Ethier s.n., kurtz t.g. markov processes characterization and convergence

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Ethier s.n., kurtz t.g. markov processes characterization and convergence

  1. 1. Markov Processes
  2. 2. Markov ProcessesCharacterizationand ConvergenceSTEWART N. ETHIERTHOMAS G. KURTZWILEY-INTERSCIENCEA JOHN WILEY & SONS, INC., PUBLICATION
  3. 3. Copyright 8 1986,2005by John Wiley ti Sons, Inc. All rights reserved.Publishedby John Wiley & Sons, Inc., Hoboken. New Jersey.Publishedsimultaneouslyin Canada.No part of this publication may be rcproduccd, stored in a retrieval system or transmittcdin any form or by any means, electronic, mechanical, photocopying, recording, scanningor otherwise, except as pcrmittcd under Sections 107 or 108 of the 1976 United StatesCopyright Act, without either thc prior written permission of the Publisher, orauthorization through paymen1of the appropriate per-copy fee to the CopyrightClearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax(978) 750-4470. Requests to the Publisher for permission should he addressed to thePermissions Deparlment, John Wiley B Sons, Inc., 111 River Strcet, Hoboken, NJ 07030,(201) 748-601I , fax (201) 748-6008.Limit of Liability/Disclaimero f Warranty: While the publisherand author have usedtheir bcstefforts in preparingthis book, they make no representationsor warranties with respect to theaccuracy or completenessof the contentsof this book and specifically disclaim any impliedwarranties of merchantability or fitness for a particular purpose. No warranty may be createdorextended by sales representativesor written salesmaterials. The adviceand strategies containedherein may not be suitablc for your situation. You should consult with a professional whereappropriate.Neither the publishernor author shall be liable for any loss of profit or any othercommercial damages, including but not limited to special, incidental, consequential. or otherdamages.For general information on our other products and servicesor for technical support, please contactour Customer Care Department within the US. at (800) 762-2974,outside the U.S. at (317)572-3993 or fax (317)572-4002.Wiley also publishes its books in a variety of electronic formats. Somecontent that appearsin printmay not be availablein electronic format. For informationabout Wiley products, visit our web site atwww.wiley.com.Libray of CongressCataloginpin-Publicationis awilable.ISBN- I3 978-0-471-76986-6ISBN-I0 0-471-76986-XPrintedin the United Statesof America1 0 9 8 7 6 5 4 3 2 1
  4. 4. The original aim of this book was a discussion of weak approximation resultsfor Markov processes. The scope has widened with the recognition that eachtechnique for verifying weak convergenceis closely tied to a method of charac-terizing the limiting process. The result is a book with perhaps more pagesdevoted to characterization than to convergence.The lntroduction illustrates the three main techniques for proving con-vergence theorems applied to a single problem. The first technique is based onoperator semigroup convergence theorems. Convergence of generators (in anappropriate sense) implies convergence of the corresponding sernigroups,which in turn implies convergence of the Markov processes. Trotter’s originalwork in this area was motivated in part by diffusion approximations. Thesecond technique, which is more probabilistic in nature, is based on the mar-tingale characterization of Markov processes as developed by Stroock andVaradhan. Here again one must verify convergence of generators, but weakcompactness arguments and the martingale characterization of the limit areused to complete the proof. The third technique depends on the representationof the processes as solutions of stochastic equations, and is more in the spiritof classical analysis. If the equations “converge,” then (one hopes) the solu-tions converge.Although the book is intended primarily as a reference, problems areincluded in the hope that it will also be useful as a text in a graduate course onstochastic processes. Such a course might include basic material on stochasticprocesses and martingales (Chapter 2, Sections 1-6). an introduction to weakconvergence (Chapter 3, Sections 1-9, omitting some of the more technicalresults and proofs), a development of Markov processes and martingale prob-lems (Chapter 4, Sections 1-4 and 8). and the martingale central limit theorem(Chapter 7, Section I). A selection of applications to particular processes couldcomplete the course.V
  5. 5. Vi PREFACEAs an aid to the instructor of such a course, we include a flowchart for allproofs in the book. Thus, if ones goal is to cover a particular section, the chartindicates which of the earlier results can be skipped with impunity. (It alsoreveals that the courseoutline suggestedabove is not entirelyself-contained.)Results contained in standard probability texts such as Billingsley (1979) orBreiman (1968) are assumed and used without reference, as are results frommeasure theory and elementary functional analysis. Our standard referencehere is Rudin (1974). Beyond this, our intent has been to make the bookself-contained (an exception being Chapter 8). At points where this has notseemed feasible, we have included complete references, frequently discussingthe needed material in appendixes.Many people contributed toward the completion of this project. CristinaCostantini, Eimear Goggin, S.J. Sheu, and Richard Stockbridge read largeportions of the manuscript and helped to eliminate a number of errors.Carolyn Birr, Dee Frana, Diane Reppert, and Marci Kurtz typed the manu-script. The National Science Foundation and the University of Wisconsin,through a Romnes Fellowship, provided support for much of the research inthe book.We are particularly grateful to our editor, Beatrice Shube, for her patienceand constant encouragement. Finally, we must acknowledge our teachers,colleagues,and friends at Wisconsin and Michigan State, who have providedthe stimulatingenvironment in which ideas germinateand flourish. They con-tributed to this work in many uncredited ways. We hope they approve of theresult.STEWARTN. ETHIERTHOMASG. KURTZSalt Lake City, UtahMadison, WisconsinAugust 198s
  6. 6. Introduction1 Operator SemigroupsDefinitions and Basic Properties, 6The Hille-Yosida Theorem, 10Cores, 16Multivalued Operators, 20Semigroups on Function Spaces, 22Approximation Theorems, 28Perturbation Theorems, 37Problems, 42Notes, 472 Stochastic Processesand Martingales12345678910Stochastic Processes, 49Martingales, 55Local Martingales, 64The Projection Theorem, 71The Doob-Meyer Decomposition, 74Square Integrable Martingales, 78Semigroups of Conditioned Shifts, 80Martingales Indexed by Directed Sets,Problems, 89Notes, 938449vii
  7. 7. viii CONTENTS3 Convergence of Probability Measures1 The Prohorov Metric, 962 Prohorov’sTheorem, 1033 Weak Convergence, 1074 Separatingand ConvergenceDeterminingSets, 1115 The Space D,[O, GO), 1166 The Compact Setsof DEIO,a), 1227 Convergencein Distribution in &[O, m), 1278 Criteria for RelativeCompactnessin DKIO,a), 1329 Further Criteria for Relative Compactnessin D,[O, oo), 14110 Convergenceto a Processin C,[O, a), 14711 Problems, 15012 Notes, 1544 Generators and Markov Processes1 Markov Processes and Transition Functions, 1562 Markov Jump Processes and Feller Processes, 1623 The MartingaleProblem: Generalitiesand SamplePath Properties, 1734 The Martingale Problem: Uniqueness, the MarkovProperty,and Duality, 1825 The MartingaleProblem: Existence, 1966 The Martingale Problem: Localization, 2167 The MartingaleProblem:Generalizations, 22I8 ConvergenceTheorems, 2259 Stationary Distributions, 23810 Perturbation Results, 253I 1 Problems, 26112 Notes, 2735 Stochastic Integral Equations1 Brownian Motion, 2752 StochasticIntegrals, 2793 StochasticIntegral Equations, 2904 Problems, 3025 Notes, 3056 Random Time Changes1 One-Parameter Random Time Changes, 3062 Multiparameter Random Time Changes, 3113 convergence, 32195155275306
  8. 8. 4 Markov Processesin Zd,3295 Diffusion Processes, 3286 Problems, 3327 Notes, 3357 InvariancePrinciplesand DiffusionApproximations1 The Martingale Central Limit Theorem, 3382 Measures of Mixing, 3453 Central Limit Theorems for Stationary Sequences, 3504 Diffusion Approximations, 3545 Strong Approximation Theorems, 3566 Problems, 3607 Notes, 3648 Examplesof Generators1 NondegenerateDiffusions, 3662 Degenerate Diffusions, 3713 Other Processes, 3764 Problems, 3825 Notes, 3859 BranchingProcesses1 Galton-Watson Processes, 3862 Two-Type Markov Branching Processes, 3923 Branching Processes in Random Environments, 3964 Branching Markov Processes, 4005 Problems, 4076 Notes, 40910 Genetic ModelsI The Wright-Fisher Model, 4112 Applications of the Diffusion Approximation, 4153 Genotypic-FrequencyModels, 4264 Infinitely-Many-AlleleModels, 4355 Problems, 4486 Notes, 45111 Density DependentPopulationProcesses1 Examples, 4522 Law of Large Numbers and Central Limit Theorem, 455337365386410452
  9. 9. 3 Diffusion Approximations, 4594 Hitting Distributions, 4645 Problems, 4666 Notes, 46712 RandomEvolutions1 Introduction, 4682 Driving Process in a Compact StateSpace, 4723 Driving Process in a Noncompact State Space, 4794 Non-Markovian Driving Process, 4835 Problems, 4916 Notes, 491Appendixes1 Convergenceof Expectations, 4922 Uniform Integrability, 4933 Bounded PointwiseConvergence, 4954 MonotoneClass Theorems, 4965 Gronwall’sInequality, 4986 The Whitney Extension Theorem, 4997 Approximation by Polynomials, 5008 Bimeasuresand Transition Functions, 5029 Tulcea’sTheorem, 50410 MeasurableSelectionsand Measurabilityof Inverses, 50611 AnalyticSets, 506ReferencesIndexFlowchart168492508521529
  10. 10. The development of any stochastic model involves !he identification of proper-ties and parameters that, one hopes, uniquely characterize a stochastic process.Questions concerning continuous dependence on parameters and robustnessunder perturbation arise naturally out of any such characterization. In fact themodel may well be derived by some sort of limiting or approximation argu-ment. The interplay between characterization and approximation or con-vergence problems for Markov processes is the central theme of this book.Operator semigroups, martingale problems, and stochastic equations provideapproaches to the characterization of Markov processes, and to each of theseapproaches correspond methods for proving convergenceresulls.The processes of interest to us here always have values in a complete,separable metric space E, and almost always have sample paths in DE(O,m),the space of right continuous E-valued functions on [O, 00) having left limits.We give DEIO, 00) the Skorohod topology (Chapter 3), under which it alsobecomes a complete, separable metric space. The type of convergence weare usually concerned with is convergence in distribution; that is, for asequence of processes { X J we are interested in conditions under whichlimn.+mE[f(X.)J = &ff(X)] for everyfg C(D,[O, 00)). (For a metric space S,C(S)denotes the space of bounded continuous functions on S. Convergence indistribution is denoted by X,=. X . ) As an introduction to the methods pre-sented in this book we consider a simple but (we hope) illuminatingexample.For each n 2 1, defineU x ) = 1 + 3x x - - , y,(x) = 3x + +-t>(.- r>.( 1 ) ( :> 1Markov Processes Characterizationand ConvergenceEdited by STEWARTN. ETHIER and THOMASG.KURTZCopyright 01986,2005 by John Wiley & Sons,Inc
  11. 11. 2 INTRODUCTIONand let U, be a birth-and-death process in b, with transition probabilitiessatisfying(2) P{K(r +h) =j + I I ~ ( t )a j } = n~,,(:)h +~ ( h )and(3)as Ado+. In this process, known as the sChlo8l model, x(r)represents thenumber of molecules at time t of a substanceR in a volume n undergoing thechemical reactions(4)with the indicated rates. (See Chapter 11, Section 1.)(5) x,,(t)= n’/*(n- yn(n1/2r)- 1). r 2 0.The problem is to show that X,convergesin distribution to a Markov processX to be characterized below.The first method we consider is based on a semigroup characterization ofX. Let En= {n‘/*(n-‘y - I) :y E Z+},and note that1 33 1Ro R, R2 +2R S 3R,We rescale and renormalize letting(6) ~ w m=Erm.(t)) I x m = XJdefinesa semigroup {T,(I)}on B(E,) with generator of the form(7) G,/(x) =: n3’2L,(1 +n -‘/‘x){f(x +n -’I4)-/(x)}+n3/2pn(1 +n -l/*x){/(x - -3/41 - ~ ~ x ~ ~ .(SeeChapter I.) Letting A(x) = 1 +3x2,p(x) =3x +x3, and(8) G~’(x)= 4/”(x) -x ~ ’ ( x ) ,a Taylor expansionshows that(9) G,f ( x )=Gf(x)+t1”~{,4,,(I +n.-‘/*x) -A( 1 +n -‘l4x)}{f(x +n -’I*) -/(x)}+n3/3{p,(1 +n-‘l4x)-I(1 +~t-I/*x)}{J(X- n-3/4) -f(x)}+ A(1 +n-l/*x) I’(1 -u){f”(x +un-”*) -r(x)} du
  12. 12. for all/€ C2(R)withf‘ E Cc(R)and all x E Em.Consequently.for such/;lim sup I G,f(x)- Gf(x)1 = 0.n-m x c E .Now by Theorem 1.1 of Chapter 8,(1 1) A E ((AGf):f€C[-00, 001n C’(R), G/E C[-aO, 001)is the generator of a Feller semigroup {T(t)}on C[-00, 001. By Theorem 2.7of Chapter 4 and Theorem I. I of Chapter 8, there exists a diffusion process Xcorresponding to (T(t)),that is, a strong Markov process X with continuoussample paths such that(12) ECJ(X(t))I *.*I = - S)S(X(d)for allfe C[-00, a03 and t 2 s 2 0.(4c: = a(X(w):u 5 s).)To prove that X, 3X (assuming convergence of initial distributions), itsuffices by Corollary 8.7 of Chapter 4 to show that (10) holds for all/in zt coreD for the generator A, that is, for allf in a subspace D of 9 ( A ) such that A isthe closure of the restriction of A to D.We claim that(13) D -= (/+ g:/I Q E C’(R),/’ E: Cc(W),(x’g)‘ E Cc(W)}is a core, and that (10) holds for all/€ D. To see that D is a core, first checkthat(14) ~ ( A ) = ( J E C [ - C Q , ~ ]nC2(R):f”~~(W),x3f’~C[-oo,oo]}.Then let h E C;(R) satisfy xI- 5 h sf E 9 ( A ) ,choose g E: D with (x’g)’ E Cc(W)and x 3 ( f - g)’ E e(R) and define(15)Thenj,, +g E D for each m,f, +g -+f, and G(fm +Q)-+ C/.a martingale problem. Observe thatand put h,(x) = h(x/m).GivenSdX) =S(0) - do) + (j-gY(Y)hm( Y1d ~ .s:The second method is based on the characterization of X as the solution ofis an {.Ffn)-martingalefor each /E B(E,) with compact support. Conse-quently, if some subsequence {A’,,,) converges in distribution to X , then, by thecontinuous mapping theorem (Corollary 1.9 of Chapter 3) and Problem 7 ofChapter 7,
  13. 13. 4 ~ O W c r I o Nis an {Pf)-martingale for eachfe C,(R), or in other words, X is a solution ofthe martingale problem for {(AG f ) : f cC,(W)}. But by Theorem 2.3 ofChapter 8, this property characterizes the distribution on Dn[O, 00) of X .Therefore,Corollary 8.16 of Chapter 4 gives X,=X (assumingconvergenceofinitial distributions),provided we can show thatLet (p(x) I ex +e-x, and check that there exist constants C , , a O such.thatC,,a<G,cp I;C,,,rp on [-a, u] foreach n 2 I and ct > 0, and Ka+-00. Letting = inf ( f 2 0: IX,,(t) I2 a},we haveIinf C P ( Y ) ~SUP Ixn(t)lka{ostsre-G.4 TIrl L a(19)ELeXP -Cn,a(?n, 8 A 73)cp(Xn(Tn, a A VJ5 QdXn(O))lby Lemma 3.2 of Chapter 4 and the optional sampling theorem. An additional(mild)assumption on the initial distributionsthereforeguarantees(18).Actually we can avoid having to verify (18) by observing that the uniformconvergence of G,f to Gf forfe C:(R) and the uniqueness for the limitingmartingale problem imply (again by Corollary 8.16 of Chapter 4) that X, =. Xin Dad[O, 00) where WA denotes the one-point compactification of R. Con-vergence in &LO, 00) then follows from the fact that X, and X have samplepaths in DRIO,00).Both of the approachesconsidered so far have involvedcharacterizationsinterms of generators. We now consider methods based on stochasticequations.First, by Theorems 3.7 and 3.10 of Chapter 5, we can characterize X as theunique solutionof the stochasticintegralequationwhere W is a standard one-dimensional, Brownian motion. (In the presentexample, the term 2JW(t) corresponds to the stochastic integral term.) Aconvergencetheory can be developed using this characterization of X,but wedo not do so here. The interested reader is referred to Kushner(1974).The final approach we discuss is based on a characterizationof X involvingrandom time changes. We observe first that U,satisfies
  14. 14. where N, and N- are independent, standard (parameter I), Poisson processes.ConsequentIy,X,satisfiesX,(r) = X,(O) +n- 3/4R+(n3/2 A,( I+n- /*X,(s))ds(22)- n-"R.(nl" 6p,(l + n-/4X,(s)) ds)+ n3l4[(A, - p&I + n - 4X,(s)) ds,where R+(u)= N+(u)- u and R_(u)= N-(u)-u are independent, centered,standard, Poisson processes. Now i t is easy to see that(23) (n /*R+(n3/2* 1, n l4R -(n32.))=.(W+,W-1,where W+ and W- are independent, standard, one-dimensional Brownianmotions. Consequently, if some subsequence {A".) converges in distribution toX, one might expect thatX(t)= X(0) + W+(4t)+ W ( 4 t )- X ( S ) ~ds.(24) s.(In this simple example, (20) and (24) are equivalent, but they will not be so ingeneral.) Clearly, (24) characterizes X,and using the estimate (18) we concludeX,-X (assuming convergence of initial distributions) from Theorem 5.4 ofChapter 6.For a further discussion of the Schlogl model and related models seeSchlogl (1972) and Malek-Mansour et al. (1981). The martingale proof ofconvergence is from Costantini and Nappo (1982), and the time change proofis from Kurtz(1981c).Chapters 4-7 contain the main characterization and convergence results(with the emphasis in Chapters 5 and 7 on diffusion processes). Chapters 1-3contain preliminary material on operator semigroups, martingales, and weakconvergence, and Chapters 8- I 2 are concerned with applications.
  15. 15. 1Operator semigroups provide a primary tool in the study of Msrkov pro-cesses. In this chapter we develop the basic background for their study and theexistence and approximation results that are used later as the basis for exis-tence and approximation theorems for Markov processes. Section 1 gives thebasic definitions,and Section 2 the Hille-Yosida theorem, which characterizesthe operators that are generators of semigroups. Section 3 concerns theproblem of verifying the hypotheses of this theorem, and Sections4 and 5 aredevoted to generalizations of the concept of the generator. Sections 6 and 7present the approximationand perturbation resuJts.Throughout the chapter, L denotesa real Banach space with norm 11 * 11.OPERATOR SEMICROUPS1. DEFINITIONS AND BASIC PROPERRESA one-parameter family { T(t):t 2 0) of bounded linear operators on aBanach space L is called a semigroupif T(0)= I and T(s+t ) = T(s)T(c)for alls, t 2 0. A semigroup(T(t))on L is said to be strongly continuousif lim,,o T(r)/=/for everyfe L;it is said to be a contraction semigroupif 11T(t)II5 1 for allt 2 0.Given a bounded linear operator B on L,defineMarkov Processes Characterizationand ConvergenceEdited by STEWARTN. ETHIER and THOMASG.KURTZCopyright 01986,2005 by John Wiley & Sons,Inc
  16. 16. 1. DmNmoNz AND EASIC ?ROPERTIES 7A simple calculation gives e") = e""e for all s,t 2 0, and hence {e"} is asemigroup, which can easily be seen to be strongly continuous. Furthermorewe haveAn inequality of this type holds in general for strongly continuous serni-groups.1.1 Propositionthere exist constants M 2 1 and o 2 0 such that(1-3) II T(t)lI 5 Me", t 2 0.Let (T(t))be a strongly continuous semigroup on L. ThenProof. Note first that there exist constants M 2 I and ro > 0 such that11 T(t)11 5 M for 0 I t s t o . For if not, we could find a sequence (t,} of positivenumbers tending to zero such that 11 T(t,)((-+ 00, but then the uniformboundedness principle would imply that sup,((T(rJfI1 = 00 for some f E L,contradicting the assumption of strong continuity. Now let o = t i log M.Given t 2 0, write t = kt, +s, where k is a nonnegative integer and 0 s s <t,; then(1.4) 0I(T(t)I(= II f(~)T(t,,)~Ils MM r; MM/O = Me".1.2 Corollaryeach$€ L, t -+ T(t)/is a continuousfunction from [0, 00) into L.Let {T(r))be a stronglycontinuoussemigroupon L.Then, for1.3 Remark Let { T(r)}be a strongly continuous semigroup on L such that(1.3) holds, and put S(t) = e-"T(r) for each t 2 0. Then {S(t)) is a stronglycontinuoussemigroupon L such that(1.7) IIW II s M, t 2 0.
  17. 17. 8 OraATORS€MIGROWSIn particular, if M = 1, then {S(t)} is a strongly continuous contraction semi-group on L.Let {S(t)} be a strongly continuous semigroup on L such that (1.7) holds,and define the norm 111 111 on L byThen 11f11 5; IIIJIII 5; Mllfll for eachfE L, so the new norm is equivalent tothe original norm; also, with respect to 111 * 111, {S(t)) is a strongly continuouscontractionsemigroupon L.Most of the results in the subsequent sections of this chapter are stated interms of strongly continuous contraction semigroups. Using these reductions,however, many of them can be reformulated in terms of noncontraction semi-groups. 0A (possibly unbounded) linear operator A on L is a linear mapping whosedomain 9 ( A ) is a subspaceof L and whose range a ( A ) lies in L. The graph ofA is given byNote that L x L is itself a Banach space with componentwise addition andscalar multiplication and norm [l(J @)[I= llfll + IIg 11. A is said to be closed if9 ( A )is a closed subspaceof L x L.The (injinitesimal) generator of a semigroup {T(c))on L is the linear oper-ator A defined by(1.10)1A , = lim ;{T(t)f-J}.1-0The domain 9 ( A ) of A is the subspaceof allJE L for which this limit exists.Before indicating some of the propertiesof generators,we briefly discuss thecalculus of Banach space-valued functions.Let A be a closed interval in (- 00, a),and denote by CJA) the space ofcontinuous functions u: A+ L. Let Cl(A)be the space of continuouslydiffer-entiablefunctionsu: A +L.If A is the finite interval [a, b], u : A + L is said to be (Rietnann)integrableover A if limd,, u(sk)(fk - t,,-I) exists, where a = to S s, 5 Il I .. 5;t,- ,s s, s f n = b and S = max (rr - f k - l); the limit is denoted by jb,u(t)dt oru(t)dt. If A = [a, a),u: A + L is said to be integrable over A if u I , ~ , ~ ,isintegrable over [a, b] for each b 2 a and limg,, Jtu(t)dt exists; again, thelimit is denoted by {A ~ ( t )dt or {; u(r)dt.We leave the proof of the followinglemma to the reader (Problem3).
  18. 18. 1. MflMTlONS AND 8ASlC PROPERTIES 91.4 Lemma (a) If u E C,jA) and JAll u(t)I1 dt < 00, then u is integrable overA and(1.1 I )In particular, if A is the finite interval [a, 61, then every function in C,(A) isintegrable over A.Let B be a closed linear operator on L. Suppose that u E CJA),u(t) E 9 ( E )for all t E A, Bu E CJA), and both u and Bu are integrable overA. Then JA U(t) dt E 9 ( B )and(1.12)(c) If u E Ci,[a, b], then(b)B u(t) dt Bu(t) dr.I =I(1.13) I$u(t)dt = u(b)- u(a).1.5 Proposition Let (T(t)}be a strongly continuous semigroup on L withgenerator A.(a) Iff€ L and t 2 0, then So T(s)fdsE 9 ( A )and(1.14)(b)(1.15)(c) Iff€ 9 ( A )and r 2 0, thenIff€ 9 ( A )and t 2 0. then T(t)/EB(A)andd--r(t)j=A T ( t ) / = T(r)AJdt(1.16) T(t)J-j= A T(.s)jds = T(s)Afds.Proof. (a) Observe thatfor all h > 0, and as h -,0 the right side of(I.17)converges to T(t)/-f:
  19. 19. 10 OPERATOR SEMlGROUPS(b) Since(1.18)for all h > 0, where A, = h-[T(h)-I], it follows that T(t)fe9 ( A )and (d/dt)+T(t)f= A T(r)/= T(t)A$ Thus, it sufices to check that(d/df)-T(r)f -- T(r)Af(assumingt > 0).But this followsfrom the identity(1.19)1-- h- h ) f - W)SI- T(t)A/= T(t -h)[A, -A]f+ [T(I -h) - T(t)]Af,valid for 0 < h 5 t.(c) This is a consequenceof(b)and Lemma 1.4(c). 01.6 Corollary If A is the generator of a strongly continuous semigroup{ T(t)}on L, then 9 ( A )is densein L and A is closed.Proof. Since Iim,,o + t - fo T(s)f ds =f for every fc L, Proposition 1.qa)implies that 9 ( A ) is dense in L. To show that A is closed, let {f,} c 9 ( A )satisfy$,4f and AS,- g. Then T(r)f,-Jn = roT(s)AJnds for each t > 0, so,letting n-+ a,we find that T(r)f-f= 6 T(s)gds. Dividing by t and letting0I-+ 0, we concludethatje 9 ( A )and Af= g.2. THE HILL€-YOSIDA THEORfMLet A be a closed linear operator on L. If, for some real 2, A - A (K A1 - A) isone-to-one, W ( l -A) = L, and (1-A)- is a bounded linear operator on L,then 1 is said to belong to the resoluent set p(A) of A, and RA = (A -A)- iscalled the resoluenr (at A) of A.2.1 Proposition Let {T(I))be a strongly continuous contraction semigroupon L with generator A. Then (0,00) c p(A) and(2.1) (A-A)-g = e-AT(tb drfor all g E L and d > 0.Proof. Let 1 > 0 be arbitrary. Define U, on L by UAg = J$e-"T(t)g df.Since(2.2)0)It U ~ g l l Lrne-"l/ T(r)sll df 9~-llgll
  20. 20. 2. THE HILLL-YOSIDA THEOREM 11foreach g E L, U Ais a bounded linear operator on L. Now given g E L,for every h > 0,so, letting h-, 0,.we find that UAg E g ( A ) and AUAg=AU,g - g, that is,(2.4) (1- A)UAg 9, 9 E L.In addition,if g E $@(A),then (using Lemma 1.4(b))(2.5) UAAg = e- "T(t)Ag dt = [A(e-"T(t)g) dt= A lme-"(t)g dt = AuAg,so(2.6) uA(A- A)g = 99 g E %A).By (2.6),A - A is one-to-one, and by (2.4),9 ( A - A) = L.Also, (A - A)- =U Aby (2.4) and (2.6), so A E p(A). Since rl > 0 was arbitrary, the proof iscomplete. 0Let A be a closed linear operator on L. Since (A - A)(p - A ) =(p - AHA - A ) for all A, p E p(A), we have (p - A)-(A - A).. = (A - A)--(p - A ) I , and a simple calculationgives the resolvent identity(2.7) RA R, = R, RA = (A - p)-(R, - RA), A, p E p(A).IfI.Ep(A)andJA-pI< I)R,II-,then(2.8)definesa bounded linear operator that is in fact (p - A ) - . In particular, thisimplies that p(A) is open in R.A linear operator A on L is said to be dissipative if IIJ j - AjII 2 Allfll forevery/€ B(A)and I > 0.2.2 lemma Let A be a dissipative linear operator on L and let 1 > 0. ThenA is closed if and only if #(A - A) is closed.Proof. Suppose A is closed. If (1;)c 9 ( A ) and (A - A)jw-+ h, then the dissi-pativity of A implies that {J.} is Cauchy. Thus, there exists/€ L such that
  21. 21. 12 OPERATORSEMICRourSL.+J and hence Al,,--+ Af - h. Since A is closed,fe 9 ( A ) and h = (A - A)J Itfollowsthat @(I - A ) is closed.Suppose*(A -A) isclosed.If {L}c 9(A),S,-J and A h 3g, then (A -A)fn-+?/- g, which equals (A - A)J, for somefo E 9(A). By the dissipativity of A,0f n d f o ,and hence/=fO E 9 ( A )and As= g. Thus, A is closed.2.3 lemma Let A be a dissipative closed linear operator on L, and putp+(A)= p(A) n (0, 00). If p+(A)is nonempty,then p+(A)= (0, a).froof. It suffices to show that p+(A)is both open and closed in (0, a).Since&A) is necessarily open in R,p+(A) is open in (0, 00). Suppose that {i"}cp+(A)and A,-+ A > 0. Given g E L,let g,, = (A - AKA, - A)-g for each ti, andnote that, because A is dissipative,(2.9) lim IIg,, -g11 = lim 11(I- Am)& - A)-g 11 5 lim 1.1-1.111g11 = 0.Hence @(A -A) is dense in L, but because A is closed and dissipative,9 ( A -A) is closed by Lemma 2.2, and therefore @(A - A) = L. Using thedissipativity of A once again, we conclude that I -A is one-to-one andII(A -A)-(I s I - . It follows that 1 B p+(A),so p+(A)is closed in (0, a),asI - a l *-.OD n-al 4required. 02.4 lemma Let A be a dissipativeclosed linear operator on L, and supposethat 9 ( A ) is dense in L and (0, 03) c p(A). Then the Yosida approximation A,of A, defined for each A > 0 by A, = RA(A -A)-, has the following proper-ties:la) For each A >0, Al is a bounded linear operator on L and {PJ}is a(b) A, A, = A, A, for all A, p > 0.(c) lim,-m A, f= Affor everyfe 9(A).strongly continuouscontraction semigroupon L.Proof.(I - A)R, = I on L and R,(A - A ) = I on $+I),it followsthat(2.10) A,=AR,-Al on L, A > O ,andFor each R > 0. let R, = (A-A)- and note that 11R, 11 5 A - I . Since
  22. 22. 2. T M HILL€-YOSIDA THEOREM 13for all t 2: 0, proving (a).Conclusion (b) is a consequenceof (2.10)and (2.7). Asfor (c),we claim first that(2.13) lim I R , f = f , SE L.d-+mNoting that llLRaf-lll = IIRAAfll s A-I(A/II 4 0 as A+ a, for eachf e 9 ( A ) , (2.13) follows from the facts that 9 ( A ) is dense in L andlll.Ra - Ill S 2 for all 1 > 0. Finally, (c) is a consequence of (2.1 I) and(2.I 3). 02.5 lemma If B and C are bounded linear operators on L such thatBC = CB and 11elB(II; I and 11efc11 5 I for all t 1 0, then(2.14) IIe"!f - elC/ It It It Bf - C/I1for everyfe L and t 2 0.Proof. The result follows from the identity= [e"e- B - C)fds.(Notethat the last equality uses the commutivity of B and C.) 0We are now ready to prove the Hille-Yosida theorem.2.6 Theorem A linear operator A on L is the generator of a strongly contin-uous contraction semigroup on L if and only if:(a) 9 ( A ) is dense in L.(b) A is dissipative.(c) a(1- A) = L for some R > 0.Proof. The necessity of the conditions (a)+) follows from Corollary 1.6 andProposition 2.1. We therefore turn to the proof of sulliciency.By (b),(c),and Lemma 2.2, A is closed and p(A) n (0, m) is nonempty, soby Lemma 2.3, (0, m) c p(A). Using the notation of Lemma 2.4, we define foreach L > 0 the strongly continuous contraction semigroup {T(c)} on L byK(t)= erAA.By Lemmas 2.4b) and 2.5,(2.16) IInw- q(t)/ll 111AJ- AJll
  23. 23. 14 OrUATOROMCROUISfor all f~ L, t 2 0, and A, p > 0. Thus, by Lemma 2.4(c), limA*mT,(t)/existsfor-all t 2 0, uniformly on bounded intervals, for allfe 9(A), hence for everyf~ B(A)= L.Denoting the limit by T(t)fand using the identity(2.17) T(s+t ) j - T(s)T(t)f=[T(s+r) - T,(s +t)Jf+ T,(s)CT,(t) - 7(01S+ CT,(s) - WJWJ;we concludethat { T(t)}is a stronglycontinuouscontractionsemigroupon L.I.5(c),It remains only to show that A is the generator of {T(t)}.By Proposition(2.18)foraltfE L, t 2 0,and R > 0. For eachfE 9 ( A )and r 2 0, the identity(2.19)together with Lemma 244, implies that G(s)AJ-r T(s)Afas A+ bc), uni-formlyin 0 5 s s t. Consequently,(2.18) yieldsT,(s)As- T(s)Af= T*(sXAJ-Af)+ cTAW - 7wl A/;(2.20)for all/€ 9 ( A ) and t 2 0. From this we find that the generator B of { T(r)}isan extension of A. But, for each 1 >0,A -B is one-to-one by the necessity of(b),and #(A -A) = L since rl E p(A). We conclude that B = A, completing theproof. 0The above proof and Proposition 2.9 below yield the followingresult as aby-product.2.7 Proposition Let {T(t)}be a strongly continuous contraction semigroupon L with generator A, and let Ad be the Yosida approximation of A (definedin Lemma 2.4). Then(2.21)so, for each fE L, liniA-,me"1/= T(r)ffor all I 2 0, uniformly on boundedintervals.1Ie"Y- T(t)fII 5 tit As-AfII, f s %4), t & 0,rt > 0,2 8 Corollary Let {T(r)}be a strongly continuouscontraction semigrouponL with generator A. For M c L,let(2.22) Ay i= { A > 0: A(A -A)- : M 4M}.If either (a)M is a closed convex subset ofL and AM is unbounded,or (b)M isa closed subspaceof L and AM is nonempty, then(2.23) T(t):M-+M, t 2 0.
  24. 24. 1. TH€ HNLE-VOSIDA THEOREM 1sProof. If A, j~> 0 and I1 -p/lI < I, then (cf.(2.8))(2.24) p ( p - A ) - = n = Of ;(*-$[A(I-A)-1]"?Consequently, if M is a closed convex subset of L, then I E AM implies(0, A] c AM, and if M is a closed subspaceof L, then A. E AM implies(0, 2 4 tA,,, .Therefore, under either (a)or (b),we have AM = (0, 00). Finally, by (2.10).(2.25) exp {IA,} = exp { - t I ) exp {tA[l(lt - A ) - ] )forall I 2 0 and I > 0, so the conclusion follows from Proposition 2.7. 02.9 Proposition Let { T(t)} and {S(t)} be strongly continuous contractionsemigroups on L with generators A and B, respectively. If A = B, thenT(t)= S(t) for all r 2 0.Proof. This result is a consequenceof the next proposition. 02.10 Proposition Let A be a dissipative linear operator on L. Suppose thatu : [0, a)-+L is continuous, ~ ( t )E Q(A) for all r > 0, Au: (0, a)-+L is contin-uous, and(2.26) u(t) = U(E) + Au(s) ds,for all t > E > 0. Then IIu(r)II 5 II40)It for all t 2 0.
  25. 25. 16 OPERATOR SEMlCROUrSwhere the first inequality is due to the dissipativity of A. The result followsfrom the continuity of Au and u by first letting max (t, -ti- ,)+ 0 and thenlettingc+ 0. 0In many applications, an alternative form of the Hille-Yosida theorem ismore useful. To state it, we need two definitionsand a lemma.A linear operator A on L is said to be closable if it has a closed linearextension. If A is closable, then the closure A of A is the minimal closed linearextension of A; more specifically, it is the closed linear operator 6 whosegraph is the closure(in L x L)of the graph of A.2.11 lemma Let A be a dissipativelinear operator on L with 9 ( AL.Then A is ciosableand L@(A -A) =9?(A -A^)forevery I > 0.dense inProof. For the first assertion, it suffices to show that if {A}c 9 ( A ) , 0,and Af,-+g E L,&heng = 0. Choose {g,} c $(A) such that g,,,--tg. By thedissipativity of A,(2.28) IIV - - 4It = lim II(A- A h , + &)I1a-m2 lim AIlgm + KII AIIgmIIn- mfor every 1 >0 and each m. Dividing by I and letting A+ 00, we find thatIIg, -g II 2 IIg, II foreach m. Letting m--, 00,we conclude that g = 0.Let 1 > 0. The inclusion @(A - A) =)@(A - A) is obvious, so ro proveequality, we need only show that 5?(I -A) is closed. But this is an immediateconsequenceof Lemma 2.2. 02.12 Theorem A linear operator A on L is closable and its closure A is thegenerator of a strongly continuouscontractionsemigroupon L if and only i f(a) 9 ( A )is dense in L.(b) A is dissipative.(c) B(1- A) is dense in L for some A > 0.Proof. By Lemma 2.11, A satisfies(a)-+) above if and only if A is closable andA’ satisfies(a)+) of Theorem 2.6. a3. CORESIn this section we introduce a concept that is of considerable importance inSections6 and 7.
  26. 26. Let A be a closed linear operator on L. A subspace D of 9 ( A ) is said to be acore for A if the closure of the restriction of A to D is equal to A (i.e., ifAJ, = A).-3.1 Proposition Let A be the generator of a strongly continuous contractionsemigroup on L. Then a subspace D of 9 ( A )is a core for A if and only if D isdense in L and w(1. - AID)is dense in L for some 1> 0.3.2 Remark A subspace of L is dense in L if and only if it is weakly dense(Rudin (l973), Theorem 3.12). 0Proof. The sufficiency follows from Theorem 2.12 and from the observationthat, if A and B generate strongly continuous contraction semigroups on Land if A is an extension of 8, then A = B. The necessity depends on Lemma2.1 1. 03.3 Proposition Let A be the generator of a strongly continuous contractionsemigroup IT([)}on L. Let Do and D be dense subspaces of L with Do c D c9 ( A ) .(Usually,Do = D.)If T(r):Do-+ D for all t 2 0, then D is a core for A.Proof. Givenf E Doand L > 0,(3.1)for n = I, 2,. ...By the strongcontinuity of { T(t)}and Proposition 2.1,(3.2)Ilim (i.- A)S, = lim - e ak/n7(:)(,l - A)/n-m n-(u k = O= lme -"T(t)(d - A)$&= (1- A ) - ( L - A)!=/:so a(>.- A ID) 3 Do.This sufices by Proposition 3.I since Dois dense in L. 0Given a dissipative linear operator A with 9 ( A )dense in L, one often wantsto show that A generates a strongly continuous contraction semigroup on L.By Theorem 2.12, a necessary and sufficient condition is that .%(A - A ) bedense in L for some A > 0. We can view this problem as one of characterizinga core (namely, g ( A ) )for the generator of a strongly continuous contractionsemigroup, except that, unlike the situation in Propositions 3.1 and 3.3, thegenerator is not provided in advance. Thus, the remainder of this section isprimarily concerned with verifying the range condition (condition (c)) ofTheorem 2.12.Observe that the followingresult generalizes Proposition 3.3.
  27. 27. 18 OrUATOR YMIGROUK3.4 Propositlon Let A be a dissipative linear operator on L,and Do a sub-space of B(A)that is dense in L. Suppose that, for eachJE Do, there exists acontinuous function u,: [O, 00)" L such that u,(O) =1; u,(t) E .@(A) for allr > 0, Au,: (0, a)-+L is continuous,and(3.3)for all t > E > 0. Then A is closable, the closure of A generates a stronglycontinuouscontraction semigroup {T(f)}on L,and T(t)J= u,(t) for allfE Doand r 2 0.Proof. By Lemma 2.11, A is closable. Fix f~ Do and denote uf by u. Letto > E > 0, and note that I:"e-u(t) dt E 9(A)and(3.4) 2loe-u(t) dt = e-Au(t) At.Consequently,(3.5)I0I"e-u(r) dt = (e-a -e-O)u(c) + loe- [Au(s) ds dt= (e-- e-O)u(c) += A I"e 3 ( t ) dt +e-u(c) -e-Ou(t,).(e-# - e-O)Au(s) dsI"Since IIu(t)(l5 llfll for all t 2 0 by Proposition 2.10, we can let 6-0 andto-+ Q) in (3.5)to obtain $; e-u(t) dr E B(2)and(3.6) (I - 2)ime-u(t)dr =J:We conclude that @(l -2)3 Do, which by Theorem 2.6 proves that 2gener-ates a strongly continuous contraction semigroup {T(r)}on L. Now for eachfE Do.(3.7) W f- W f=Im4mfor all t > E > 0. Subtracting (3.3) from this and applying Proposition 2.100once again,we obtain the second conclusion of the proposition.The next result shows that a suficient condition for A to generate is that Abe triangulizable. Of course, this is a very restrictive assumption, but it isoccasionallysatisfied.
  28. 28. 3. CORES 193.5 Proposition Let A be a dissipative linear operator on L, and supposethat L,, L,, L 3 ,...is a sequence of finite-dimensionalsubspaces of 9 ( A )suchthat u."-,L, is dense in L. If A : L , 4 L, for n = I, 2, . ..,then A is closableand the closure of A generates a strongly continuous contraction semigroupon L.Proof. For n = 1, 2, .. .,(A - AWL,) L, for all 1 not belonging to the set ofeigenvalues of AIL., hence for all but at most finitely many L > 0. Conse-quently,(A - AWU,", ,L,) = u:=,L,for all but at most countably many L > 0and in particular for some A > 0. Thus, the conditions of Theorem 2.12 aresatisfied. C3We turn next to a generalization of Proposition 3.3 in a different direction.The idea is to try to approximate A sufficiently well by a sequence of gener-ators for which the conditions of Proposition 3.3 are satisfied. Before statingthe result we record the followingsimple but frequently useful lemma.3.6 Lemma Let A,, A 2 , .. I and A be linear operators on L, Do a subspaceof L, and A > 0. Suppose that, for each g E Do, there existsJ, E g(A,)nd(A)for n = 1.2,. . .such that g, = ( A - A,)f,+gasn-+ 60 andlim [[(A,- A)Ll[= 0.n-.m(3.8)Then *(A - A) 3 Do.Proof. Given g E Do, choose {f,} and {g,} as in the statement of thelemma, and observe that limn-mII(A - A)J, -g,II -- 0 by (3.8). It follows that0limn+mI(( A - A)f, - g 11 = 0, giving the desired result.3.7 Proposition Let A be a linear operator on L and Do and D, densesubspaces of L satisfying Do c 9 ( A ) c D, c L. Let 111 . 111 be a norm on D,.For n = 1,2, . ..,suppose that A, generates a strongly continuous contractionsemigroup IT&)) on L and d ( A )c O(A,). Suppose further that there existw 2 0 and a sequence {&,} c (0, 60) tending to zero such that, for n = 1.2, ...,and(3.11) T,(t):Do+ 9(A), r 2 0.Then A is closable and the closure of A generates a strongly continuouscontraction semigroupon L.
  29. 29. 20 OPERATOISMCROUPSProof. Observe first that O(A)is dense in L and, by (3.9) and the dissipativityof each A,, A is dissipative. It therefore sufices to verify condition (c) ofTheorem 2.12.Fix 1 > o.Given g E Do, let(3.12)for each m, n 2 1 (cf. (3.1)). Then, for n = 1, 2, ..., (A - An)fm,,-+e-T(f)(A - An)g dt = g as m-r 00, so there exists a sequence {m,f ofpositiveintegerssuch that (A -A,,)S,,,-+ gas n--, 03. Moreover,(3.13) It(An -.Alfm., n II 111fm. n 111M 2k = OIllg1115 enm,-1 C e- Wa&h- 0 as n+mby (3.9) and (3.10), so Lemma 3.6 gives the desired conclusion. 03.8 Corollary Let A be a linear operator on L with B(A) dense in L, and letIll * 111 be a norm on 9 ( A ) with respect to which 9 ( A ) is a Banach space.For n = 1, 2, ..., let T. be a linear 11 ))-contraction on L such thatT,: 9(A)-+ 9 ( A ) , and define A, = n(T, - I). Suppose there exist w 2 0 and asequence {t,} c (0, a)tending to zero such that, for n = 1, 2, ...,(3.9) holdsand(3.14)Then A is closable and the closure of A generates a strongly continuouscontraction semigroupon L.Proof. We apply Proposition 3.7 with Do = D, = 9(A). For n = I, 2,. , .,exp (t.4,):9 ( A )+9 ( A )and(3.15) 111~ X P(tAn) I m A ) 111 S ~ X P{ -nil exp {nt111T.(@(A)111f s ~ X P{allfor all t 2 0, so the hypothesesof the proposition are satisfied. 04. MULTlVAlUED OPERATORSRecall that if A is a linear operator on L,then the graph g(A) of A is asubspace of L x L such that (0,g) E g(A) implies g = 0. More generally, weregard an arbitrary subset A of L x L as a multiualued operator on L withdomain 9 ( A ) = {/: (J g) E A for some g } and range *(A) = (g: (JI g ) e A forsome/}. A c L x L is said to be linear if A is a subspace of L x L. If A islinear, then A is said to be sinyfe-uaiuedif (0, g) E A impliesg = 0; in chis case,
  30. 30. 4. MULTIVALUED OPERATORS 21A is a graph of a linear operator on L, also denoted by A, so we write Af = g if(Jg) E A. If A c L x L is linear, then A is said to be dissipariue if(I lf- g II 2 R (I.fII for all (5g) E A and R > 0 ; the closure A’ of A is of coursejust the closure in L x L of the subspace A. Finally, we define1 - A = ((JAf- g): (Jg) E A } for each 1> 0.Observe that a (single-valued)linear operator A is closable if and only if theclosure of A (in the above sense) is single-valued. Consequently. the term“closable” is no longer needed.We begin by noting that the generator of a strongly continuous contractionsemigroup is a maximal dissipative (multivalued)linear operator.4.1 Proposition Let A be the generator of a strongly continuous contractionsemigroup on L. Let B c L x L be linear and dissipative, and suppose thatA c 8.Then A = B.Proof. Let U;g) E B and 1 > 0. Then ( f . 1.- g) E I - B. Since A E p(A),there exists h E 9 ( A ) such that Ah - Ah = AJ- g. Hence (h, If--g) E1 - A c A - B. By linearity, (1-h, 0)E I - B, so by dissipativity, J = h.0Hence g = Ah, so (J; g) E A.We turn next to an extension of Lemma 2.1 1.4.2 Lemma Let A t L x L be linear and dissipative.Then-(4.1) A0 = {(SI 8) E A’: 9 E @A)}is single-valued and cR(A - A ) = 9(1 - A)for every 1 > 0.Proof. Given (0,g) E A,, we must show that g = 0. By the definition of A,,there exists a sequence {(g., h,)] c A such that g,-+g. For each n,(g,, h, + l,g) E A by the linearity of A, so II Ag, - h,, - Ag I1 2 dIIg, II for every1. > 0 by the dissipativity of A’. Dividing by 1 and letting A- a,we find thatIlg,, - gll 2 )lg. I1for each n. Letting n-, a,we conclude that g = 0.The proof of the second assertion is similar to that of the second assertionof Lemma 2.I I. 0The main result of this section is the following version of the Hille-Yosidatheorem.4.3 Theorem Let A c L x L be linear and dissipative, and define A. by(4.1). Then A. is the generator of a strongly continuous contraction semigroupon 9 ( A )if and only if 9?(R - A) 2 9 ( A )for some A > 0.-Proof. A, is single-valued by Lemma 4.2 and is clearly dissipative, so by theHille-Yosida theorem (Theorem- 2.6), A, generates a strongly continuous-contraction semigroup on 9 ( A ) if and only if 9 ( A , ) is dense in 9 ( A ) and@(I. - A,) = 9 ( A ) for some A > 0. The latter condition is clearly equivalent to
  31. 31. 22 OPERATOR SEMIGROUPS9 ( L - A)=3 a(A)for some A >0. which by Lemma 4.2 is equivalent to41(1 - A) 3 d(A)for some 1->0. Thus, to complete the proof,-it suffices toshow that 9 ( A o ) is dense in 9 ( A ) assuming that 5?(A - A,) = B(A)for some1 > 0.By Lemma 2.3, Se(1- A,)= 9 ( A ) for every A >O, so 9(1- A ) =9 ( R - A)3 9 ( A ) for every R > 0. By the dissipativity of A, we may regard(A - A)- as a (single-valued)bounded linear operator on .@(A - A) of normat most L- for each 1> 0. Given cf;g) E A and R > 0, Af -g e @R - A)and/E 9 ( X )c 9 ( A ) c W(A-A),so g E g(A- X),and therefore IIA(d - A)-f--/Il= II(A - A)-gll 5 1-IIgII. Since 9(A) is dense in O(A),itfollowsthat(4.2)----lim A(L - A)-y=S, fE 9 ( ~ ) .I - m(Note that this does not follow from-(2.13).) But clearly, (A- A)-:&(A - A0)+ 9(Ao), that is, (A - A)-:9(A)-+ 9(Ao), for all L > 0. In view0of (4.2), this completesthe proof.Milltivalued operators arise naturally in several ways. For example, thefollowingconcept is crucial in Sections6 and 7.For n = 1, 2, ..., let L,, in addition to L, be a Banach space with normalso denoted by 11 * 11, and let n,: L-. L, be a bounded linear transformation.Assume that sup, IIn,,II < 00. If A, c L, x L, is linear for each n 2 I, theextended limit of the sequence {A,} is defined by(4.3) ex-lim A, = {U;g) c L x L:there exists u,,8,) E A, for eachn-mn 2 1 such that IIf, -rrJll+ 0 and 11g, - n,g 113 O}.We leave it to the reader to show that cx-lim,,, A, is necessarily closed inL x L (Problem 11).To see that ex-lim,,,A, need not be single-valued even if each A, is, letL, = L, a, = I, and A, = B +nC for each n 2 1, where B and C are boundedlinear operators on L.If/ belongs to N(C),the null space of C, and h E L,then A,,(f+ (I/n)h)+ Bf+ Ch,so{(ABf+ Ch):JeN(C),h E L}c ex-lim A,.(4.4)n-mAnother situation in which multivalued operators arise is described in thenext section.5. SEMIGROUPS ON FUNCTION SPACESIn this section we want to extend the notion of the generator of a semigroup,but to do so we need to be able to integrate functions u: [O, a)+L that are
  32. 32. 5. SEMICROUIS ON FUNCllON SPACES 23not continuous and to which the Riemann integral of Section 1 does notapply. For our purposes, the most efficient way to get around this difficulty isto restrict the class of Banach spaces L under consideration. We thereforeassume in this section that L is a “function space” that arises in the followingway.Let (M,a)be a measurable space, let r be a collection of positive mea-sures on A, and let 2‘ be the vector space of .,#-measurable functionsf suchthat(5.1)Note that 11. [I is a seminorm on Y but need not be a norm. LetN = { f ~9’:llfll = 0) and let L be the quotient space 9 / N ,that is, L is thespace of equivalence classes of functions in 9,wheref- g if I[/- gll = 0. Asis typically the case in discussions of Lp-spaces, we do not distinguishbetweena function in Y and its equivalenceclass in L unless necessary.L is a Banach space, the completenessfollowing as for E-spaces. In fact, if vis a o-finite measureon A’, 1 s q 5 ao,p-’ +q-’ = 1, andIlSIl --= SUP If1dP < m.r c r I(5.2)where (1 . 11, is the norm on U(v), then L = E(v). Of course, if r is the set ofprobability measures on A, then L = B(M, A),the space of bounded 4-measurable functionson M with the sup norm.Let (S, 9,v) be a a-finite measure space, let f:S x M -+R be 9’x A-measurable, and let g: S+ 10, 00) be 9’-measurable. If Ilf(s, .)[I5 g(s) for alls E S and g(s)v(ds) < m, then(5.3)and we can define j f ( s , .)v(ds) E L to be the equivalence class of functions in2’equivalent to h, where(5.4)With the above in mind, we say that u : S-+ L is measurable if there existsan Y x A-measurable function u such that u(s, .) E u(s) for each s E S.Wedefine a semigroup (T(t)}on t to be measurable if T( * )J is measurable as afunction on ([O, m), a[O,00)) for each/€ L. We define thefull generaror A’ ofa measurablecontraction semigroup (T(r)}on L by
  33. 33. We note that A is not, in general,single-valued.For example, if L = B(R)withthe sup norm and T(t)f(x)s f ( x +t), then (0, g) E A^ for each y E B(R) that iszero almost everywherewith respect to Lebesguemeasure.5.1 Proposition Let L be as above, and let {T(r)}be a measurable contrac-tion semigroup on L.Then the full generator A^ of {T(t))is linear and dissi-pative and satisfiesfor all h E W(A-A)and A > 0. IfT(s) e-"T(t)h dt = I"e-"T(s +t)h dt0(5.7)for all h E L, 1 > 0, and s 2 0, then 5#(1 - 2)= L for every 1 > 0.Proof. Let V; g) E A,A=- 0, and h = y- g. Then(5.8) lme-"T(r)hdr = A dpe-"T(r)fdt - e-"T(t)g dr= 1 r e-"T(t)fdt -1 e-" T(s)gds dt=JConsequently, IlflI s A- 11 h 11, proving dissipativity,and (5.6)holds.g = 4.j- h. Then(5.9) T(s)gds = 1Assuming (5.7), let h E L and A >0, and define f- e-"T(t)hdt andlme-"T(s +u)h du ds - T(s)h ds= I en*ime-"T(u)hdu ds - T(s)hds= elSIle-"T(u)h du -1."e-AuT(u)hdu+ T(s)hds - T(s)hds= Wf-ffor all t 2 0,soU;g) E Aand h = Af-g E SI(A - A). 0
  34. 34. 5. SEMKROUrJONFUNCllONWACES 25The following proposition,which is analogousto Proposition I.s(a), gives auseful description of someelementsof 2.5.2 Proposition Let L and (T(t))be as in Theorem 5.1, let h B t and u 2 0,and supposethat(5.10)forall I z 0.Then(5.1 1)T(t)lT(s)hds = 1T(t +s)h ds(lT(s)hds, T(u)h-h E A’.)p d . Put1=Zt;T(s)hds. Then= I”‘T(s)hds -1T(SPds=6‘T(s)(T(u)h-h)dsfor all r 2 0. 0In the present context,given a dissipative closed linear operator A c L x L,it may be possible to find measurable functions u: KO, a)-+L andu: [O, oo)+ tsuch that (u(t), u(t)) E A for every t >0 and(5.13) u(t) = u(0)+ 4s)ds, t ;I0.lOne would expect u to be continuous, and since A is closed and linear, it isreasonableto expect thatfor all t > 0. With these considerations in mind, we have the following multi-valued extension of Proposition 2.10. Note that this result is in fact valid forarbitraryL.
  35. 35. 26 OIflAlOISEMKiROUrS5.3 Proposition Let A c L x L be a dissipative closed linear operator.Suppose u: [O, a)-,L is continuous and (sou(s) ds, u(t) -u(0))E A for eacht > 0.Then(5.15)for all t 2 0. Given I > 0, defineIIu(4II s II 40)II(5.16) l= e-&u(t) dt, g = 1 e-*"(u(t) -40))dr.Then cf,g) E A and y- g = u(0).Proof. Fix r 2 0,and for each E > 0, put u,(t) = ti-(5.17)Since (u,(r), & - I ( & +e) -~ ( 1 ) ) )E A, it follows as in Proposition 2.10 thatIIu,(t)II S llu8(0)ll.Letting&-+0, we obtain (5.15).(5.18) j = e-*qt) dt = 1 e-*l$ u(s) ds dt,so U;8) E A by the continuity of u and the fact that A is closed and linear.Theequation 1f-g = u(0)follows immediately from the definitionoffand g. 0u(s) ds. Thenu,(t) = ~(0)+ E-(u(s +E ) -u(s))ds.Integrating by parts,Heuristically,if {S(r)}has generator 8 and {T(t)}has generator A +B, then(cf. Lemma 6.2)(5.19)for all t 2 0. Consequently,a weak form of the equation u, = (A +B)uis(5.20)We extend Proposition 5.3 to this setting.T(t)f=S(t)f+ r S ( r -s)AT(s)/ds0u(t) = S(t)u(O) +5S(t -s)Au(s)ds.05.4 Proposition Let L be as in Proposition 5.1, let A c L x L be a dissi-pative closed linear operator, and let {S(t)}be a strongly continuous, measur-able, contraction semigroup on L. Suppose u: [O, 00)- L is continuous,u: LO, 00)- L is bounded and measurable,and
  36. 36. 5. SEMICROWS ON FUNCnON SPAACES 27(5.21)for all r z 0. If(5.22)for every t > 0, and(5.23)for all q. r, r 2 0, then (5.15)holds for all I z 0.S(q +r)D(s) ds = S(q) S(r)o(s)dsc5.5 Remark The above result holds in an arbitrary Banach space under theassumption that u is strongly measurable, that is, u can be uniformly approx-0imated by measurable simple functions.Proof. Assume first that u: [O, m)-+ L is continuously differentiable,u: [O, a)--+L is continuous, and (u(t),41))E A for all t z 0. Let 0 = to < t , <(5.24)< t, = t. Then, as in the proof of Proposition 2.10,nIIu(t)I1= II 40)II + 1cI14tO I1 - II44- I ) Ill
  37. 37. 28 O?ERATORJEMK;ROU‘Swhere s’= t,- I and s” = t, for r,- I ss < r,. Since the integrand on the right isbounded and tends to zero as max (t, -ti, 0, we obtain (5.15) in this case.In the generalcase, fix t 2 0, and for each E >0, putu(s) ds, u,(t) = e-I“’lsb+‘= & - I 1S(r +S)U(O) ds +& - I(5.25)Then(5.26) u,(t) = u(r +s) dsU#) = e-IS(t +s -r)dr)dr ds= ~ - l S ( t )(dS(s)u(O) ds +6- I s’5’S(t +s -r)u(r)dr ds0 0+ I’r S ( t -r)u(r+s) dr ds0 01= S(t)[.s-I S(s)u(O)ds +6 - l 5’I’S(s - r)u(r)dr ds0 0+ S(t -r)ua(r)dr.By the specialcase already treated,(5.27) II u,(t)I1S )Ie -and lettingE--, 0, we obtain (5.15) in general. 06. APPROXIMATION THEOREMSIn this section, we adopt the following conventions. For n = 1, 2, ...,L,,inaddition to L,is a Banach space (with norm also denoted by I[6 11) and n,:L+ L,,is a bounded linear transformation. We assume that sup,,IIn, II < 00.We writef.-+fiff. E t,,foreach n 2 1,Je L, and lirn,-= [If, - a, Ill = 0.6.1 Theorem For n = I, 2,. ,.,let (T,(t)) and { T(r))be strongly continuouscontraction semigroups on L, and L with generators A, and A. Let D be acore for A. Then the following are equivalent:(3intervals.For each1E L,T,(t)n,f-+ T(r)ffor all t 2 0, uniformly on bounded
  38. 38. 6. APWOXIMATION THEOREMS 29(b)(c)For eachf E L, T,(l)n,J+ T(t)ffor all t 2 0.For each f~ D, there exists 1,E Q(A,) for each n 2 I such thatj,,-.Jand A,f,--+ Af(i.e., {(J AS):/€ D ) c e~-Iim,,+~A,,).The proof of this result depends on the following two lemmas, the first ofwhich generalizesLemma 2.5.6.2 Lemma Fix a positive integer n. Let {S,(r)} and {S(t)} be stronglycontin-uous contraction semigroups on L, and L with generators B,, and B. Let/E 9(B)and assume that n,,S(s)j~g(B,,) for all s 2 0 and that B,n,S( * )j:[O, 00) -+L,, is continuous. Then, for each t 2 0,(6.1)and therefore(6.2)S,(t)n, f - n,,S(f)j= S,,(C- sWB, n,,- n, B)S(s)fds,LIISn(t)n,f - n, S(tV It 5 II(B,n n - n, B)s(s)/II ds.Plod. It suffices to note that the integrand in (6.1) is -(d/ds)S,(t - s)n,S(s)/for 0 s s ,< t. 06.3 Lemma Suppose that the hypotheses of Theorem 6.1 are satisfiedtogether with condition (c) of that theorem. For n = 1, 2,. ..and R > 0, let Atand A be the Yosida approximations of A, and A (cf. Lemma 2.4). ThenA: n, f-+Ayfor everyfe L and R > 0.Proof. Fix R > 0. Let /E D and g =(A - A)f By assumption, there exists1;E B(A,)for each n 2 I such that /;--+fand Ad,-+AJ and therefore (A - A,)S,-+g. Now observethat(6.3) I1A:nng- nnA"gl1= II[AZ(R - AJ-1 - Rf]n,g -n,[RZ(R - A)- - Af-JgII= A2(1(R - An)- ring - nn(A - A)-eIts R211(R - A n F 1ring -Lit + RItSn - nn(R - A)-gII5 LIInng - ( A - An)/nII + nZII/n - nSIIfor every n 2 I. Consequently, 11A: n,g - R, Ag II -+ 0 for all g E - At,,).But &(A - AID) is dense in L and the linear transformations Ain,, - n,AL,n = I, 2,. ..,are uniformly bounded,so the conclusion of the lemma follows.0Proof of Theorem 6.1. (a *b) Immediate.
  39. 39. 30 OPERATOR SEMICROWS(b =5 c) Let 1 > 0.fE 4W), and g = (A - A)A so that f= e-"T(t)gdt. For each n 2 1, put fn = jz e-"X(t)n,,g dr E B(A,). By (b) and thedominated convergence theorem,S,-.l; so since (A - An)f, = n,g-+ g = (A-A)J we also have A,,&-, A/:(c =.a) For n = 1, 2,. ..and A >0, let {Ti(t)}and {T(r))be the strong-ly continuouscontraction semigroupson t,and L generated by the YosidaapproximationsA: and A. Given/€ D, choose {jJas in (c).Then(6.4) T,(l)nn f- nm T(tlf= UtKnn f-L) + CUt)f,- T$l)LI+ Ti(tMS,.-n, n+"CWnf - n, T A W ]+ nnCT?t).f- T(l)fJforevery n 2 I and t 2 0. Fix to 2 0. By Proposition 2.7 and Lemma 6.3,lim SUP 11 X(t).t, - T,"(t)LII5 lim to 11An S,- Aijn 11n- w 0 sI sfo n-m(6.5)lim to{ IIAn S. - nn MII + IInn(AS- AWIIn - m+ IInnAY- AfnnfII + I I A ~ ~ ~ . ~ - L ) I I Is K~oIlAf- AYII,where A= sup,((It,((.Using Lemmas 6.2, 6.3, and the dominated con-vergence theorem,we obtain(6.6) lim sup 11 T;(t)n,f -n, Ta(r)fIIn-m OLILIOs lim II(R."n. - n,A")T"s)Jl/ ds = 0.n-mApplying(6.5), (6.6). and Proposition 2.7 to (6.4), we find that(6.7) SUP I1T,(t)nnf -n, T(t)fll S 2Kr011A!f- AfII.I - C O O s r s t oSince I was arbitrary, Lemma 2.4(c) shows that the left side of (6.7) is zero.But this is valid for allfe D,and sinceD is dense in L, it holds for allJe L.0There is a discrete-parameter analogue of Theorem 6.1, the proof of whichdependson the followinglemma.6.4 lemma Let B be a linear contraction on L.Then(6.8) IIBY- en(8-Yll 5 J;;IIBJ-JIIfor allfs L and n = 0, 1,. ...
  40. 40. 6. APFUOXIMATION THEOREMS 31Proof. Fix/€ L and n 2 0. Fork = 0, I,. ..,(6.9)Therefore(6.10)(Note that the last equality follows from the fact that a Poisson random0variable with parameter n has mean n and variance n.)6.5 Theorem For n = I, 2,. ..,let T,, be a linear contraction on L,, let E, bea positive number, and put A, = E;(T,, - I). Assume that Iim,,,&, = 0. Let{ T(t)}be a strongly continuous contraction semigroup on L with generator A,and let D be a core for A. Then the following are equivalent:(a)intervals.(b)(c)For each/€ L, T!,!Cnln,/-tT(t)ffor all t 2 0, uniformly on boundedFor each/€ L, T!,!%,, f- T(t)/for all t 2 0.For each / E D,there exists S. E L, for each n 2 I such that h4/and Anf,-+ AJ(i.e., ((JA ~ ) : / ED}c ex-limn.,, A,).Proof. (a b) Immediate.(b 3C ) Let A > 0,/ E B(A),and g = (A - AM; so that f = jg e-"f(t)edt. For each n 2 I, put(6.1I )
  41. 41. 32 OPERATORSMCROUISBy (b) and the dominated convergence theorem,L-+J and a simple calcu-lation shows that(6.12) (1-AalL = nag -trlE,naga3+ - 1 +e-Aca) e-A*cnT~+n,gk = Ofor every n 2 1, so (A -A,).& -,g =(A -A ) j It followsthat A,,S,-+Af:(6.13) T!IbJn,J- n, T(r)f(c*a) Givenfe 0,choose {fa} as in (c). Thenand by Theorem6.1,(6.15)Consequently,(6.16) lim sup 11 T~laln,J-n, T(r)f11= 0.But this is valid for allfE D, and sinceD is dense in L, it holds for allfE L.lim sup I(exp {&a[ i ] ~ a } n a1-na VIUII =0.a-m OSCSIOn-m 05151006.6 Corollary Let {V(t):f 2 0) be a family of linear contractions onL with V(0)= I, and let {T(r)} be a strongly continuous contractionsemigroup on L with generator A. Let D be a core for A. If lims40~ - * [ V ( & ) f - f j= Affor every/€ D, then, for eachfe L, V(r/n)y-+ T(t)ffor allr r:0, uniformly on bounded intervals.Proof. It sunices to show that if {tn) is a sequence of positive numbers suchthat in-* r 2 0, then V(t,,/n)"+ T(t)ffor everyfe t.But this is an immediateconsequenceof Theorem 6.5 with T.= V(tJn) and E, = tJn for each n 2 I. 0
  42. 42. 6. APPROXlMATltM THEOREMS 336.7 Codary Let {T(t)), (S(t)}, and (V(r)} be strongly continuous contrac-tion semigroups on L with generators A, B, and C,respectively. Let D be acore for A, and assume that D c 9(B) n 9(C)and that A = B + C on D.Then, for each/ E L.(6.I 7)for all r 2 0, uniformly on bounded intervals. Alternatively, if (E,} is asequenceof positive numbers tending to zero, then, foreach/€ L,(6.18)for all t 2 0, uniformly on bounded intervals.Proof. The first result follows easily from Corollary 6.6 with V(t)IS(c)U(t)0for all t 2 0. The second followsdirectly from Theorem 6.5.6.8 Corollary Let (T(t)}be a stronglycontinuouscontraction semigrouponL with generator A. Then, for each / E L,(I -(r/n)A)-"J- T(t)ffor all I 2 0,uniformly on bounded intervals. Alternatively,if {en} is a sequence of positivenumbers tending to zero, then, for each f e t,(I -E,,A)-~"~Y--+T(t)Jfor allt ;r 0, uniformly on bounded intervals.Proof. The first result is a consequence of Corollary 6.6. Simply takeV(i)= (I - tA)- for each f 2 0, and note that if E > 0 and 1 = E - , thenwhere AI is the Yosida approximation of A (cf. Lemma 2.4). The second result0follows from (6.19) and Theorem 6.5.We would now like to generalizeTheorem 6.1 in two ways. First, we wouldlike to be able to use some extension A, of the generator A, in verifying theconditions for convergence. That is, given U;g) E A, it may be possible to findu,,g,) E A, for each n 2 1 such that /.-/ and g,+ g when it is not possible(or at least more diflicult) to find u,,g,) E A, for each n 2 1. Second, wewould like to consider notions of convergence other than norm convergence.For example, convergence of bounded sequences of functions pointwise oruniformly on compact sets may be more appropriate than uniform con-vergencefor some applications.An analogous generalization of Theorem 6.5 isalso given.
  43. 43. 34 N TORS EM CROUPSLet LIM denote a notion of convergence of certain sequences f,E L,,n = 1,2,...,to elementsf€ L satisfying the followingconditions:(6.20) LIMf, =f and LIM g, =g implyLIM (aJ;+Pg,) = cf+ /?g for all a, /3 E R.(6.21) LIMf:) = f k ) for each k 2 1 andlim sup ll/!hJ -J, 11 V llj4kJ-/[I = 0 imply LIMA, =/:There exists K >0 such that for eachfe L,there is asequenceA, E L, with Ilf.11 s KIIfII, n = 1, 2,.. .,satisfyingLIML =f.h-m r Z 1 ,(6.22)If A, c L, x L, is linear for each n 2 1, then, by analogy with (4.3).we define(6.23) ex-LIM A, = (U;g) E L x L:there exists ( f . ,8,) E A,for each n 2 1 such that LIMA, =/and LIM g, = g}.6.9 Theorem For n = 1, 2,. .., let A, c L, x L, and A c L x L be linearand dissipative with 9 ( A - A,) = L, and 9 ( A -A) = L for some (hence all)A > 0, and let {T,(r)} and {T(t)) be the-corresponding strongly continuouscontraction semigroups on 9(A,) and 9(A). Let LIM satisfy (6.20H6.22)together with(6.24) LIMf, = 0implies LIM (A -A,)-% = 0 for all 1>0.(1) If A c ex-LIM A,, then, for each U;g) E A, there exists u,,9,) E A,for each n z 1 such that sup, /If. 11 < 00, sup, IIg, II< 00, LIM J, =f,LIM 8,= g, and LIM T,(t)J, = T(r)ffor all t 2 0.(b) If in addition {x(r)}extends to a contraction semigroup (alsodenoted by {x(t)})on L, for each n 2 1, and if(6.25) LIMA = 0implies LIM T,(r)f. = 0 for all t 2 0,then, for eachfe B(A),LIMJ;=/implies LIM x(t)f. = T(t)/for all t 2 0.-6.10 Remark Under the hypotheses of the theorem, ex-LIM A, is closedin L x L (Problem 16). Consequently, the conclusion of (a) is valid for allUI Q)E A’. 0Proof. By renorming L,, n = 1, 2,...,if necessary, we can assume K = 1 in(6.22).Let 2’denote the Banach spa& (naLILJx L with norm given byI I ( { L J s f)II= SUPnz1111; IIV IIf II, and let(6.26)
  44. 44. 6. APFROXlMATlON THKMFMS 35Conditions (6.20)and (6.21) imply that Yois a closed subspacc of 9,andCondition (6.22) (with K = 1) implies that, for each/€ L, there is an element( { f n } , / ) 6 9 0 with II({fn}*AII= IIJll.Let(6.27) d = {[({fn}*jh ({gn}. 911 E 9 X 9:Un.gn) An for eachn 2 1 and U;g)E A}.Then Iis linear and dissipative, and @(A - .d)= Y for all 1 > 0. The corre-sponding strongly continuous semigroup {.T(f)} on 9(d)is given by-(6.28)We would like to show that(6.29)To do so, we need the following observation. If V; g) E A, 1 > 0, h = AJ- g,((hn), h) E Y o . and(6.30) (f"* 9,) = ((A - A n ) *k9 - h n )for each n z I, thenTo prove this, since A c ex-LIM A,,, choose c/"., 8,) E A, for each n 2 1 suchthat LIM3, =f and LIM 3, = g. Then LIM (h, -(ly",- 8,))= 0, so by (6.24),LIM (1- A,)-h, -f, = 0. It follows that LIMf, = LIM (A - A,,)-*h, =LIMA =f and LIM g,, = LIM (@, -h,) = V-h = g. Also, sup, IIj, II s1-I SUP, IIh, 11 < 00 and SUP. IIgn II 5 2 SUP, IIh n 11 -= 00. Consequently, [({h),n,((9,). g)] belongs to 9,x Y o ,and it clearly also belongs to d .Given ({h,},h) E Y oand rl > 0, there exists c(,g) E A such that ly- g = h.Define u,,g,) E A, for each n z 1 by (6.30). Then (A - d)-({h,,},h) =( { f n } , J ) E 90by (6.31)v SO(6.32) (1- d ) - :9 0 3 Y o , L > 0.By Corollary 2.8, this proves (6.29).To prove (a), let (1g) E A, A > 0, and h = Af- g. By (6.22). there exists({h,}, h) E Y owith ll({h,,}, h)II = IIh 11. Define (h,g,) E A, for each n 2 1 by(6.30). By (6.31). (6.29), and (6.28), ({T,,(t)f,,}, T(t)f) E Y ofor all t 2 0, so theconclusion of (a)is satisfied.As for (b), observe that, by (a) together withI_(6.25), LIML =fB B(A)implies LIM T(t)/,-- T(t)ffor all t 2 0. Letfs d(A)and choose {$&I} c B(A)such that II/I -/[I s 2-& for each k 2 1. Put Po = 0, and by (6.22), choose
  45. 45. for each k 2 1. Since(6.34)andfor each n 2 1 and k 2 1, (6.21) implies that(6.36)Q, mLIM 1u!~==A LZM T,(t)Cut)= T(t)J;I Iso the conclusionof (b)followsfrom (6.25). 06.11 Theorem For n = 1, 2,..., let T, be a linear contraction on L,, letE, > 0, and put A, = &;(T, -I). Assume that limn-mc,, = 0. Let A c L x Lbc linear and dissipative with 9?(1 -A) = L for some (henceall) 1 > 0, and letIT(t)} be the corresponding strongly continuous contraction semigroup on9(A).Let LIM satisfy(6.20)-(6.22),(6.24),and(6.37) lim JjhII =0 implies LIM 2= 0.W If A c ex-LIM A,, then, for each U;g) E A, there exists f,,E L,for each n 2 1 such that sup,Ilf,jl < 00, sup,)IA,J,(I < 00, LIMA -I;LIM AJ, =g, and LIM ch!&= T(r)/for all r z 0.(6.38) LIMJ, = 0 implies LIM T!/-y, = 0 for all t 2 0,then for eachftz 9(A),LIMA Efimplies LIM c/"!f,,= T(t)ffor all r 2 0.(bJ If in addition-Proof. Let U;g) E A. By Theorem 6.9, there cxistsI; E L, for each n L 1 suchthat SUp,!lfn I1 < a, sup,II Af,Il < 00, LIMf, -S, LIM AS, = g, andLJM e"X = T(r)Jfor all t 2 0.Since(6.39)
  46. 46. 7. NRTUROATION THEOREMS 37for all t 2 0, we deduce from (6.37) that(6.40)The conclusion of(a)therefore follows from (6.14)and (6.37).The proof of (b)is completelyanalogousto that of Theorem 6.9(b). 07. PERTURBATION THEOREMSOne of the main results of this section concerns the approximation of semi-groups with generators of the form A + B,where A and B themselves generatesemigroups.(By definition, O(A+ B)= O(A)n 9(B).)First, however, we givesome suflicient conditions for A + B to generate a semigroup.7.1 Theorem Let A be a linear operator on L such that A’ is single-valuedand generates a strongly continuous contraction semigroup on L.Let B be adissipative linear operator on L such that 9 ( B )3 9(A). (In particular, 6 issingle-valuedby Lemma 4.2.) Ifwhere 0 5 a c I and /I2 0, then A + B is single-valued and generates astrongly continuouscontraction semigroup on L. Moreover, A + B = A + 8.Proof. Let y 2 0 be arbitrary. Clearly, 9 ( A +yB)= 9 ( A ) is dense in L. Inaddition, A + yB is dissipative. To see this, let A be the Yosida approx-imation of A’ for each p > 0, so that A, = p[p(p - .$)-I -11. If/€ d ( A )andA > 0. thenby Lemma 24c)and the dissipativity of yB.
  47. 47. I f j e 9(A),then there exists (f.} c 9 ( A ) such thatf.+/and AS,-+ 26 BY(7.1), {Bf;)is Cauchy, s o f ~9(B)and BS,+ BJ Hence 9(J)t 9(B)and (7.1)extends to(7.3)In addition,if/€ 9(A)and if (I,)is as above, then(7.4)implying that A -t- yB is a dissipativeextension of A +ys.(7.5) T = { y 2 0: 4?(6 - A -yb)= L for some (henceall) 6 > 0).To complete the proof, it sufficesby Theorem 2.6 and Proposition 4.1 to showthat 1 E r.Noting that 0 E r by assumption,it is enough to show that(A+yg)f= lim A& +y lim Bf. = lim ( A +yB)/,= (A +yE)Ja a aLet1 - ayy E r n Lo, 1) implies [y, y -+ 7 )c rTo prove (7.6), let y E r n [O, I), 0 5 E < (2a)-(l -ay), and L > 0. Ifg E B(A),then/= (I- A -y@- g satisfies(7.7)by (7.3), that is,(7.8)and consequently,(7.9) IIB(L-A-;.B)-gli ~ [ 2 a ( l-q)-+/?(~- a y ) - ~ - ] l l g l l .Thus, for I suficiently large, IIE&(A -A -B)-II < 1, which implies ,thatI -11lgsrr 5: all 4.31+811f11 dl(A+rb)fll +aril mr +PlifliIlj3Jll 5 -aY)-JJ(A +ytr>/n +P(1-aY)-llJIl,- A - yb)- is invertible.We concludethat(7.10) B(6-A -(y -k e)B) 3 .@((A - A -(y 4- 6)&1 - A - yB)-)=@(I - &&I -A- yB)-)= Lforsuch 6, so y +E E r,implying(7.6) and completingthe proof. 07.2 Corollary If A generates a strongly continuous contraction semigroupon L and E is a bounded linear operator on L,then A + B generates astronglycontinuoussemigroup {T(t))on L such that(7.1 I) 11 T(r)i)5 e"", r 2 0.Proof. Apply Theorem 7.1 with B - [IB 11I in place of B. El
  48. 48. Before turning to limit theorems, we state the following lemma, the proof ofwhich is left to the reader (Problem 18). For an operator A, letM ( A ) 5 {fe.$@(A):Af = 0) denote the null space of A.7.3 Lemma Let B generate a strongly continuous contraction semigroup{S(t))on L, and assume that(7.12) tim A e-"S(r)(dr = Pf exists for all (e L.Then the following conclusions hold :a-o+(a) P is a linear contraction on L and P2= P.(b) S(r)P = PS(r)= P for all t 2 0.(c) @P) = XCB).-(d) N(P) = W(E).7.4 Remark If in the lemma(7.13) B = y - ( Q - I),where Q is a linear contraction on L and y > 0, then a simple calculationshows that (7.12) is equivalent tom(7.14) lim (I - p) 1 pkQL/=Pf exists for all /E L. 0p - l - k = O7.5 Remarkholds andIf in the lemma lim,+mS(r)( exists for every /E L, then (7.12)(7.15) Pf = lim S(i)J / E L.t-mIf E is as in Remark 7.4 and if limk-mQYexists for every (E L. then (7.14)holds (in fact,so does (7.15)) and(7.16) Pf= lim Q? (E L.k-mThe proofs of these assertionsare elementary. 0For the following result, recall the notation introduced in the first para-graph of Section 6,as well as the notion of the extended limit of a sequenceofoperators (Section4).7.6 Theonm Let A c L x L be linear, and let B generate a strongly contin-uous contraction semigroup {S(t)}on L satisfying(7.12). Let D be a subspace
  49. 49. 40 OPERATORS€MIGROWSof 9 ( A )and D a corefor B. For n = 1,2,. ..,let A, be a linear operator on L,and let a, > 0. Supposethat limn,man= 00 and that(7.17) {U;g) E A:fE D}c ex-lim A,,n - e(7.18) ((h,Bh): h E D) t ex-Jim a;A,.Define C = (U;fg):U;g) E A, f~ D} and assume that {(Ag) E c:g E 0)issingle-valued and generates a strongly continuous contraction semigroup{~ ( c ) } on 6.n-oD(a) If A, is the generator of a strongly continuous contraction semi-group {F(t)}on Lafor each n 2 1, then, for eachfe 6,x(t)nJ--r T(t)fforall 2 0, uniformly on bounded intervals.(b) If A, = E,-I(T, -I) for each n 2 1, where T. is a linear contractionon L, and E, >0, and if lim,,,~, =0, then, for eachfE D, T!%, f-. T(f)ffor all f 2 0, uniformly on bounded intervals.Proof. Theorems6.1 and 6.5 are applicable,provided we can show that(U;g) E C:g E 6)c ex-Jim A, n (b x 6).(7.19) ( n - r n )Since ex-lim,,, A, is closed, it sufficesto show that C c ex-limn,, A,. GivenU;g) B A with ftz D, choosef . E 9(An) for each n 2 1 such that fa- f andA,f,-, g. Given h E D, choose h, E B(A,) for each n 2 I such that h,+ h anda,- A, h, +Bh. Then f . +a, h, -+f and A,cf, +a; h,)3 g +Bh. Conse-quently,(7.20) {U;g +Bh):U;g) E A, f E D, h E D} c ex-lim A,.But sinceex-limn,, A, is closed and since, by Lemma 7.3(d),(7.21)for all g E L, we conclude that(7.22)1-4)7 -Pg - g E M ( P )= 9 ( B )= 9t(B(n*){U;Pg):V;g) E A, f e D) c ex-lim A,,n-mcompletingthe proof. 0We conclude this section with two corollaries. The first one extends theconclusions of Theorem 7.6, and the other describesan important special caseof the theorem.7.7 Corollary Assume the hypotheses of Theorem 7.qa) and suppose that(7.15) holds. If h E M(P) and if {t,} c 10, GO) satisfies tima,, t.u, = 00,
  50. 50. 7. PERTUIIATION THEOREMS 41then T,,(r,)n,h-+ 0. Consequently, for each f E P-(6) and 6 E (0, I),%(r)n,f-+ T(r)P/;uniformly in b s t g 6-.Assume the hypotheses of Theorem 7.6(b), and suppose that either(i) lim,,,a,q, = 0 and (7.15) holds, or (ii) lim,,-.,a,,c, = y > 0 and (7.16)holds (where Q is as in (7.13)).If h E N(P)and if {&,) c (0, 1,. ..} satisfiesk,a, E, = m, then TFn,h -+ 0. Consequently, for eachf E P - (6)and6 E (0, I), T!"%J-, T(~)PJuniformly in b s r 5 6 - .Proof. We give the proof assuming the hypotheses of Theorem 7.6(a), theother case being similar. Let b E J(r(P),let (t,} be as above, and let E > 0.Choose s 2 0 such that II S(s)hII 5 c/2K, where K = supnrI 11n, 11, and let s, =sAr,a, for each n 2 I. Thenfor all n suficiently large by (7.18) and Theorem 6.1. If J E L, thenf - Pf E .N(P), so 7Jrn)n,(J- Pf)+ 0 whenever {t,} c LO, 00) satisfiest, = r # 0. If f e P-(d), this, together with the conclusion of the theorem0applied to PJ completesthe proof.7.8 Corollary Let ll,A, and B be linear operators on L such that B generatesa strongly continuous contraction semigroup {S(r))on L satisfying (7.12).Assume that 9(n)n 9 ( A ) n 9 ( E )is a core for B. For each a sufkiently large,suppose that an extension of ll +aA +aE generates a strongly continuouscontraction semigroup { T,(r)} on L. Let D be a subspaceof(7.24) (/E 9(n)n 9 ( A ) n .N(B):there exists h E Q(n)n 9 ( A ) n 9(B) with Bh = - A / } ,and defineThen C is dissipative, and if ((J8) E c:g E 0).which is therefore single-valued, generates a strongly continuous contraction semigroup (T(r))on 6,then, for eachJE D, lima+,., x(r)/=T(r)/for all t 2 0, uniformly on boundedintervals.Proof.limn+ma, = GO, and apply Theorem 7.qa) with L, = L, n, = I, A replaced by(7.26) (U;n/+ A h ) : / € D, h E 9(n)n 9 ( A )n 9(B), Bh = -A!},A, equal to the generator of {T*(r)},a, replaced by af. andD = WJ)n 9 ( A ) n 9(B).Since A,,cf+ a;h) = nf+Ah +a i l l h when-ever/€ D, h E 9(n)n 9 ( A ) n 9(B),Bh = -AS, and n 2 1, and since limn--Let {a,} be a sequence of (sufficientlylarge)positive numbers such that
  51. 51. 42 OIEIATORSMGROUISa,-2A,h = Bh for all h E D, we find that (7.17) and (7.18) hold, so the theoremis applicable. The dissipativity of C followsfrom the dissipativity of ex-lim,,,A". 07.9 Remark (a) Observe that in Corollary 7.8 it is necessary that PAf= 0(b) Let /E 9 ( A ) satisfy PAf= 0. To actually solve the equationfor allfE D by Lemma 7.3(d).Bh = -Affor h, supposethat(7.27) II(s(t)- p)g11 dt < 00, g E L.Then h -" limA-o+(A - B)-Af= j; (S(t) - P)A/dt belongs to 9(B)(sinceB is closed) and satisfies Bh = -A$ Of course, the requirement that hbelong to 9(n)A 9 ( A ) must also be satisfied.(c) When applying Corollary 7.8, it is not necessary to determine Cexplicitly. instead, suppose a linear operator Co on b can be found suchthat Cogenerates a strongly continuous contraction semigroup on b andCoc C.Then {V; g) E (f:g E b} = Coby Proposition 4.1.(d) See Problem 20 for a generalization and Problem 22 for a closelyrelated result. 08.1.2.3.4.PROBLEMSDefine {T(r)}on &R) by T(t)J(x)=/(x +I). Show that {T(t)}is a strong-ly continuouscontractionsemigroupon t,and determine its generator A.(In particular,this requires that 9 ( A )be characterized.)Define {T(r)}on c(R) byfor each r >0 and T(0)= I. Show that {T(t)}is a strongly continuouscontraction semigroupon L,and determineits generator A.Prove Lemma 1.4.Let (T(r)}be a strongly continuous contraction semigroup on L withgenerator A, and let/€ 9(A2).(a) Prove thatJo
  52. 52. a m o m 43(b) Show that IIASII 5 411AJII 11/11.Let A generate a strongly continuous semigroup on L. Show that fl.i I9 ( A " )is dense in L.Show directly that the linear operator A = fd2/dxzon L satisfies condi-tions (a)-@)of Theorem 2.6 when 9 ( A )and L are as follows:(a) g ( ~ )= { f ~C2[0,11:a,f"(i)-(- l)&f(i) = 0, i= 0, I}.L = CCO, 11. ao.Po. a I ,PI 2 0, a. I-Po 7 0, al i-PI > 0.(b) L@(A)-= {fe CCO, 00):ao/"(0)- Bof(O) = 0)L = CCO, 001, ao,Po 2 0, a. +Do > 0.(c)Hint: Look .for solutions of A ,-4/"= g of the form f ( x )=exp { -a x } & ) .Show that CF(R) is a core for the generators of the semigroups of Prob-lems 1 and 2.In this problem, every statement involving k, I, or n is assumed to holdfor all k,I, n 2 1.be a sequence of closed subspaces of L. Let0,. M,,and MP be bounded linear operators on L. Assume that u, andMp)map L,into L, ,and that for some fl, >0, IIMPII <fi, and9 ( A ) -- C,(Pa), L = Qua).Let L, c L, c L, c *5.6.7.8.lim I(Mf""- M,1) = 0.r))" mSuppose that the restriction of A, Ithat there exist nonnegativeconstants dlk( (= a(,), & I , and y such that(8.4)Mf"Uj to L, is dissipative andf E t,II u h U J - UI UJll s ad11 UJll + IIUJII),(8.7)Define A =(8.8)If 9 ( A ) is dense in L, show that A is single-valued and generates astronglycontinuouscontractionsemigroupon L.I Mj[I,on1OD W~ ( A I= {I. u Ln: 1 fijllujflI < 00 .n = l j = J
  53. 53. 4 (wMToII6McROWSHint: Fix A > 3y and apply Lemma 3.6.Show first that for g E 9 ( A ) andf n * (a - AA-’g,n1-1(8.9) (a-Y)IlUd~ll IIuhgll + (fikJ+r,akj)lluj/;ll.Denoting by p the positive measure on the set of positive integers thatgives mass P h to k,observe that the formula(8.10)definesa positive bounded linear operatoron L’(p)of norm at most 27.9. As an application of Corollary 3.8, prove the following result, whichyields the conclusionof Theorem 7.1under a different set of hypotheses.Let A and E generate strongly continuous contraction semigroups{T(r))and {S(t)}on L. Let D be a dense subspace of L and 111 * 111 a normon D with respect to which D is a Banach space. Assume that 111fIII 2 11/11for allfc D. Supposethere existsp 2 0 such that(8.11) D =W2);IIA’Ill S rlllflll, fQ D;(8.12)(8.13) T(t):D-, 0, S(t): D-, D, t 2 0;(8.14) 111W )111 s e’, 111S(0 111 s e”’, 2 0.Then the closure of the restriction of A +B to D is single-valued andgeneratesa strongly continuouscontractionsemigroupon L.We remark that only one of the two conditions (8.11) and (8.12) isreally needed.See Ethier (1976).10. Define the bounded linear operator E on L =C([O, 13 x [O, 11) byBf(x,y) =(8.15)f(x, z) dz, and defineA c L x L byA = {Ut/,=+W:SEC2(C0,13 x CO, 11)n W ? AfA0, Y) =f3, y) =0 for all y E LO, 11,h E Jlr(B)).Show that A satisfiesthe conditionsof Theorem 4.3.11. Show that ex-lim,,, A,, defined by (4.3X is closedin L x L.12. Does the dissipativity of A, for each n 2 1 imply the dissipativity ofex-lim,,, A,?13. In Theorem 6.1 (and Theorem 6.5).show that (a)-+) are equivalent to thefollowing:
  54. 54. a. raocmts 45(d) There exists 1 > 0 such that (A - A,)-’n,,g+(1 - A ) - ’ g for allg E L.14. Let L, {L,,},and In,) be as in Section 6. For each n 2 1, let {T,(t)) be acontraction semigroup on L,, or, for each n 1 I, let (T,(r)} be defined interms of a linear contraction T, on L, and a number E, > 0 by 7Jr) =E, = 0. Let { T(t)}be a contraction semigroup on L, let J g E L, and suppose that lim,4mT(t)j= 8 andfor all t 2 0; in the latter case assume that(8.16) lim sup I[7Jr)nJ- n, T(r)jII = 0for every ro > 0. Show that(8.17)if and only if(8.18)IS. Using the results of Problem 2 and Theorem 6.5, prove the central limittheorem. That is, if X,,X,,... are independent, identically distributed,real-valued random variables with mean 0 and variance I, show thatn- c;=I X , converges in distribution to a standard normal randomvariable as n-+ 00. (Define TJ(x) = E u ( x +n-’”X,)] and c, = n-’,)Under the hypotheses of Theorem 6.9, show that ex-LIM A,, is closed inL x L.17. Show that (6.21) implies(6.37) under the following(very reasonable)addi-tional assumption.(8.19) If j,E L, for each n 2 1 and if, for some no 2 1,j,= 0for all n 2 no, then LIMS, = 0.Prove Lemma 7.3 and the remarks followingit.Under the assumptionsof Corollary 6.7, prove (6.18) using Theorem 7.6.Hinr: For each n 2 I, define the contraction operator T,on L x L byn-. w 0 SI 610lim sup 11 T,(t)nJ- nnT(t)fll= 0n-w 120lim sup IIT,(r)n,g - n, T(r)g)I = 0.n-m t a O16.18.19.(8.20)20. Corollary 7.8 has been called a second-order limit theorem. Prove thefollowingkth-order limit theorem as an application of Theorem 7.6.Let A,,, A , , ...,A, be linear operators on L such that A, generates astrongly continuous contraction semigroup {S(c)} on L satisfying (7.12).Assume that 5% = n $ - 0 9 ( A , ) is a core for A,. For each a suficiently
  55. 55. 16 OPEUTORS€MICIOUPSlarge, suppose that an extension of Cf=oajAjgeneratesa strongly contin-uous contraction semigroup { 7Jf)) on L.Let D be a subspaceof(8.21) {fo E 9:there exist fl,fz,.. ., f , - l E .9 withIm110AL-m+j/;=O for m = O , . .. ,k- 1 ,and defineIk - Il = O(8.22) C = {(fo, PAj&): fo E D,f,,...,&-, as above .Then C is dissipative and if {U;g) E c:g E 61,which is therefore single-valued, generates a strongly continuouscontraction semigroup {T(r)}on6, then, for eachfE 6,lima-,,,, lf&)f=T(t)f for all t 2 0, uniformly onbounded intervals.21. Prove the followinggeneralization of Theorem 7.6.Let M be a closed subspace of L,let A t L x L be linear, and let B,and B, generate strongly continuous contraction semigroups (S,(t)} and{S,(r)} on M and L,respectively,satisfying(8.23) lim R 1e-A"S,(t)fdr = P,f exists for all ffs M,(8.24) lim R e-"S,(f)fdt-= P,f exists for all f E L.Assume that @P,) c M.Let D be a subspace of 9(A), D,a core for B,,and D,a core for B,. For n = 1. 2,. ..,let A, be a linear operator on L,and let a,, /In> 0. Suppose that lim,-ma, = 00,(8.25) (U;g) E A : ~ ED}c ex-lim A,,(8.26) {(h, B,h): h c D,}c ex-lim a;A,,(8.27) {(k,B, k): k E D2}c ex-lim A,.Define C = {U;P I P ,9): (Jg) E A,fe D} and assume that {Ug) e c:g Eb} generates a strongly continuous contraction semigroup {~ ( t ) fon D.Then conclusions(a)and (b)of Theorem 7.6 hold.22. Prove the followingmodificationof Corollary 7.8.Let n, A. and B be linear operators 0 1 1 L such that 8 generates astrongiy continuous contraction semigroup {S(C)) on L satisfying(7.12).Assume that 9(n)n D(A) n B(B) is a core for B. For each a sufkientlylarge, suppose that an extension of ll +aA +a2B generates a stronglyA-O+A-O+ c/I, = 00, andn-mn-mn-m
  56. 56. 9. NOTES 47continuous contraction semigroup { T#)) on t.Let D be a subspace of9(n)n 9 ( A ) n N(B)with m P ) c 6, and define C = {(JPA/):/E D}.Then C is dissipative. Suppose that c generates a strongly continuouscontraction semigroup{ V(r)} on D,and thatm(8.28) lim L [ e-"U(r)fdt = P,f exists for every f e 6.A - O + JOLet Do be a subspace of {/E D:there exists h E 9(n)n 9 ( A ) n 9 ( B )with Bh = - A t } , and define(8.29) Co = {(J P o P n f + P , P A h ) : / € Do,h E 9(n)n 9 ( A ) n 9(B), Bh = -AS).Then C, is dissipative, and if {U;8) E co:g E a,} generates a stronglycontinuous contraction semigroup { T(r))on 6,. then, for each /E Do,Iirnadm T&)f= T(r)/for all t 2 0, uniformly on bounded intervals.23. Let A generate a strongly continuous semigroup {T(t)} on L, letB(t):L-4 L, t 2 0, be bounded linear operators such that (B(t)} isstronglycontinuousin t L 0 (i.e., t-+ B(r)fiscontinuousfor eachJE L).(a) Show that for each f~ L there exists a unique u: [O, o o ) ~Lsatisfying(8.30) ~ ( t )= T(t)f+ T(t - s)B(s)u(s)ds.(b) Show that if B(t)g is continuously differentiable in c for each g E L,andf E 9 ( A ) ,then the solution of (8.30)satisfies(8.31)a-u(t) = Au(r) + B(t)u(t).at9. NOTESAmong the best general references on operator semigroups are Hille andPhillips (1957),Dynkin (1965),Davies (1980),Yosida (1980).and Pazy (1983).Theorem 2.6 is due to Hille (1948)and Yosida (1948).To the best of our knowledge, Proposition 3.3 first appeared in a paper ofWatanabe (1968).Theorem 4.3 is the linear version of a theorem of Crandall and Liggett(1971). The concept of the extended limit is due to Sova (1967) and Kurtz(1969).Sufficient conditions for the convergence of semigroups in terms of con-vergence of their generators were first obtained by Neveu (1958). Skorohod(l958), and Trotter (1958).The necessary and suflicient conditionsof Theorems
  57. 57. 48 OrUATORSMCIOUPS6.1 and 6.5 were found by Sova (1967) and Kurtz (1969). The proof given herefollows Goldstein (1976). Hasegawa (1964) and Kato (1966) found necessaryand sufficient conditions of a different sort. Lemma 6.4 and Corollary 6.6 aredue to Chernoff (1968). Corollary 6.7 is known as the Trotter (1959) productformula. Corollary 6.8 can be found in Hille (1948). Theorems 6.9 and 6.11were proved by Kurtz (1970a).Theorem 7.1 was obtained by Kato (1966) assuminga < and in general byGustafson (1966). Lemma 7.3 appears in Hille (1948). Theorem 7.6 is due toEthier and Nagylaki (1980) and Corollary 7.7 to Kurtz (1977). Corollary 7.8was proved by Kurtz (1973) and Kertz (1974); related results are given inDavies (1980).Problem 4(b) is due to Kallman and Rota (1970), Problem 8 to Liggett(1972), Problem 9 to Kurtz (see Ethier (1976)), Problem 13 to Kato (1966), andProblem 14 to Norman (1977). Problem 20 is closely related to a theorem ofKertz(1978).
  58. 58. 2This chapter consists primarily of background material that is needed later.Section I defines various concepts in the theory of stochastic processes, inparticular the notion of a stopping time. Section 2 gives a basic introductionto martingale theory including the optional sampling theorem, and local mar-tingales are discussed in Section 3, in particular the existence of the quadraticvariation or square bracket process. Section 4 contains additional technicalmaterial on processes and conditional expectations, including a Fubinitheorem. The DoobMeyer decomposition theorem for submartingales isgiven in Section 5, and some of the special properties of square integrablemartingalesare noted in Section 6. The semigroupof conditioned shifts on thespace of progressiveprocesses is discussed in Section 7. The optional samplingtheorem for martingalesindexed by a metric lattice is given in Section 8.STOCHASTIC PROCESSESAND MARTINGALES1. STOCHASTIC PROCESSESA stochastic process X (or simply a process) with index set 1 and state space(E, a)(a measurable space) defined on a probability space (Cl, 9,P) is afunction defined on 1 x Q with values in E such that for each r E 1,X(t, .): R-+ E is an E-valued random variable, that is, {UJ: X(f, UJ) E r}E .Ffor every E a.We assume throughout that E is a metric space with metric r49Markov Processes Characterizationand ConvergenceEdited by STEWARTN. ETHIER and THOMASG.KURTZCopyright 01986,2005 by John Wiley & Sons,Inc
  59. 59. 50 STOCHfiTIC PROCESS AND MARTINGALESand that 1is the Bore1 a-algebra B(E).As is usually done, we write X(t) andX(t,* ) interchangeably.In this chapter, with the exception of Section 8, we take N = [O, 00). We areprimarily interested in viewing X as a “random” function of time. Conse-quently, it is natural to put further restrictions on X. We say that X ismeasurable if X:[O, 00) x f2-t E is g[O, 00) x $-measurable. We say that Xis (almost surely) continuous (right continuous, lefz continuous) if for (almost)every o E R, X(., w) is continuous (right continuous, left continuous). Notethat the statements “ X is measurable” and “X is continuous” are not parallelin that “X is measurable” is stronger than the statement that X( .,w) ismeasurable for each o E R. The function X(-,a)is called the sample path ofthe process at w.A collection (S,}E {F,,t E LO, 00)) of 0-algebras of sets in F is a fir-tration if 9,c $,+, for t, s E [O, m). Intuitively 9,corresponds to the infor-mation known to an observer at time t. In particular, for a process X wedefine (4:) by 9;= a(X(s):s 5 c); that is, 9: is the information obtainedby observingX up to time t.We occasionally need additional structure on {9J.We say {S,}is rightcontinuous if for each t L 0, SI=,sit,. = r)a,04tlt,. Note the filtration{F,+}is always right continuous (Problem 7). We say (9,)is complete if(a,9,P)is completeand { A E 9: P(A) = 0)c So,A process X is adapted to a filtration {S,)(or simply {F,}-adapted)if X(r)is 6,-measurable for each t L 0. Since6,is increasing in I, X is {$,}-adaptedif and only if 9; c S,for each t 2 0.A process X is {.F,}-progressive (or simply progressive if (9,)= (9:))iffor each t 2 0 the restriction of X to [O,t] x R is &[O,t] x 9,-measurable.Note that if X is {4F,}-progressive,then X is (FJ-adapted and measurable,but the converse is not necessarily the case (see Section 4 however). However,every righf (left) continuous (9J-adapted process is {.F,}-progressive(Problem 1).There are a variety of notions of equivalence between two stochastic pro-cesses. For 0 s f , < t2 < * - * < f,, let p,,,....,-be the probability measure ong ( E ) x - .* x 9 ( E ) induced by the mapping (X(t,),...,X(c,))- Em,that is,p I , * . ..,, ~ r )= P{(X(t,),...,X(t,)) E r}, r E a ( E ) x - - x @(E). The prob-ability measures {p,,,.., , m 2 1, 0 5 t , < * e . < t,} are called the Jinite-dimensional distributions of X. If X and Y are stochastic processes with thesame finite-dimensionaldistributions, then we say Y is a version of X (and X isa version of Y).Note that X and Y need not be defined on the same probabil-ity space. If X and Y are defined on the same probability space and for eachc 2 0, P(X(t)= Y(t)}= 1, then we say Y is a modijication of X. (We areimplicitly assuming that (X(t), Y(t)) is an E x E-valued random variable,which is always the case if E is separable.) If Y is a modification of X,thenclearly Y is a version of A’. Finally if there exists N E 9 such that ON)= 0and X(-,w ) = Y(a , w ) for all w $ N, then we say X and Y are indistinguish-able. If X and Y are indistinguishable,then clearly Y is a modification of X.
  60. 60. 1. STOCHASTIC m o m m SIA random variable T with values in [O, GO] is an {9,}-stopping time if{I s t } E 9,for every t 2 0. (Note that we allow I = 00.) If I < 00 as., we sayI isfinite as. If T s 7 < 00 for some constant T, we say T is bounded. In somesense a stopping time is a random time that is recognizable by an observerwhose informationat time t is 9,.If r is an {PI)-stoppingtime, then for s < r, {T s s} E 9,c 9,,{T < t } =U,(z I; I - l/n} E 9,and (I = t } = {I5 t } - (z< t } e 9,.If T is discrete(i.e.,if there exists a countableset D c [O, 003 such that {IE D)= a),then I isan (9,)-stoppingtime if and only if {I = t } E S,for each t E D n [O, m).1.1 Lemma A [O, 001-valued random variable T is an {Pl+)-stoppingtime ifand only if {I < t} E 9,for every t 2 0.Proof. If { t < t } e 9,for every t z 0, then {I < t +n - I } E St+,-,for n 2 mand { 7 <11 = on{?< t + n u ) E flm91+m-,= .(PI+. The necessity wasobserved above. 01.2 PropositionThen the following hold.Let t l rT ~ ,... be {SF,}-stopping times and let c E [O,oo).(a) rl +c and A c are {9,}-stoppingtimes.(b) sup, I, is an {.F,}-stoppingtime.(c) minks,. rkis an {9,}-stoppingtime for each n 2 1.(d) If (9,)is right continuous, then inf,r,, and I,-are {F,}-stoppingtimes.Proof. We prove (b) and (d) and leave (a) and (c) to the reader. Note that{sup,,I" s t } = on{z, s t } E: PI so (b) follows. Similarly {inf,,?, e t ) =U,{I, < I} E P I ,so if (9,)is right continuous, then inf,?, is a stopping timeby Lemma 1.1. Sinceiimn4rnT,, = ~up,,,inf,,~,,,~,and limn-* z, = inf,sup,,,r,,(d)follows. 0-By Proposition 1.2(a) every stopping time I can be approximated by asequenceof bounded stopping times, that is, limn-mT A n = I. This fact is veryuseful in proving theorems about stopping times. A second equally usefulapproximation is the approximation of arbitrary stopping times by a nonin-creasing sequenceof discrete stoppingtimes.1.3 Propositionand suppose thatand defineFor n = 1, 2,..., let 0 = r: < tl < * - * and limk-rntl: = 00,sup&+ - I;) = 0. Let I be an {F,+}-stoppingtime
  61. 61. 52 STOCHASTIC PRCK€SSES AND MAWlNCALEsThen t, is an {S,}-stopping time and limndm7, = 7. If in addition {I:} t{t;"), then t, 2 tn+l.Recall the intuitive description of 9,as the information known to anobserver at time t. For an (9,)-stopping time 7, the a-algebra9,should havethe sameintuitivemeaning. For technical reasons S,is defined by(1.3)Similarly, PC+is defined by replacing 9,by 9,+.See Problem 6 for somemotivation as to why the definition is reasonable. Given an E-valued processX,define X(a0) c xo for some fixed xo E E.9,= { A E 9:A n ( 7 s t } E 9,for all t 2 0).1.4 Proposition Let t and u be {9,}-stoppingtimes, let y be a nonnegative9,-measurable random variable, and let X be an ($r,}-progressive E-valuedprocess. Define X and Y by Xr(r)= X(7At) and Y(t)= X(7 +r), and define9,= F I h ,and MI = f,,,,t 2 0. (Recall that r h t and .c +r are stoppingtimes.)Then the followinghold:(4 .Fris a u-algebra.(b) T and 7 A u are SP,-measurable.(c) If t 5 usthen F,c F..(d) X(t)is fr-measurablc.(e) {Y,} is a filtration and X is both {gJ-progressive and(f) {Ju;)is a filtration and Y is {J1PIj-progressive.0 7 +- y is an {fJ-stopping time.{#,}-progressive.Proof. (3 Clearly 0and 0 are in PI,since9,is a u-algebra and {r 5 I } EF,.If A A (7 S c} EP,,then A n {t s, t } = (t 5 t ) - A n (7 s t ) E .F,,and hence A E implies A B 9,.Similarly Ak A {s 5; t } E s,,k = I,&. .., implies (UrA,) n (7 s t } = U&(Akn {T I; t } )E S,, andhence f,is closed under countableunions.(1.4) {TAU s c } n {T s r } = { T A U5 c A t } n {T s r }(b) For each c 2 0 and t 2 0,= ( { T 5 c A t } u {a I;cArj) n (t 5 t ) E F,.Hence { f A u 5 c] E .Frand r A d is S,-measurable, as is 7 (takeu = 1).
  62. 62. 1. STOCHASTIC moassEs 53(c) If A E .Ft,then A n {a S t } = A n { t < t } n {IT s t } E 9,for allr 2 0. Hence A E 9#.(d) Fix t 2 0. By (b), T A t is .F,-measurable. Consequently the mappingo - r (t(o)Ar, o)is a measurable mapping of (a,9,)into ([O, r ] x Q,a[O,t ] x 9,)and since X is IF,}-progressive, (s, a)-+X(s, w) is a measur-able mapping of ([0, t ] x R, a[O,t3 x 9,)into (E, 1(E)).Since X ( t A t ) isthe composition of these two mappings, it is .F,-measurable. Finally, forP E @E), {X(r)E r}n { 7 s t } = {X(TA I ) E T} n {T s t } E .F,and henceBy (a) and (c), (Y,} is a filtration, and since 9,c 9,by (c), X is(9,)-progressive if it is {Y,}-progressive.To see that X is (Y,}-progressive,we begin by showing that if s 5 t and H E a[O,t ] x .Fs,then(1.5) H n (10,t ] x { T A t 2 S } ) E taco, t ] x Flh,= a[O,13 x 9,.To verify this, note that the collection X,,, of H E: a[O,t ] x 9,satisfying (1.5) is a a-algebra. Since A E 9,implies A n { T A t 2 s} E F,,,,it followsthat if B E a[O,13 and A E 9,.then(1.6) (B x A) n ([0,t ] x { T A C2 s)){x(t)E rjE 9,.(el= B x ( A n { T A Cz s}) E a[O,C] x Y,,so B x A Ea[O,r] x YS.(1.7) {(s, W ) E LO, t3 x R: x ( T ( ~ ) A ~ ,0))E r}But the collection of B x A of this form generatesFinally, for r E d ( E )and t 2 0,= {(S,W):X(T(W)AS,W)E~,~ ( w ) A r5;sst)= ({(s,w):T(w)At 5 s 5; I } n([O, r ] x {X(TA I)E r}))u {(s, 0):x(s, E r-, s < t(w)A C)since(1.8) {(s, 0)): ~ ( w ) A tI,s t }and since the last set on the right in (1.7) is in a[O,I ] x Y, by (1.5).(0 Again {HI}is a filtration by (a) and (c). Fix r 2 0. By part (e) themapping (s, u)-+X((t(w)+t)As, w) from ([O, 003 x Q, a [ O , 003 x F,,,)
  63. 63. 54 STOCHASTIC CIIOCESSLS AND MARTINGALESinto (E, @E)) is measurable, as is the mapping (u, a)-+(r(w)+ u, 0) from([O, t] x fi, a[O,t] x gFt+J into ([0, 003 x Q, S[O, 003 x gr+J.Themapping (u, a)+ X(T(O)+u, o)from ([O, t ] x Q, a[O,r ] x Yr+Jinto(E, A?(&) is a compositionof the first two mappings so it too is measurable.SinceZ1= F,+,,Y is {X1j-progressive,C@ Let y. = [ny]/n. Note that ( 7 +y. s t } n {y, = k/n} ={ T 5 t -k/n} n (7. = k/n} E 91-r,m,since (7. = k/n} E 9,.Consequently,{ T +y. S t } E 9,.Since 7 +y = SUPAT +7,). part (g) follows by Proposi-tion 1.2(b). 0Let X be an E-valued process and let rE S(E).Thefirst entruncetime intor is defined by(1.9) Te(I‘) = inf (t: X(t)E r}(where inf 0 = m), and for a [O, m]-valued random variable a, the firstentrance time into I‘after u is defined by(1.10) Te(r,0 ) = inf {t 2 u:X(r)e r}.For each w r s n and O S S 5 t, let Fx(s, t, w ) c E be the closure of{X(u, a):s ,< u I; t}. Thejrst contact time with ris defined by(1.1 1) Tc(r)= inf { t : F,(O, t) n r # 0)and thejrst contact time with I’after a by(1.12) q(r,a)= inf { t 2 a: Fx(a,I ) n r it a}.Thefirst exit time from r (after a) is the first entrance time of Iy (after cr).Although intuitively the above times are “recognizable”to our observer, theyare not in general stopping times (or even random variables).We do, however,have the followingresult, which is sufficientfor our purposes.1.5 Proposition Suppose that X is a right continuous, {.F,}-adapted, E-valued process and that d is an {@,}-stoppingtime.(a) If r is closed and X has left limits at each t > 0 or if r is compact,(b) If ris open, then re(r,0 )is an (b,+)-stoppingtime.then Tc(r,a)is an {4tl}-stoppingtime.Proof. Using the right continuityof X,if ris open,(1.13) {t,(r,U ) < t ) = u {x(s)E r)n {U c S} E F,,a 6 0 n l O . 0implying part (b). For n = 1. 2,. ..let r, = {x:r(x, r)< l/n}. Then, under theconditionsofpart (a),zc(I‘, Q) = limn-mre(r,,,a),and(1.14) {rc(r.4s-r}~ P ( ( ~ ~ t } n { x ( t ) ~ r } ) u n . { ~ ~ ( r . , a ) < t }o

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