.3.1

1 + V/1-

k~l/2)(B) 2

~1)(B)2

k~1/2)'2 1+ {1-

ki 1i'2"

D

If (1.3.1) holds, then we may take 7 - R and S -- 2R so

that (1.3.7) holds. In particular, for K - 0 it follows that k~1 / 2 ) ' - k~l/V~ and k~l)' = k~/R. Then Theorem 1.3.4 implies 2

R(1 + 1 1 -

kI12/R 2)

which improves the corresponding lower bound of )~1 given in Theorem 1.3.2. In applications, it is usually easy to estimate A0 and A1 in "local domains" (e.g. compact domains when E is a locally compact topological space). Thus, it is useful to comparing the spectral gap A1 with the corresponding local quantities. For any A E ~ , let RA :-- ess~ sup

J(dx • A ~) (dx)

If J(- • A ~) is not absolutely continuous with respect to #, we set RA -- oo. Next, for B e ~ with #(B) > 0, define # B ( A ) : = # ( A ~ B ) / # ( B ) and

/~1 (B)"-- inf { EB(f , f) "-- -~ 1/B •

If(x) -- f(y)[2J(dx, dy)" # B ( f )

-- 0,#B(f 2) -- 1}. T h e o r e m 1.3.5 Let K = 0 and # a probability measure. For any A E ~,~ with #(A) e (0, 1) we have A0(Ar (1.3.10) #(A) " If RA < oo then for any B D A with #(B) < 1, we have

~1~

)~1(B){Ao(AC)#(B) - 2RAp(B~)} 2)~1 (B)H- #(B)2(A0(A ~) + 2RA) "

(1.3.11)

40

Chapter 1

Pro@

Poincar6 Inequality and Spectral Gap

Let f E o~ with #(f2) = 1 and flA = 0, we have #(f2) _ #(f)2 _ 1 - #(flA~) 2 /> 1 -- #(f2)#(A~) = #(A).

Then

)~1 ~

8 ( f , f)

8 ( f , f)

~(g) - ~(f) 2

~(A)

This implies (1.3.10) by taking infimum over f. Next, for any f with #(f) = 0 and # ( f 2 ) = 1, let r "- # ( f 2 1 s ) . We have

8 ( f , f) ~ 8 B ( f l B , f i B ) >/)~I(B)#(B)-I[#(f21B)

- # ( B ) - I # ( f l B ) 2]

= )~1 ( B ) # ( B ) - 1 [ # ( f 2 1 B ) - # ( B ) - I # ( f I B ~ ) 2]

(1.3.12)

A I ( B ) # ( B ) - I [ r - #(B)-I#(f21B~)#(BC)] -- A I ( B ) # ( B ) - 2 [ r - #(BC)]. On the other hand, since

IflA~(X) -- flA~(Y)I ~ I f ( x ) - f(Y)l + 1A•

-- flA(Y)I,

we have

E

(I1A~(X) -- flA~(y))2J(dx, dy) •

~ S~.,(si,,)-s (y)),.(d,,.d,,l+~ i,.,.(SI.,(,,)-SI, (,,))'.(d,,.d,,) = 4 g ( f , f) +

4 fAxA c f ( x ) 2 j ( d x , dy) ~< 4#(f, f ) + 4RA#(f21A).

Therefore,

1 f2 8 ( f , f) ) ~Ao(A~)#(1A~) -- R A # ( f 2 1 A ) >f ~Ao(A~)(1 - r) - RAr. Combining this with (1.3.12) we arrived at 8 ( f , f)

inf max

rE[0,1]

AI(B) 1 (r - #(B~)) Ao(A~)(1 #(B) 2 '2

r) - rRA

AI(B) #(B) 2 (ro - #(SC)), where ro satisfies AI(B) #(B) 2 (ro - #(Be)) =

Ao(Ar

- ro) - roRA,

}

1.3

Poinca% inequalities for jump processes

that is

41

)~o(AC)#(B) 2 + 2)~I(B)#(B c) e (0, 1). r0 = 2~1(B) + #(B)2)~o(A c) + 2 R A # ( B ) 2

Then the proof is completed.

[:]

Now, we consider countable Markov chains. Let E be countable and (qij) a regular (i.e. totally stable, conservative and generating a unique q-process) and irreducible Q-matrix, reversible with respect to a probability measure # := (#~)~>0, see e.g. [38] for details. Then the corresponding measure J is given by J(i, j) := #iqij for i ~ j and J(i, i) := 0. We simply consider the case where E - Z+ or ~d and take ~(i, j) "- qi V qj. C o r o l l a r y 1.3.6 Consider the birth-death processes on Z+ with birth rates bi and death rates ai. Let ~/(i, j) - ( a i + bi) V (aj + bj) (i 7~ j). Then k~~) > 0 if and only if there exists c > 0 such that #iai

~/(i i -

1) ~ ~> c E

'

~j'

i~> 1.

(1.3.13)

j)i

c2 1 In this case we have k~~)t >t c and hence )~1 ~ -~ for c~ - -~.

Proof.

Take A - Ii "- {i, i + 1,... } for a fixed i ~> 1. We have /.ti qij

J(~) (i' J) - (qi V qj)" #iai

[(ai + bi) V (ai-i --

-~-bi-1)] a

#ibi

[(ai + bi) V (ai+l -t-bi+l)] a

='#i5i

if

j-i-l,

='#ibi

if

j-i+1.

#idi

~ c.

K]

42

Chapter 1

Poincar4 Inequality and Spectral Gap

C o r o l l a r y 1.3.7 Consider a reversible, irreducible, conservative Q-matrix (qij) on E - Z d. Assume that XI(En) > 0 for big n >~ 1, where En "- {i " lil n

x/qi V qj

j6E

for some c > 0 and some n >~ 1, then )~1 > O. Proof.

For any A c E c, by the symmetry of #iqij in i , j , we have

c#(A) -~. Therefore, (1.3.11) yields that AI>0. C o r o l l a r y 1.3.8 In the situation of Corollary 1.3.7, assume further that the Dirichlet form is regular. Let L r ~ #iqij(r - r If there exists a J

function r > 0 such that lim Lr

I~,-~ r

< 0, then )~1 > O.

Proof. There exists n >~ 1 and c > 0 such that Lr ~ - c r on E c. Let m > n and Enm := Em \ En, Trim := inf{t ~> 0 : zt r Enm}, where {xt}t>~o is the associated Markov chain. We have for Cure := r Lento(i) -

nr

~

r


O.

Therefore, )~o(E~m) >/ c. Since the form is regular, we obtain A0(E~) = lim )~o(Enm) >~ c and hence A1 > 0 by (1.3.11) and AI(En) > 0 for all n. [3]

m---~c~

1.3

Poincar6 inequalities for jump processes

43

A c r i t e r i o n for b i r t h - d e a t h p r o c e s s e s

1.3.3

In this subsection, we present a criterion of the spectral gap for birth-death processes based on a discrete Hardy inequality. Let En := { 0 , ' " ,n} for n >~ 0. Define k

6n "- sup ~ ( # j b j ) -1 E k>/n j=n

#J'

n >~ O.

j>/k+l

Theorem 1.3.9 (~n 1 ~ )~0(Enc) ~ (45n) - 1 . To prove Theorem 1.3.9, let us introduce the following discrete Hardy inequality. 1.3.10 Let {ai}i~>0 and {~i}i~>0 be two positive sequences with E O~j < (:X:). Define J

Lemma

i

~ i f (i) 2 - 1}, i>~0

j=0

i>~0

i

i>/O

.

j >/i

We have B 0

2 OLi .

j=0

Then A >~ sup io ~o

ai i--io

~

~;1 =B.

j-o

On the other hand, let f with ~ f(i)2/~i < oc and let ni " - ~ i>/0

j--0

~;1 ,

i>~1.

Since for any a, b > 0 one has b-a

v~' we obtain (no "= 0)

1

< 2 ( v ~ - ~/~.1),

i~>1,

(1.3.15)

44

Chapter 1

(5

Poinca% Inequality and Spectral Gap

~ OLk)l/2).

OLi ( 1/2 ( g~k) 1/2 ~ 2 (~-~"OLk)k~i

(1.3.16)

k)i+l

By Schwartz inequality (1.3.15), the definition of B and (1.3.16), we obtain

~>o

j=o

~>o

k=0

i < 2 ~ a~ v/h-7~ f(j)2/~j v/-n-~ ~>o j=o

i~O (k~>~i~k)l/2 j=O

,v ~I

c,

~o k >~j ~o This implies that A ~< 4B.

V1

Proof of Theorem 1.3.9. Let ai "-- ~i+n+l, ~i "- #i+nbi+n,i ~ 0, we have B - 5n. Next, for any f with fiE** = 0, let g ( i ) " - f ( i + l + n ) - f ( i + n ) , i >~O.

i

Then ~ g(j) = f(i + n + 1), i.e. f with f[E~ -- 0 and g are determined by

j=0

each other. Hence 1.= A

inf

E)3ig(i)2 " ~ a i ~>o ~>o

= inf { E

~g(y) j=o

= 1

#ibi(f(i + 1) - f(i)) 2" flE~ = O, ~

i >~n = inf{8(f, f)"

i >~n+l flEn

--

0, #(f2) _ 1} - Ao(EnC).

Then the desired result follows from Lemma 1.3.10.

#if(i) 2

1}

1.4

Poincar@ inequality for diffusion processes

Theorem

1.3.11

45

if and only if 50 >

We have )~1 > 0

(equivalently, 5n

0

> 0

for all n >~0). More precisely,

1

~0(E~)

~0 50 ~

/-to

1 ~ )~1 ~ )~0(E~) ~ 450

Proof. By (1.3.10) and Theorem 1.3.9, it suffices to note that )~1 since for any non-constant f E ~ ( E ) , 8(f,f) p ( f 2 ) _ p(f)2 ~

~(f' f) >/~0(E~). #((f - f (0))2)

Poincar@ inequality

1.4

~ )~0(E~)

for d i f f u s i o n

processes

We first consider the one-dimensional case, then move to the general diffusion processes on ]~d and Riemannian manifolds. 1.4.1

The one-dimensional

case

d2 d Consider the differential operator L "- a(x)-~-~2 + b(x)-~--~, where a ( > 0) and ~.at]

b are continuous functions on R. Let

b(~) dr e I~, C(x) .- fo x a(r)

xER.

It is easy to see that L is symmetric with respect to the measure #(dx) :=

eC(x) ~~a'x-------Tdx"For any interval (a, ~), where a e R,/3 e (a, c~], consider m0(a, 13) "- inf

af'2d#

9 f e C 1 [o~,/3),

f(~) - 0, jfZ

f 2 d # = 1}.

This quantity is called the first mixed eigenvalue of L on (a, 13) with condition at a and Neumann condition at /3 if/3 < c~. If/3 < c~ closure of ( L , ~ ) with ~ " - {f e C2[a,13] 9 f(a) = 0, f ' ( ~ ) essential self-adjoint operator on L2((a, /3); #) and m0(a,/3) is the quantity such that

Dirichlet then the 0} is an smallest

L f = -mo(a, ~) f for some non-constant f e C2[a,~] with f ( a ) - 0, f ' ( ~ ) = 0. Indeed, since on finite intervals there holds a Sobolev inequality (which, of course, implies the compactness of the corresponding diffusion semigroup), the spectrum of L is discrete, see Chapter 3 for details. To study m0(c~,/3), we make use of the following weighted Hardy inequality due to [147].

46

Chapter 1

Poincar@Inequality and Spectral Gap

Proposition 1.4.1

Let # and u be two Borel measures on [a, oc) where a is a constant, and let oc > p >1 1. Let A be the smallest positive constant C > 0 such that { ~

fa x f (t)dt] p#(dx) } l/p (45(a, 3)) -1 .

Proof.

Let p - 2. Then the desired result follows from Proposition 1.4.1 for 1[~,~)]# in place of # and ~ " - l[~,~)a#. W] Next, let us consider the spectral gap of L on (a, ~), where a e [-c~, c~),/3 e (a, c~]. Assume that #((c~,/~)) < c~, consider

)~I(C~, f3) := inf

Z af'2d# 9 f E C I ( a , Z ) ,

f d # = 0,

f2d# = 1 .

We call Al(a, fl) the spectral gap of L on (c~,/3) (with Neumann boundary conditions). If c~ and ~ are finite then L defined on { f E C~(c~, t3)" f'(c~) = f'(~) = 0} is essentially self-adjoint on L2((c~,/3); #) and its closure has the spectral gap m0(c~, fl), which is the smallest A > 0 such that

L f = -)~f for some non-constant f e C~(a,/~) with f'(a) - f'(fl) - O . Similarly, we define the first mixed eigenvalue of L on an interval with Dirichlet condition at the right end point and Neumann at the left:

mto(a, fl) "- inf

Z a f'2d# " f e cl(a, fl), f(Z)

O,

f2d# = 1 .

1.4

47

Poinca% inequality for diffusion processes

Assume that tt((a,/~)) < c~ a n d / ~ e-C(s)#((a, s))ds < oo.

L e m m a 1.4.3

We have inr (m~( , r) V m0(~', t3)) ~ )~1 (o~, ~) ~(~,~)

sup

A m0( , Z)),

~e(~,~)

and the equalities hold if )~l(a, 13) is an eigenvalue which is in particular the case when c~ and ~ are finite. Proof.

For any r E (a, 13) and any n >~ 1, let gn and hn be smooth functions

such that gn > 0 in (a, r), hn > 0 in (r, ~), g ~ ( r ) - h n ( r ) - O, and / r z h2nd# = 1 f '

g~d#-

ag n,2 d# ~ m~o(a, r ) + -n1, frz ah n,2 d# 0 if and only if ~(0, ec) < ec while ,'~1(--00, 00) > 0 if and only if 5(0, c~) V 5'(-oo, O) < c~.

48

Chapter 1

Poincar~ Inequality and Spectral Gap

Proof.

Similarly to Theorem 1.4.2, one has 5'(a, r) -1 /~ m~o(a, r) ~ (45'(a, r)) -1. Then the proof is completed by Theorem 1.4.2 and Lemma 1.4.3. U] Theorems 1.4.2 and 1.4.4 provide sharp criteria for mo(a,/3) and Al(a, 13) t o be positive. But the upper bounds are at least four times of the corresponding lower bounds. To obtain sharp estimates, we introduce the following variational formulae of m0(a,/3) and )~1(a, ~). T h e o r e m 1.4.5

In the situation of Lemma 1.4.3 and let a E I~. We have mo(a, ~) >~

sup inf - L f /~$1 (a,Z) (a,Z) f '

(1.4.1)

sup inf - L f , sup inf - L f } feS2(a,r) (~,~) f feSl(~,Z) (~,~) f ' and the equalities hold when/3 < c~, where

)~1(O~,~) ~

sup min {

re(aft)

~l(a,~)'-{f6C2[a,~)'f(a)=0,

f'>0

in

~2(a, r) "= { f E C2[a, r] . f (r) - O, f' < O in

(1.4.2)

[a,~)},

(a, r] }.

Proof.

By Lemma 1.4.3, we only need to prove (1.4.1). Since both sides of (1.4.1) are decreasing in ~ and m0(a, oe) - lim m0(a,n), we only consider the case where ~ < o~. In this case, the spectrum of L, with Dirichlet condition at a and Neumann condition at ~, is discrete. Let fl be the eigenfunction with respect to m0(a,/3), then one may assume that f~ > 0 in [a, ~). Hence m0(a,/3) -

Lfl fl

~< the right-hand side of (1.4.1).

Next, for any f E ~l(a,/3) with L f ~o be the L-diffusion process reflecting at/3, and let Ta be its hitting time at a. We have (up to Ta)

df(xt) - dMt + L f ( x t ) d t - f'(13)lz(xt)dLt for some martingale Mt and increasing process Lt. Thus, L f 0 be such that fl ~ R f (R exists since f~ > 0 in [a, ~)), we have

e-mo(a,Z)t f l (x ) _ EX f l (xtAr, ) ~ REX f (xtAr~ ) < Re-Ct f (x) for all t > 0 and all x E (a, ~). This implies m0(a, ~) ~> c.

U]

1.4

Poincar~ inequality for diffusion processes

49

To apply Theorem 1.4.5, one has to choose a proper test function f such that L f 0. To meet this requirement, let us take

I(f)(x) "- ~ x e-C(r)dr ~~ eC(u)f (u) du

for f ~ ~0(a,/3) : - { f > 0"

~

Z f d p < c~}. Then I ( f ) E ~ l (a, ~) and

-LI(f)

I(f)

S

= I(f) > O.

Thus, Theorem 1.4.5 implies the following consequence immediately. C o r o l l a r y 1.4.6

In the situation of Lemma 1.4.3, we have too(a,/3) >~

sup inf f /eo%(~,Z) (~,Z) I(f)"

In particular, if13 < oc then each strictly positive function f c C[a,/3] provides a non-trivial lower bound of too(a, ~). Finally, we present the following two simple examples to illustrate the above results. d2 d E x a m p l e 1.4.1 Consider L "- dx 2 (x + 1)a~xx on [0, c~), where a E ( - 1 , c~). Then )~1(0, c~) > 0 if and only if a >~ 0, and when a >~ 0 we have 1 )~1(0, c~) ~> ~ which is sharp for a - 0. The equivalence of/~1(0, (X)) > 0 and a ~> 0 follows from Theorem 1.4.4. For a >~ 0, let f(x) - ex/2 - 1. Then f C ~1(0, c~) and

Proof.

1 x/2

Lf 0 if and only if

(a-

a ~> 2. Moreover, letting f(x) - (1-~-X)1/2 we have L f - - ~1 (l -+-X)1/2+(a-2) 1 - i f for a / > 2, then 1

)~1(0, C~) ~ m0(0, (:x:))~ ~, which is sharp for a - 2. Indeed, when a - 2 we have for f ~ ( x ) " - (1 + x) ~,

_ 1 ~ 8 9 f o f~(x)2#(dx) - ( f o f~(x)#(dx)) 2 - 4"

/~1(0, oo) ~ lim

1.4.2

f~

+ x)2f~2(x)#(dx)

S p e c t r a l g a p for d i f f u s i o n p r o c e s s e s o n ][~d

Consider the operator

d d L "- E aijOiOj + E biOi, i,j---1 i=1

(1.4.3)

where a(x) := (aij (x)) is strictly positive definite for each x E ]Rd, aij E C2(IR d) f o r l ~ < i , j ~ < d , and d

bi "= y ~ (aij Oj V + Oj aij ) ,

1 < i 0) for each x ~- y. Consider d

L(x, y) " - ~

i,j--1 d § i--1

02

02

02

(aij(x) OxiOxj + a~j(y) OyiOyj + 2Cij(x, y) OxiOyj )

(

o

+

(1.4.4)

o)

Then L generates a unique diffusion process (xt, yt) on ]i~d • I~ d up to the time (see e.g. [Iii]) T := inf{t >~ O:xt = Yt}. Letting xt = yt for t >~ T, due to the strong Markov property of the L-diffusion process, we conclude that the marginal processes xt and Yt are generated by L. We call (xt, Yt) a coupling of the L-diffusion process with coupling time T, see [48] for more details. Next, let p be a positive function on R d • R d which is C2-smooth outside the diagonal {(x, x ) ' x E I~d} and p(x, x) = 0 for x c I~ d. We assume that

Ix- yl

sup < ce, 0 min ( v / a i ( x ) v-/ a i ( y ) )

2

+ 4(ja (x)a(y) Z,

2 +4min~/ i

~)

~( ) ~( _)a~ax,~v,

.~x, ~-~ _1 ~, ~d { (~a,(x~- ~o,(~)~ (,- (x,- ~,~ ) X

i=1

IX -- y12 ~

+ (b~(x) - b~(y))(x~ - y~)}. Then, for f E o~1 (0, co) with f " ~< 0 and c~f" +/~f' 0, one has N

L f o p < c~(p)f" (p) +/~(p)f'(p) 0. Thus, on {r < p < R},

L ( f o p) = f'(p)Lp + f"(p)F(p, p) ~ 4~(r) m-.oc ffl Dm m~cc j ~ N ODm

(g2)

f

lim [

g(Nm, Vg)dA

m--~oc J$2 ~ ODm

,(g2)

lim m---,oc

([Vg[2 + g L g ) d # -

-~),

~ Dm

where dA denotes the (d-1)-dimensional measure induced by #. Therefore, by (1.4.17) one has Aes~ f> suPr>0 (4~(r)) -1 and hence results in Theorem 1.4.11 follow. K] C o r o l l a r y 1.4.13 (1) Consider the situation of Theorem 1.4.11 and assume that i n f a l > 0. Then 9 (a)

lim ~(r)

< 0 implies ~(r) < oc for all r > ro, hence also implies

)~1 > 0 provided 1 c ~ ( L ) .

60

Chapter 1

(b)

lim

Poincar~ Inequality and Spectral Gap

_ - c ~ implies ~(r) --. 0 as r ~ oc, hence also implies

~(r)

Cress(L) - O. (2) Consider the situation of Theorem 1.4.12. Then: (a) lim ~(r) < 0 implies ~(r) < oc for all r > ro, hence also implies r----~oo

AI>O.

(b)

lim ~(r) -

-oc

implies ~(r) --+ 0 as r -~ oc, hence also implies

r--+oo

Cress(L) - 0 . It is the case if the Ricci curvature is bounded below and (VV, Vp) --, -cx~ outside cut(o) as p ~ cx~.

Proof. We only prove the first assertion since the second can b e reduced to the case where a l = a2 - 1 according to T h e o r e m 1.4.12, and the last sentence follows by noting t h a t A p is b o u n d e d above according to the Laplacian comparison theorem. Assume t h a t c " - inf a l > 0 and there exist rl > r0 and 5(rl) > 0 such t h a t

~l(r)

=

~(r)

c~ exp

~(r)

< - 5 ( r l ) for r ~> rl, we have ~+(r) - 0

~ - 5 ( r l ) for r ~> r l . T h e n for any s > r l ,

t el(U) du dt ~< 1 1 0~1(U) (~(rl)

--el(t) exp

I

exp

-

1 C~I(U)

du

dt~
c + 1. Define ~ ( r ) " -

/0

h(s)ds, r E IR.

Since f E ~l(Lb), there is a sequence {gn} C C~C(M) such that gn --~ f and Lbgn ~ Lbf in Li(#) as n ~ ec. Letting fn "- ~(gn), we have I]fnll~ g~ since pb is sub-Markovian. By (1.4.24), #(hPs) < e-[4(p-1)/(Pr)-c~]t + C ( t , r , p , a ) # ( ( f

- s)+) p

~l

n

with #M• the Riemannian distance on M x M. Next, let gn E C ~ ( M x M) such that 0 ~< gn < 1, gn(X,y) -- 0 if p(x,y)