2. Markov Processes

3. Markov Processes
Characterizationand Convergence
STEWART N. ETHIER
THOMAS G. KURTZ
WILEY-
INTERSCIENCE
A JOHN WILEY & SONS, INC., PUBLICATION

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5. The original aim of this book was a discussion of weak approximation results
for Markov processes. The scope has widened with the recognition that each
technique for verifying weak convergenceis closely tied to a method of charac-
terizing the limiting process. The result is a book with perhaps more pages
devoted 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 on
operator semigroup convergence theorems. Convergence of generators (in an
appropriate sense) implies convergence of the corresponding sernigroups,
which in turn implies convergence of the Markov processes. Trotter’s original
work in this area was motivated in part by diffusion approximations. The
second technique, which is more probabilistic in nature, is based on the mar-
tingale characterization of Markov processes as developed by Stroock and
Varadhan. Here again one must verify convergence of generators, but weak
compactness arguments and the martingale characterization of the limit are
used to complete the proof. The third technique depends on the representation
of the processes as solutions of stochastic equations, and is more in the spirit
of classical analysis. If the equations “converge,” then (one hopes) the solu-
tions converge.
Although the book is intended primarily as a reference, problems are
included in the hope that it will also be useful as a text in a graduate course on
stochastic processes. Such a course might include basic material on stochastic
processes and martingales (Chapter 2, Sections 1-6). an introduction to weak
convergence (Chapter 3, Sections 1-9, omitting some of the more technical
results 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 could
complete the course.
V

6. Vi PREFACE
As an aid to the instructor of such a course, we include a flowchart for all
proofs in the book. Thus, if one's goal is to cover a particular section, the chart
indicates which of the earlier results can be skipped with impunity. (It also
reveals that the courseoutline suggestedabove is not entirelyself-contained.)
Results contained in standard probability texts such as Billingsley (1979) or
Breiman (1968) are assumed and used without reference, as are results from
measure theory and elementary functional analysis. Our standard reference
here is Rudin (1974). Beyond this, our intent has been to make the book
self-contained (an exception being Chapter 8). At points where this has not
seemed feasible, we have included complete references, frequently discussing
the needed material in appendixes.
Many people contributed toward the completion of this project. Cristina
Costantini, Eimear Goggin, S.J. Sheu, and Richard Stockbridge read large
portions 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 in
the book.
We are particularly grateful to our editor, Beatrice Shube, for her patience
and constant encouragement. Finally, we must acknowledge our teachers,
colleagues,and friends at Wisconsin and Michigan State, who have provided
the stimulatingenvironment in which ideas germinateand flourish. They con-
tributed to this work in many uncredited ways. We hope they approve of the
result.
STEWARTN. ETHIER
THOMASG. KURTZ
Salt Lake City, Utah
Madison, Wisconsin
August 198s

7. Introduction
1 Operator Semigroups
Definitions and Basic Properties, 6
The Hille-Yosida Theorem, 10
Cores, 16
Multivalued Operators, 20
Semigroups on Function Spaces, 22
Approximation Theorems, 28
Perturbation Theorems, 37
Problems, 42
Notes, 47
2 Stochastic Processesand Martingales
1
2
3
4
5
6
7
8
9
10
Stochastic Processes, 49
Martingales, 55
Local Martingales, 64
The Projection Theorem, 71
The Doob-Meyer Decomposition, 74
Square Integrable Martingales, 78
Semigroups of Conditioned Shifts, 80
Martingales Indexed by Directed Sets,
Problems, 89
Notes, 93
84
49
vii

8. viii CONTENTS
3 Convergence of Probability Measures
1 The Prohorov Metric, 96
2 Prohorov’sTheorem, 103
3 Weak Convergence, 107
4 Separatingand ConvergenceDeterminingSets, 111
5 The Space D,[O, GO), 116
6 The Compact Setsof DEIO,a), 122
7 Convergencein Distribution in &[O, m), 127
8 Criteria for RelativeCompactnessin DKIO,a), 132
9 Further Criteria for Relative Compactness
in D,[O, oo), 141
10 Convergenceto a Processin C,[O, a), 147
11 Problems, 150
12 Notes, 154
4 Generators and Markov Processes
1 Markov Processes and Transition Functions, 156
2 Markov Jump Processes and Feller Processes, 162
3 The MartingaleProblem: Generalitiesand Sample
Path Properties, 173
4 The Martingale Problem: Uniqueness, the Markov
Property,and Duality, 182
5 The MartingaleProblem: Existence, 196
6 The Martingale Problem: Localization, 216
7 The MartingaleProblem:Generalizations, 22I
8 ConvergenceTheorems, 225
9 Stationary Distributions, 238
10 Perturbation Results, 253
I 1 Problems, 261
12 Notes, 273
5 Stochastic Integral Equations
1 Brownian Motion, 275
2 StochasticIntegrals, 279
3 StochasticIntegral Equations, 290
4 Problems, 302
5 Notes, 305
6 Random Time Changes
1 One-Parameter Random Time Changes, 306
2 Multiparameter Random Time Changes, 311
3 convergence, 321
95
155
275
306

9. 4 Markov Processesin Zd,329
5 Diffusion Processes, 328
6 Problems, 332
7 Notes, 335
7 InvariancePrinciplesand DiffusionApproximations
1 The Martingale Central Limit Theorem, 338
2 Measures of Mixing, 345
3 Central Limit Theorems for Stationary Sequences, 350
4 Diffusion Approximations, 354
5 Strong Approximation Theorems, 356
6 Problems, 360
7 Notes, 364
8 Examplesof Generators
1 NondegenerateDiffusions, 366
2 Degenerate Diffusions, 371
3 Other Processes, 376
4 Problems, 382
5 Notes, 385
9 BranchingProcesses
1 Galton-Watson Processes, 386
2 Two-Type Markov Branching Processes, 392
3 Branching Processes in Random Environments, 396
4 Branching Markov Processes, 400
5 Problems, 407
6 Notes, 409
10 Genetic Models
I The Wright-Fisher Model, 411
2 Applications of the Diffusion Approximation, 415
3 Genotypic-FrequencyModels, 426
4 Infinitely-Many-AlleleModels, 435
5 Problems, 448
6 Notes, 451
11 Density DependentPopulationProcesses
1 Examples, 452
2 Law of Large Numbers and Central Limit Theorem, 455
337
365
386
410
452

10. 3 Diffusion Approximations, 459
4 Hitting Distributions, 464
5 Problems, 466
6 Notes, 467
12 RandomEvolutions
1 Introduction, 468
2 Driving Process in a Compact StateSpace, 472
3 Driving Process in a Noncompact State Space, 479
4 Non-Markovian Driving Process, 483
5 Problems, 491
6 Notes, 491
Appendixes
1 Convergenceof Expectations, 492
2 Uniform Integrability, 493
3 Bounded PointwiseConvergence, 495
4 MonotoneClass Theorems, 496
5 Gronwall’sInequality, 498
6 The Whitney Extension Theorem, 499
7 Approximation by Polynomials, 500
8 Bimeasuresand Transition Functions, 502
9 Tulcea’sTheorem, 504
10 MeasurableSelectionsand Measurabilityof Inverses, 506
11 AnalyticSets, 506
References
Index
Flowchart
168
492
508
521
529

11. 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 robustness
under perturbation arise naturally out of any such characterization. In fact the
model 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 provide
approaches to the characterization of Markov processes, and to each of these
approaches 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 also
becomes a complete, separable metric space. The type of convergence we
are usually concerned with is convergence in distribution; that is, for a
sequence of processes { X J we are interested in conditions under which
limn.+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 in
distribution 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, define
U x ) = 1 + 3x x - - , y,(x) = 3x + +-t>(.- r>.( 1 ) ( :> 1
Markov Processes Characterizationand Convergence
Edited by STEWARTN. ETHIER and THOMASG.KURTZ
Copyright 01986,2005 by John Wiley & Sons,Inc

12. 2 INTRODUCTION
and let U, be a birth-and-death process in b, with transition probabilities
satisfying
(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 the
number of molecules at time t of a substanceR in a volume n undergoing the
chemical 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 process
X to be characterized below.
The first method we consider is based on a semigroup characterization of
X. Let En= {n‘/*(n-‘y - I) :y E Z+},and note that
1 3
3 1
Ro R, R2 +2R S 3R,
We rescale and renormalize letting
(6) ~ w m=Erm.(t)) I x m = XJ
definesa 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

13. 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 2.7
of Chapter 4 and Theorem I. I of Chapter 8, there exists a diffusion process X
corresponding to (T(t)),that is, a strong Markov process X with continuous
sample 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), it
suffices by Corollary 8.7 of Chapter 4 to show that (10) holds for all/in zt core
D for the generator A, that is, for allf in a subspace D of 9 ( A ) such that A is
the 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 check
that
(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 s
f 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 that
and put h,(x) = h(x/m).Given
SdX) =S(0) - do) + (j-gY(Y)hm( Y1d ~ .
s:
The second method is based on the characterization of X as the solution of
is an {.Ffn)-martingalefor each /E B(E,) with compact support. Conse-
quently, if some subsequence {A’,,,) converges in distribution to X , then, by the
continuous mapping theorem (Corollary 1.9 of Chapter 3) and Problem 7 of
Chapter 7,

14. 4 ~ O W c r I o N
is an {Pf)-martingale for eachfe C,'(R), or in other words, X is a solution of
the martingale problem for {(AG f ) : f cC,'(W)}. But by Theorem 2.3 of
Chapter 8, this property characterizes the distribution on Dn[O, 00) of X .
Therefore,Corollary 8.16 of Chapter 4 gives X,=X (assumingconvergenceof
initial distributions),provided we can show that
Let (p(x) I ex +e-x, and check that there exist constants C , , a O such.that
C,,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 have
Iinf C P ( Y ) ~SUP Ixn(t)lka
{ostsr
e-G.4 T
Irl L a
(19)
ELeXP -Cn,a(?n, 8 A 73)cp(Xn(Tn, a A VJ
5 QdXn(O))l
by 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 uniform
convergence of G,f to Gf forfe C:(R) and the uniqueness for the limiting
martingale problem imply (again by Corollary 8.16 of Chapter 4) that X, =. X
in 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 sample
paths in DRIO,00).
Both of the approachesconsidered so far have involvedcharacterizationsin
terms 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 the
unique solutionof the stochasticintegralequation
where W is a standard one-dimensional, Brownian motion. (In the present
example, the term 2JW(t) corresponds to the stochastic integral term.) A
convergencetheory can be developed using this characterization of X,but we
do not do so here. The interested reader is referred to Kushner(1974).
The final approach we discuss is based on a characterizationof X involving
random time changes. We observe first that U,satisfies

15. where N, and N- are independent, standard (parameter I), Poisson processes.
ConsequentIy,X,satisfies
X,(r) = X,(O) +n- 3/4R+(n3/2 A,( I+n- '/*X,(s))ds(22)
- n-"'R.(nl" 6'p,(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 -(n3'2.))=.(W+,W-1,
where W+ and W- are independent, standard, one-dimensional Brownian
motions. Consequently, if some subsequence {A'".) converges in distribution to
X, one might expect that
X(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 in
general.) Clearly, (24) characterizes X,and using the estimate (18) we conclude
X,-X (assuming convergence of initial distributions) from Theorem 5.4 of
Chapter 6.
For a further discussion of the Schlogl model and related models see
Schlogl (1972) and Malek-Mansour et al. (1981). The martingale proof of
convergence is from Costantini and Nappo (1982), and the time change proof
is 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 1-3
contain preliminary material on operator semigroups, martingales, and weak
convergence, and Chapters 8- I 2 are concerned with applications.

16. 1
Operator semigroups provide a primary tool in the study of Msrkov pro-
cesses. In this chapter we develop the basic background for their study and the
existence and approximation results that are used later as the basis for exis-
tence and approximation theorems for Markov processes. Section 1 gives the
basic definitions,and Section 2 the Hille-Yosida theorem, which characterizes
the operators that are generators of semigroups. Section 3 concerns the
problem of verifying the hypotheses of this theorem, and Sections4 and 5 are
devoted to generalizations of the concept of the generator. Sections 6 and 7
present the approximationand perturbation resuJts.
Throughout the chapter, L denotesa real Banach space with norm 11 * 11.
OPERATOR SEMICROUPS
1. DEFINITIONS AND BASIC PROPERRES
A one-parameter family { T(t):t 2 0) of bounded linear operators on a
Banach space L is called a semigroupif T(0)= I and T(s+t ) = T(s)T(c)for all
s, 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 all
t 2 0.
Given a bounded linear operator B on L,define
Markov Processes Characterizationand Convergence
Edited by STEWARTN. ETHIER and THOMASG.KURTZ
Copyright 01986,2005 by John Wiley & Sons,Inc

17. 1. DmNmoNz AND EASIC ?ROPERTIES 7
A simple calculation gives e'"')' = e""e'' for all s,t 2 0, and hence {e'"} is a
semigroup, which can easily be seen to be strongly continuous. Furthermore
we have
An inequality of this type holds in general for strongly continuous serni-
groups.
1.1 Proposition
there 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. Then
Proof. Note first that there exist constants M 2 I and ro > 0 such that
11 T(t)11 5 M for 0 I t s t o . For if not, we could find a sequence (t,} of positive
numbers tending to zero such that 11 T(t,)((-+ 00, but then the uniform
boundedness 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 Corollary
each$€ L, t -+ T(t)/is a continuousfunction from [0, 00) into L.
Let {T(r))be a stronglycontinuoussemigroupon L.Then, for
1.3 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 strongly
continuoussemigroupon L such that
(1.7) IIW II s M, t 2 0.

18. 8 OraATORS€MIGROWS
In 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 by
Then 11f11 5; IIIJIII 5; Mllfll for eachfE L, so the new norm is equivalent to
the original norm; also, with respect to 111 * 111, {S(t)) is a strongly continuous
contractionsemigroupon L.
Most of the results in the subsequent sections of this chapter are stated in
terms of strongly continuous contraction semigroups. Using these reductions,
however, many of them can be reformulated in terms of noncontraction semi-
groups. 0
A (possibly unbounded) linear operator A on L is a linear mapping whose
domain 9 ( A ) is a subspaceof L and whose range a ( A ) lies in L. The graph of
A is given by
Note that L x L is itself a Banach space with componentwise addition and
scalar multiplication and norm [l(J @)[I= llfll + IIg 11. A is said to be closed if
9 ( 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)
1
A , = lim ;{T(t)f-J}.
1-0
The 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 the
calculus of Banach space-valued functions.
Let A be a closed interval in (- 00, a),and denote by CJA) the space of
continuous 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)integrable
over 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 or
u(t)dt. If A = [a, a),u: A + L is said to be integrable over A if u I , ~ , ~ ,is
integrable over [a, b] for each b 2 a and limg,, Jtu(t)dt exists; again, the
limit is denoted by {A ~ ( t )dt or {; u(r)dt.
We leave the proof of the followinglemma to the reader (Problem3).

19. 1. MflMTlONS AND 8ASlC PROPERTIES 9
1.4 Lemma (a) If u E C,jA) and JAll u(t)I1 dt < 00, then u is integrable over
A and
(1.1 I )
In particular, if A is the finite interval [a, 61, then every function in C,(A) is
integrable 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 over
A. 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 with
generator 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, then
Iff€ 9 ( A )and t 2 0. then T(t)/EB(A)and
d
--r(t)j=A T ( t ) / = T(r)AJ
dt
(1.16) T(t)J-j= A T(.s)jds = T(s)Afds.
Proof. (a) Observe that
for all h > 0, and as h -,0 the right side of(I.17)converges to T(t)/-f:

20. 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). 0
1.6 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$,4f and AS,- g. Then T(r)f,-Jn = roT(s)AJnds for each t > 0, so,
letting n-+ a,we find that T(r)f-f= 6 T(s)gds. Dividing by t and letting
0I-+ 0, we concludethatje 9 ( A )and Af= g.
2. THE HILL€-YOSIDA THEORfM
Let A be a closed linear operator on L. If, for some real 2, A - A (K A1 - A) is
one-to-one, W ( l -A) = L, and (1-A)-' is a bounded linear operator on L,
then 1 is said to belong to the resoluent set p(A) of A, and RA = (A -A)-' is
called the resoluenr (at A) of A.
2.1 Proposition Let {T(I))be a strongly continuous contraction semigroup
on L with generator A. Then (0,00) c p(A) and
(2.1) (A-A)-'g = e-A'T(tb dr
for all g E L and d > 0.
Proof. Let 1 > 0 be arbitrary. Define U, on L by UAg = J$e-"T(t)g df.
Since
(2.2)
0)
It U ~ g l l Lrne-"'l/ T(r)sll df 9~-'llgll

21. 2. THE HILLL-YOSIDA THEOREM 11
foreach 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 is
complete. 0
Let 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, this
implies that p(A) is open in R.
A linear operator A on L is said to be dissipative if IIJ j - AjII 2 Allfll for
every/€ B(A)and I > 0.
2.2 lemma Let A be a dissipative linear operator on L and let 1 > 0. Then
A 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

22. 12 OPERATORSEMICRourS
L.+J and hence Al,,--+ Af - h. Since A is closed,fe 9 ( A ) and h = (A - A)J It
followsthat @(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 put
p+(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"}c
p+(A)and A,-+ A > 0. Given g E L,let g,, = (A - AKA, - A)-'g for each ti, and
note 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 the
dissipativity of A once again, we conclude that I -A is one-to-one and
II(A -A)-'(I s I - ' . It follows that 1 B p+(A),so p+(A)is closed in (0, a),as
I - a l *-.OD n-al 4
required. 0
2.4 lemma Let A be a dissipativeclosed linear operator on L, and suppose
that 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,=A'R,-Al on L, A > O ,
and
For each R > 0. let R, = (A-A)- ' and note that 11R, 11 5 A - I . Since

23. 2. T M HILL€-YOSIDA THEOREM 13
for all t 2: 0, proving (a).Conclusion (b) is a consequenceof (2.10)and (2.7). As
for (c),we claim first that
(2.13) lim I R , f = f , SE L.
d-+m
Noting that llLRaf-lll = IIRAAfll s A-'I(A/II 4 0 as A+ a, for each
f e 9 ( A ) , (2.13) follows from the facts that 9 ( A ) is dense in L and
lll.Ra - Ill S 2 for all 1 > 0. Finally, (c) is a consequence of (2.1 I) and
(2.I 3). 0
2.5 lemma If B and C are bounded linear operators on L such that
BC = 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/I1
for 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.) 0
We 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 and
Proposition 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, so
by Lemma 2.3, (0, m) c p(A). Using the notation of Lemma 2.4, we define for
each L > 0 the strongly continuous contraction semigroup {T'(c)} on L by
K(t)= erAA.By Lemmas 2.4b) and 2.5,
(2.16) IInw- q(t)/ll 111AJ- AJll

24. 14 OrUATOROMCROUIS
for all f~ L, t 2 0, and A, p > 0. Thus, by Lemma 2.4(c), limA*mT,(t)/exists
for-all t 2 0, uniformly on bounded intervals, for allfe 9(A), hence for every
f~ 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) yields
T,(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)}is
an 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 the
proof. 0
The above proof and Proposition 2.9 below yield the followingresult as a
by-product.
2.7 Proposition Let {T(t)}be a strongly continuous contraction semigroup
on L with generator A, and let Ad be the Yosida approximation of A (defined
in Lemma 2.4). Then
(2.21)
so, for each fE L, liniA-,me'"1/= T(r)ffor all I 2 0, uniformly on bounded
intervals.
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 semigroupon
L 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 is
a closed subspaceof L and AM is nonempty, then
(2.23) T(t):M-+M, t 2 0.

25. 1. TH€ HNLE-VOSIDA THEOREM 1s
Proof. 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 t
A,,, .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. 0
2.9 Proposition Let { T(t)} and {S(t)} be strongly continuous contraction
semigroups on L with generators A and B, respectively. If A = B, then
T(t)= S(t) for all r 2 0.
Proof. This result is a consequenceof the next proposition. 0
2.10 Proposition Let A be a dissipative linear operator on L. Suppose that
u : [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.

26. 16 OPERATOR SEMlCROUrS
where the first inequality is due to the dissipativity of A. The result follows
from the continuity of Au and u by first letting max (t, -ti- ,)+ 0 and then
lettingc+ 0. 0
In many applications, an alternative form of the Hille-Yosida theorem is
more 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 linear
extension. If A is closable, then the closure A of A is the minimal closed linear
extension of A; more specifically, it is the closed linear operator 6 whose
graph 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 ( A
L.Then A is ciosableand L@(A -A) =9?(A -A^)forevery I > 0.
dense in
Proof. 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 the
dissipativity of A,
(2.28) IIV - - 4It = lim II(A- A h , + &)I1
a-m
2 lim AIlgm + KII AIIgmII
n- m
for every 1 >0 and each m. Dividing by I and letting A+ 00, we find that
IIg, -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 prove
equality, we need only show that 5?(I -A) is closed. But this is an immediate
consequenceof Lemma 2.2. 0
2.12 Theorem A linear operator A on L is closable and its closure A is the
generator 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 and
A’ satisfies(a)+) of Theorem 2.6. a
3. CORES
In this section we introduce a concept that is of considerable importance in
Sections6 and 7.

27. Let A be a closed linear operator on L. A subspace D of 9 ( A ) is said to be a
core for A if the closure of the restriction of A to D is equal to A (i.e., if
AJ, = A).
-
3.1 Proposition Let A be the generator of a strongly continuous contraction
semigroup on L. Then a subspace D of 9 ( A )is a core for A if and only if D is
dense 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). 0
Proof. The sufficiency follows from Theorem 2.12 and from the observation
that, if A and B generate strongly continuous contraction semigroups on L
and if A is an extension of 8, then A = B. The necessity depends on Lemma
2.1 1. 0
3.3 Proposition Let A be the generator of a strongly continuous contraction
semigroup IT([)}on L. Let Do and D be dense subspaces of L with Do c D c
9 ( 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)
I
lim (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. 0
Given a dissipative linear operator A with 9 ( A )dense in L, one often wants
to show that A generates a strongly continuous contraction semigroup on L.
By Theorem 2.12, a necessary and sufficient condition is that .%(A - A ) be
dense in L for some A > 0. We can view this problem as one of characterizing
a core (namely, g ( A ) )for the generator of a strongly continuous contraction
semigroup, except that, unlike the situation in Propositions 3.1 and 3.3, the
generator is not provided in advance. Thus, the remainder of this section is
primarily concerned with verifying the range condition (condition (c)) of
Theorem 2.12.
Observe that the followingresult generalizes Proposition 3.3.

28. 18 OrUATOR YMIGROUK
3.4 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 a
continuous function u,: [O, 00)" L such that u,(O) =1; u,(t) E .@(A) for all
r > 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 strongly
continuouscontraction semigroup {T(f)}on L,and T(t)J= u,(t) for allfE Do
and r 2 0.
Proof. By Lemma 2.11, A is closable. Fix f~ Do and denote uf by u. Let
to > 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)
I0
I'"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) ds
I'"
Since IIu(t)(l5 llfll for all t 2 0 by Proposition 2.10, we can let 6-0 and
to-+ 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 each
fE Do.
(3.7) W f- W f=I'm4m
for all t > E > 0. Subtracting (3.3) from this and applying Proposition 2.10
0once again,we obtain the second conclusion of the proposition.
The next result shows that a suficient condition for A' to generate is that A
be triangulizable. Of course, this is a very restrictive assumption, but it is
occasionallysatisfied.

29. 3. CORES 19
3.5 Proposition Let A be a dissipative linear operator on L, and suppose
that L,, L,, L 3 ,...is a sequence of finite-dimensionalsubspaces of 9 ( A )such
that u."-,L, is dense in L. If A : L , 4 L, for n = I, 2, . ..,then A is closable
and the closure of A generates a strongly continuous contraction semigroup
on L.
Proof. For n = 1, 2, .. .,(A - AWL,) L, for all 1 not belonging to the set of
eigenvalues 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 > 0
and in particular for some A > 0. Thus, the conditions of Theorem 2.12 are
satisfied. C3
We 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 stating
the 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 subspace
of 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 and
lim [[(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 the
lemma, and observe that limn-mII(A - A)J, -g,II -- 0 by (3.8). It follows that
0limn+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, dense
subspaces 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 contraction
semigroup IT&)) on L and d ( A )c O(A,). Suppose further that there exist
w 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 continuous
contraction semigroupon L.

30. 20 OPERATOISMCROUPS
Proof. Observe first that O(A)is dense in L and, by (3.9) and the dissipativity
of each A,, A is dissipative. It therefore sufices to verify condition (c) of
Theorem 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 of
positiveintegerssuch that (A -A,,)S,,,-+ gas n--, 03. Moreover,
(3.13) It(An -.Alfm., n II 111fm. n 111
M 2
k = O
Illg1115 enm,-1 C e- Wa&h
- 0 as n+m
by (3.9) and (3.10), so Lemma 3.6 gives the desired conclusion. 0
3.8 Corollary Let A be a linear operator on L with B(A) dense in L, and let
Ill * 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 that
T,: 9(A)-+ 9 ( A ) , and define A, = n(T, - I). Suppose there exist w 2 0 and a
sequence {t,} c (0, a)tending to zero such that, for n = 1, 2, ...,(3.9) holds
and
(3.14)
Then A is closable and the closure of A generates a strongly continuous
contraction 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{all
for all t 2 0, so the hypothesesof the proposition are satisfied. 0
4. MULTlVAlUED OPERATORS
Recall that if A is a linear operator on L,then the graph g(A) of A is a
subspace of L x L such that (0,g) E g(A) implies g = 0. More generally, we
regard an arbitrary subset A of L x L as a multiualued operator on L with
domain 9 ( A ) = {/: (J g) E A for some g } and range *(A) = (g: (JI g ) e A for
some/}. A c L x L is said to be linear if A is a subspace of L x L. If A is
linear, then A is said to be sinyfe-uaiuedif (0, g) E A impliesg = 0; in chis case,

31. 4. MULTIVALUED OPERATORS 21
A 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 course
just the closure in L x L of the subspace A. Finally, we define
1 - A = ((JAf- g): (Jg) E A } for each 1> 0.
Observe that a (single-valued)linear operator A is closable if and only if the
closure 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 contraction
semigroup is a maximal dissipative (multivalued)linear operator.
4.1 Proposition Let A be the generator of a strongly continuous contraction
semigroup on L. Let B c L x L be linear and dissipative, and suppose that
A 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) E
1 - 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 every
1. > 0 by the dissipativity of A’. Dividing by 1 and letting A- a,we find that
Ilg,, - 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 assertion
of Lemma 2.I I. 0
The main result of this section is the following version of the Hille-Yosida
theorem.
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 semigroup
on 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 the
Hille-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

32. 22 OPERATOR SEMIGROUPS
9 ( L - A)=3 a(A)for some A >0. which by Lemma 4.2 is equivalent to
41(1 - A) 3 d(A)for some 1->0. Thus, to complete the proof,-it suffices to
show that 9 ( A o ) is dense in 9 ( A ) assuming that 5?(A - A,) = B(A)for some
1 > 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 norm
at 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 view
0of (4.2), this completesthe proof.
Milltivalued operators arise naturally in several ways. For example, the
followingconcept is crucial in Sections6 and 7.
For n = 1, 2, ..., let L,, in addition to L, be a Banach space with norm
also denoted by 11 * 11, and let n,: L-. L, be a bo'unded linear transformation.
Assume that sup, IIn,,II < 00. If A, c L, x L, is linear for each n 2 I, the
extended 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 each
n-m
n 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 in
L x L (Problem 11).
To see that ex-lim,,,A, need not be single-valued even if each A, is, let
L, = L, a, = I, and A, = B +nC for each n 2 1, where B and C are bounded
linear 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-m
Another situation in which multivalued operators arise is described in the
next section.
5. SEMIGROUPS ON FUNCTION SPACES
In 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

33. 5. SEMICROUIS ON FUNCllON SPACES 23
not continuous and to which the Riemann integral of Section 1 does not
apply. For our purposes, the most efficient way to get around this difficulty is
to restrict the class of Banach spaces L under consideration. We therefore
assume in this section that L is a “function space” that arises in the following
way.
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 such
that
(5.1)
Note that 11. [I is a seminorm on Y but need not be a norm. Let
N = { f ~9’:llfll = 0) and let L be the quotient space 9 / N ,that is, L is the
space of equivalence classes of functions in 9,wheref- g if I[/- gll = 0. As
is typically the case in discussions of Lp-spaces, we do not distinguishbetween
a function in Y and its equivalenceclass in L unless necessary.
L is a Banach space, the completenessfollowing as for E-spaces. In fact, if v
is a o-finite measureon A’, 1 s q 5 ao,p-’ +q-’ = 1, and
IlSIl --= 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 of
probability 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 all
s 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 in
2’equivalent to h, where
(5.4)
With the above in mind, we say that u : S-+ L is measurable if there exists
an Y x A-measurable function u such that u(s, .) E u(s) for each s E S.We
define a semigroup (T(t)}on t to be measurable if T( * )J is measurable as a
function on ([O, m), a[O,00)) for each/€ L. We define thefull generaror A’ of
a measurablecontraction semigroup (T(r)}on L by

34. We note that A is not, in general,single-valued.For example, if L = B(R)with
the sup norm and T(t)f(x)s f ( x +t), then (0, g) E A^ for each y E B(R) that is
zero 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 satisfies
for all h E W(A-A)and A > 0. If
T(s) e-"T(t)h dt = I"e-"T(s +t)h dt
0
(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
=J
Consequently, IlflI s A- '11 h 11, proving dissipativity,and (5.6)holds.
g = 4.j- h. Then
(5.9) T(s)gds = 1
Assuming (5.7), let h E L and A >0, and define f- e-"T(t)hdt and
lme-'"T(s +u)h du ds - T(s)h ds
= I en*ime-"T(u)hdu ds - T(s)hds
= el'
SI
l
e-'"T(u)h du -1."e-AuT(u)hdu
+ T(s)hds - T(s)hds
= Wf-f
for all t 2 0,soU;g) E Aand h = Af-g E SI(A - A). 0

35. 5. SEMKROUrJONFUNCllONWACES 25
The following proposition,which is analogousto Proposition I.s(a), gives a
useful 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)ds
for all r 2 0. 0
In 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 and
u: [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.
l
One would expect u to be continuous, and since A is closed and linear, it is
reasonableto expect that
for 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 for
arbitraryL.

36. 26 OIflAlOISEMKiROUrS
5.3 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 each
t > 0.Then
(5.15)
for all t 2 0. Given I > 0, define
IIu(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 that
IIu,(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.The
equation 1f-g = u(0)follows immediately from the definitionoffand g. 0
u(s) ds. Then
u,(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)/ds
0
u(t) = S(t)u(O) +5'S(t -s)Au(s)ds.
0
5.4 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

37. 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)ds
c
5.5 Remark The above result holds in an arbitrary Banach space under the
assumption 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,
n
IIu(t)I1= II 40)II + 1cI14tO I1 - II44- I ) Ill

38. 28 O?ERATORJEMK;ROU‘S
where s’= t,- I and s” = t, for r,- I ss < r,. Since the integrand on the right is
bounded 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, put
u(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) ds
U#) = e-I
S(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 ds
0 0
+ I’r S ( t -r)u(r+s) dr ds
0 0
1= S(t)[.s-I S(s)u(O)ds +6 - l 5’I’S(s - r)u(r)dr ds
0 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. 0
6. APPROXIMATION THEOREMS
In this section, we adopt the following conventions. For n = 1, 2, ...,L,,in
addition 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 continuous
contraction semigroups on L, and L with generators A, and A. Let D be a
core for A. Then the following are equivalent:
(3
intervals.
For each1E L,T,(t)n,f-+ T(r)ffor all t 2 0, uniformly on bounded

39. 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 that
j,,-.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 of
which 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,
L
IISn(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. 0
6.3 Lemma Suppose that the hypotheses of Theorem 6.1 are satisfied
together with condition (c) of that theorem. For n = 1, 2,. ..and R > 0, let At
and A' be the Yosida approximations of A, and A (cf. Lemma 2.4). Then
A: n, f-+Ayfor everyfe L and R > 0.
Proof. Fix R > 0. Let /E D and g =(A - A)f By assumption, there exists
1;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)-'eIt
s R211(R - A n F 1ring -Lit + R'ItSn - nn(R - A)-'gII
5 LIInng - ( A - An)/nII + nZII/n - nSII
for every n 2 I. Consequently, 11A: n,g - R, A'g 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.
0
Proof of Theorem 6.1. (a *b) Immediate.

40. 30 OPERATOR SEMICROWS
(b =5 c) Let 1 > 0.fE 4W), and g = (A - A)A so that f= e-"'T(t)g
dt. For each n 2 1, put fn = jz e-"X(t)n,,g dr E B(A,). By (b) and the
dominated 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 Yosida
approximationsA: 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)fJ
forevery 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 11
n- w 0 sI sfo n-m
(6.5)
lim to{ IIAn S. - nn MII + IInn(AS- AWII
n - m
+ IInnAY- AfnnfII + I I A ~ ~ ~ . ~ - L ) I I I
s 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)fII
n-m OLILIO
s lim II(R."n. - n,A")T"s)Jl/ ds = 0.
n-m
Applying(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 o
Since 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.
0
There is a discrete-parameter analogue of Theorem 6.1, the proof of which
dependson the followinglemma.
6.4 lemma Let B be a linear contraction on L.Then
(6.8) IIBY- en(8-'Yll 5 J;;IIBJ-JII
for allfs L and n = 0, 1,. ...

41. 6. APFUOXIMATION THEOREMS 31
Proof. 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 random
0variable 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, be
a 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 bounded
For 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)e
dt. For each n 2 I, put
(6.1I )

42. 32 OPERATORSMCROUIS
By (b) and the dominated convergence theorem,L-+J and a simple calcu-
lation shows that
(6.12) (1-AalL = nag -trlE,nag
a3
+ - 1 +e-Aca) e-A*cnT~+'n,g
k = O
for 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). Then
and by Theorem6.1,
(6.15)
Consequently,
(6.16) lim sup 11 T~l'aln,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 OSCSIO
n-m 051510
0
6.6 Corollary Let {V(t):f 2 0) be a family of linear contractions on
L with V(0)= I, and let {T(r)} be a strongly continuous contraction
semigroup 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 all
r r:0, uniformly on bounded intervals.
Proof. It sunices to show that if {tn) is a sequence of positive numbers such
that in-* r 2 0, then V(t,,/n)"'+ T(t)ffor everyfe t.But this is an immediate
consequenceof Theorem 6.5 with T.= V(tJn) and E, = tJn for each n 2 I. 0

43. 6. APPROXlMATltM THEOREMS 33
6.7 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 a
core 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 a
sequenceof 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 semigroupon
L 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 positive
numbers tending to zero, then, for each f e t,(I -E,,A)-~"'~Y--+T(t)Jfor all
t ;r 0, uniformly on bounded intervals.
Proof. The first result is a consequence of Corollary 6.6. Simply take
V(i)= (I - tA)-' for each f 2 0, and note that if E > 0 and 1 = E - ' , then
where AI is the Yosida approximation of A (cf. Lemma 2.4). The second result
0follows from (6.19) and Theorem 6.5.
We would now like to generalizeTheorem 6.1 in two ways. First, we would
like to be able to use some extension A, of the generator A, in verifying the
conditions for convergence. That is, given U;g) E A, it may be possible to find
u,,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, we
would like to consider notions of convergence other than norm convergence.
For example, convergence of bounded sequences of functions pointwise or
uniformly on compact sets may be more appropriate than uniform con-
vergencefor some applications.An analogous generalization of Theorem 6.5 is
also given.

44. 34 N TORS EM CROUPS
Let 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 imply
LIM (aJ;+Pg,) = cf+ /?g for all a, /3 E R.
(6.21) LIMf:) = f k ) for each k 2 1 and
lim sup ll/!hJ -J, 11 V llj4kJ-/[I = 0 imply LIMA, =/:
There exists K >0 such that for eachfe L,there is a
sequenceA, E L, with Ilf.11 s KIIfII, n = 1, 2,.. .,satisfying
LIML =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 linear
and 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 continuous
contraction 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 (also
denoted 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 closed
in L x L (Problem 16). Consequently, the conclusion of (a) is valid for all
UI Q)E A’. 0
Proof. 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 by
I I ( { L J s f)II= SUPnz1111; IIV IIf II, and let
(6.26)

45. 6. APFROXlMATlON THKMFMS 35
Conditions (6.20)and (6.21) imply that Yois a closed subspacc of 9,and
Condition (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 each
n 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, then
To prove this, since A c ex-LIM A,,, choose c/"., 8,) E A, for each n 2 1 such
that 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 s
1-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 the
conclusion 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

46. for each k 2 1. Since
(6.34)
and
for each n 2 1 and k 2 1, (6.21) implies that
(6.36)
Q, m
LIM 1u!'~==A LZM T,(t)Cut)= T(t)J;
I I
so the conclusionof (b)followsfrom (6.25). 0
6.11 Theorem For n = 1, 2,..., let T, be a linear contraction on L,, let
E, > 0, and put A, = &;'(T, -I). Assume that limn-mc,, = 0. Let A c L x L
bc linear and dissipative with 9?(1 -A) = L for some (henceall) 1 > 0, and let
IT(t)} be the corresponding strongly continuous contraction semigroup on
9(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 c'h!&= 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 such
that SUp,!lfn I1 < a, sup,II Af,Il < 00, LIMf, -S, LIM AS, = g, and
LJM e'"X = T(r)Jfor all t 2 0.Since
(6.39)

47. 7. NRTUROATION THEOREMS 37
for 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). 0
7. PERTURBATION THEOREMS
One 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 generate
semigroups.(By definition, O(A+ B)= O(A)n 9(B).)First, however, we give
some 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-valued
and generates a strongly continuous contraction semigroup on L.Let B be a
dissipative linear operator on L such that 9 ( B )3 9(A). (In particular, 6 is
single-valuedby Lemma 4.2.) If
where 0 5 a c I and /I2 0, then A + B is single-valued and generates a
strongly 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. In
addition, 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 )and
A > 0. then
by Lemma 24c)and the dissipativity of yB.

48. 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 show
that 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)J
a a a
Let
1 - ay
y E r n Lo, 1) implies [y, y -+ 7 )c r
To prove (7.6), let y E r n [O, I), 0 5 E < (2a)-'(l -ay), and L > 0. If
g 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 ,that
I -
11lgsrr 5: all 4.31+811f11 dl(A+rb)fll +aril mr +Plifli
Ilj3Jll 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)-')
= L
forsuch 6, so y +E E r,implying(7.6) and completingthe proof. 0
7.2 Corollary If A generates a strongly continuous contraction semigroup
on L and E is a bounded linear operator on L,then A + B generates a
stronglycontinuoussemigroup {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

49. Before turning to limit theorems, we state the following lemma, the proof of
which is left to the reader (Problem 18). For an operator A, let
M ( 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 calculation
shows that (7.12) is equivalent to
m
(7.14) lim (I - p) 1 pkQL/=Pf exists for all /E L. 0
p - l - k = O
7.5 Remark
holds and
If in the lemma lim,+mS(r)( exists for every /E L, then (7.12)
(7.15) Pf = lim S(i)J / E L.
t-m
If 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-m
The proofs of these assertionsare elementary. 0
For 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 sequenceof
operators (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

50. 40 OPERATORS€MIGROWS
of 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)is
single-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)ffor
all 2 0, uniformly on bounded intervals.
(b) If A, = E,-I(T, -I) for each n 2 1, where T. is a linear contraction
on L, and E, >0, and if lim,,,~, =0, then, for eachfE D, T!'%, f-. T(f)f
for 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,. Given
U;g) B A with ftz D, choosef . E 9(An) for each n 2 1 such that fa- f and
A,f,-, g. Given h E D', choose h, E B(A,) for each n 2 I such that h,+ h and
a,- '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-m
completingthe proof. 0
We conclude this section with two corollaries. The first one extends the
conclusions of Theorem 7.6, and the other describesan important special case
of 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,

51. 7. PERTUIIATION THEOREMS 41
then 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,. ..} satisfies
k,a, E, = m, then TFn,h -+ 0. Consequently, for eachf E P - '(6)and
6 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), the
other 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. Then
for all n suficiently large by (7.18) and Theorem 6.1. If J E L, then
f - Pf E .N(P), so 7Jrn)n,(J- Pf)+ 0 whenever {t,} c LO, 00) satisfies
t, = r # 0. If f e P-'(d), this, together with the conclusion of the theorem
0applied to PJ completesthe proof.
7.8 Corollary Let ll,A, and B be linear operators on L such that B generates
a 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 +a'E generates a strongly continuous
contraction 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 define
Then 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 bounded
intervals.
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. and
D = 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

52. 42 OIEIATORSMGROUIS
a,-2A,h = Bh for all h E D', we find that (7.17) and (7.18) hold, so the theorem
is applicable. The dissipativity of C followsfrom the dissipativity of ex-lim,,,
A". 0
7.9 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 equation
for 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)(since
B is closed) and satisfies Bh = -A$ Of course, the requirement that h
belong to 9(n)A 9 ( A ) must also be satisfied.
(c) When applying Corollary 7.8, it is not necessary to determine C
explicitly. instead, suppose a linear operator Co on b can be found such
that Cogenerates a strongly continuous contraction semigroup on b and
Coc 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 closely
related result. 0
8.
1.
2.
3.
4.
PROBLEMS
Define {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) by
for each r >0 and T(0)= I. Show that {T(t)}is a strongly continuous
contraction semigroupon L,and determineits generator A.
Prove Lemma 1.4.
Let (T(r)}be a strongly continuous contraction semigroup on L with
generator A, and let/€ 9(A2).
(a) Prove that
Jo

53. a m o m 43
(b) Show that IIASII' 5 411A'JII 11/11.
Let A generate a strongly continuous semigroup on L. Show that fl.i I
9 ( 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 C'CO, 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 hold
for all k,I, n 2 1.
be a sequence of closed subspaces of L. Let
0,. M,,and MP' be bounded linear operators on L. Assume that u, and
Mp)map L,into L, ,and that for some fl, >0, IIMP'II <fi, and
9 ( A ) -- C,(Pa), L = Qua).
Let L, c L, c L, c *
5.
6.
7.
8.
lim I(Mf""- M,1) = 0.
r))" m
Suppose that the restriction of A, I
that there exist nonnegativeconstants dlk( (= a(,), & I , and y such that
(8.4)
Mf"Uj to L, is dissipative and
f 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 a
stronglycontinuouscontractionsemigroupon L.
I Mj[I,on
1
OD W
~ ( A I= {I. u Ln: 1 fijllujflI < 00 .
n = l j = J

54. 4 (wMToII6McROWS
Hint: Fix A > 3y and apply Lemma 3.6.Show first that for g E 9 ( A ) and
f n * (a - AA-’g,
n
1-1
(8.9) (a-Y)IlUd~ll IIuhgll + (fikJ+r,akj)lluj/;ll.
Denoting by p the positive measure on the set of positive integers that
gives 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, which
yields 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 norm
on D with respect to which D is a Banach space. Assume that 111fIII 2 11/11
for 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 and
generatesa strongly continuouscontractionsemigroupon L.
We remark that only one of the two conditions (8.11) and (8.12) is
really needed.See Ethier (1976).
10. Define the bounded linear operator E on L =C([O, 13 x [O, 11) by
Bf(x,y) =
(8.15)
f(x, z) dz, and defineA c L x L by
A = {Ut/,=+W:SEC2(C0,13 x CO, 11)n W ? A
fA0, 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 of
ex-lim,,, A,?
13. In Theorem 6.1 (and Theorem 6.5).show that (a)-+) are equivalent to the
following:

55. a. raocmts 45
(d) There exists 1 > 0 such that (A - A,)-’n,,g+(1 - A ) - ’ g for all
g E L.
14. Let L, {L,,},and In,) be as in Section 6. For each n 2 1, let {T,(t)) be a
contraction semigroup on L,, or, for each n 1 I, let (T,(r)} be defined in
terms 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,4m
T(t)j= 8 and
for all t 2 0; in the latter case assume that
(8.16) lim sup I[7Jr)nJ- n, T(r)jII = 0
for 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 limit
theorem. That is, if X,,X,,... are independent, identically distributed,
real-valued random variables with mean 0 and variance I, show that
n- c;=I X , converges in distribution to a standard normal random
variable 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 in
L 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,= 0
for 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 by
n-. w 0 SI 610
lim sup 11 T,(t)nJ- nnT(t)fll= 0
n-w 120
lim sup IIT,(r)n,g - n, T(r)g)I = 0.
n-m t a O
16.
18.
19.
(8.20)
20. Corollary 7.8 has been called a second-order limit theorem. Prove the
followingkth-order limit theorem as an application of Theorem 7.6.
Let A,,, A , , ...,A, be linear operators on L such that A, generates a
strongly 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

56. 16 OPEUTORS€MICIOUPS
large, 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 with
I
m
110
AL-m+j/;=O for m = O , . .. ,k- 1 ,
and define
I
k - I
l = 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)}on
6, then, for eachfE 6,lima-,,,, 'lf&)f=T(t)f for all t 2 0, uniformly on
bounded 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 E
b} 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 a
strongiy 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 sufkiently
large, suppose that an extension of ll +aA +a2B generates a strongly
A-O+
A-O+ c
/I, = 00, and
n-m
n-m
n-m

57. 9. NOTES 47
continuous contraction semigroup { T#)) on t.Let D be a subspace of
9(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 continuous
contraction semigroup{ V(r)} on D,and that
m
(8.28) lim L [ e-"U(r)fdt = P,f exists for every f e 6.
A - O + JO
Let 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 strongly
continuous 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, let
B(t):L-4 L, t 2 0, be bounded linear operators such that (B(t)} is
stronglycontinuousin 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 ) ~L
satisfying
(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).
at
9. NOTES
Among the best general references on operator semigroups are Hille and
Phillips (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 of
Watanabe (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

58. 48 OrUATORSMCIOUPS
6.1 and 6.5 were found by Sova (1967) and Kurtz (1969). The proof given here
follows Goldstein (1976). Hasegawa (1964) and Kato (1966) found necessary
and sufficient conditions of a different sort. Lemma 6.4 and Corollary 6.6 are
due to Chernoff (1968). Corollary 6.7 is known as the Trotter (1959) product
formula. Corollary 6.8 can be found in Hille (1948). Theorems 6.9 and 6.11
were proved by Kurtz (1970a).
Theorem 7.1 was obtained by Kato (1966) assuminga < and in general by
Gustafson (1966). Lemma 7.3 appears in Hille (1948). Theorem 7.6 is due to
Ethier and Nagylaki (1980) and Corollary 7.7 to Kurtz (1977). Corollary 7.8
was proved by Kurtz (1973) and Kertz (1974); related results are given in
Davies (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), and
Problem 14 to Norman (1977). Problem 20 is closely related to a theorem of
Kertz(1978).

59. 2
This chapter consists primarily of background material that is needed later.
Section I defines various concepts in the theory of stochastic processes, in
particular the notion of a stopping time. Section 2 gives a basic introduction
to martingale theory including the optional sampling theorem, and local mar-
tingales are discussed in Section 3, in particular the existence of the quadratic
variation or square bracket process. Section 4 contains additional technical
material on processes and conditional expectations, including a Fubini
theorem. The DoobMeyer decomposition theorem for submartingales is
given in Section 5, and some of the special properties of square integrable
martingalesare noted in Section 6. The semigroupof conditioned shifts on the
space of progressiveprocesses is discussed in Section 7. The optional sampling
theorem for martingalesindexed by a metric lattice is given in Section 8.
STOCHASTIC PROCESSES
AND MARTINGALES
1. STOCHASTIC PROCESSES
A 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 a
function 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 .F
for every E a.We assume throughout that E is a metric space with metric r
49
Markov Processes Characterizationand Convergence
Edited by STEWARTN. ETHIER and THOMASG.KURTZ
Copyright 01986,2005 by John Wiley & Sons,Inc

60. 50 STOCHfiTIC PROCESS AND MARTINGALES
and that 1is the Bore1 a-algebra B(E).As is usually done, we write X(t) and
X(t,* ) interchangeably.
In this chapter, with the exception of Section 8, we take N = [O, 00). We are
primarily 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 is
measurable if X:[O, 00) x f2-t E is g[O, 00) x $-measurable. We say that X
is (almost surely) continuous (right continuous, lefz continuous) if for (almost)
every o E R, X(., w) is continuous (right continuous, left continuous). Note
that the statements “ X is measurable” and “X is continuous” are not parallel
in that “X is measurable” is stronger than the statement that X( .,w) is
measurable for each o E R. The function X(-,a)is called the sample path of
the 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 $,+, for t, s E [O, m). Intuitively 9,corresponds to the infor-
mation known to an observer at time t. In particular, for a process X we
define (4:) by 9;= a(X(s):s 5 c); that is, 9: is the information obtained
by observingX up to time t.
We occasionally need additional structure on {9J.We say {S,}is right
continuous if for each t L 0, SI=,sit,. = r)a,04tlt,. Note the filtration
{F,+}is always right continuous (Problem 7). We say (9,)is complete if
(a,9,P)is completeand { A E 9: P(A) = 0)c So,
A process X is adapted to a filtration {S,)(or simply {F,}-adapted)if X(r)
is 6,-measurable for each t L 0. Since6,is increasing in I, X is {$,}-adapted
if and only if 9; c S,for each t 2 0.
A process X is {.F,}-progressive (or simply progressive if (9,)= (9:))if
for 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 on
g ( 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 the
same finite-dimensionaldistributions, then we say Y is a version of X (and X is
a 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 each
c 2 0, P(X(t)= Y(t)}= 1, then we say Y is a modijication of X. (We are
implicitly 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,then
clearly Y is a version of A’. Finally if there exists N E 9 such that ON)= 0
and 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.

61. 1. STOCHASTIC m o m m SI
A 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 say
I isfinite as. If T s 7' < 00 for some constant T, we say T is bounded. In some
sense a stopping time is a random time that is recognizable by an observer
whose 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 is
an (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 if
and 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 m
and { 7 <11 = on{?< t + n u ' ) E flm91+m-,= .(PI+. The necessity was
observed above. 0
1.2 Proposition
Then 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 time
by 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 a
sequenceof bounded stopping times, that is, limn-mT A n = I. This fact is very
useful in proving theorems about stopping times. A second equally useful
approximation is the approximation of arbitrary stopping times by a nonin-
creasing sequenceof discrete stoppingtimes.
1.3 Proposition
and suppose that
and define
For n = 1, 2,..., let 0 = r: < tl < * - * and limk-rntl: = 00,
sup&+ - I;) = 0. Let I be an {F,+}-stoppingtime

62. 52 STOCHASTIC PRCK€SSES AND MAWlNCALEs
Then 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 an
observer at time t. For an (9,)-stopping time 7, the a-algebra9,should have
the sameintuitivemeaning. For technical reasons S,is defined by
(1.3)
Similarly, PC+is defined by replacing 9,by 9,+.See Problem 6 for some
motivation as to why the definition is reasonable. Given an E-valued process
X,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 nonnegative
9,-measurable random variable, and let X be an ($r,}-progressive E-valued
process. Define X' and Y by Xr(r)= X(7At) and Y(t)= X(7 +r), and define
9,= F I h ,and MI = f,,,,t 2 0. (Recall that r h t and .c +r are stopping
times.)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 } E
F,.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,, and
hence 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).

63. 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 all
r 2 0. Hence A E 9#.
(d) Fix t 2 0. By (b), T A t is .F,-measurable. Consequently the mapping
o - 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 ) is
the composition of these two mappings, it is .F,-measurable. Finally, for
P E @E), {X(r)E r}n { 7 s t } = {X(TA I ) E T} n {T s t } E .F,and hence
By (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 E
a[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 generates
Finally, 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) the
mapping (s, u)-+X((t(w)+t)As, w) from ([O, 003 x Q, a [ O , 003 x F,,,)

64. 54 STOCHASTIC CIIOCESSLS AND MARTINGALES
into (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.The
mapping (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). 0
Let X be an E-valued process and let rE S(E).Thefirst entruncetime into
r 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 first
entrance 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, they
are 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 . 0
implying part (b). For n = 1. 2,. ..let r, = {x:r(x, r)< l/n}. Then, under the
conditionsofpart (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

65. 2 mnwAus 55
Under slightly more restrictive hypotheses on {F,},a much more general
result than Proposition 1.5 holds. We do not need this generality, so we simply
state the result without proof.
1.6 Theorem Let IS,}be complete and right continuous, and let X be an
E-valued {St,)-progressive process. Then for each r E @E), r,(Q is an
(9,)-stoppingtime.
Proof. See,for example, Elliott (1982). page 50. 0
2. MRTINGALES
A real-valued process X with E[IX(t)lJ e 00 for all r 2 0 and adapted to a
filtration (S,}is an {.%,}-martingale if
(2.1) ECX(t +s)IS,]= X(r), t, s 2 0.
is an {SF,}-submarringaleif
(2.2) ECWt +s)l.!FJ 2 XO), r, s 2 0,
and is an {SFIP,)-supermartingakif the inequality in (2.2) is reversed. Note that
X is a supermartingaleif -X is a submartingale,and that X is a martingale if
both X and - X are submartingales. Consequently, results proved for sub-
martingales immediately give analogous results for martingales and super-
martingales. If {@,} = {Sf}we simply say X is a martingale (submartingale,
supermartingale).
Jensen's inequality gives the following.
2.1 Proposition (a) Suppose X is an {.!F,}-martingale, cp is convex, and
&[lcp(X(t))(]< 00 for all t 2 0,Then cp 0 X is an (9,}-submartingale.
(b) Suppose X is an (9,)-submartingale, cp is convex and nonde-
creasing, and &[lcp(X(t))l]< 00 for all t 2 0. Then cp 0 X is an
(9,)-submartingale.
Note that for part (a) the last inequality is in fact equality, and in part (b) the
0last inequality followsfrom the assumption that cp is nondecreasing.
2.2 Lemma Let r , and r2 be {.!F,}-stopping times assuming values in
Itl,t 2 , ...,tm} c [O, 00). If X is an {9,}-submartingale,then
(2.4) ECX(r2)lFI,12 x(flA rd.

66. 56 STOCHASTIC PROCESSES AND MARTINGALES
Proof. Assume t, K tz < .* * < t,. We must show that for every A E 9,,
(2.5)
Since A = uysl(A n ( t l = ti}), it is sufficientto show that
n
The following is a simple application of Lemma 2.2. Let x t = x VO,
2.3 Lemma Let X be a submartingale, T > 0. and F c [O, TJ be finite.
Then for each x > 0,
(2.9) I;x - ' E [ X + ( T ) ]
and
Proof.
Then
Let T = min {r E F: X(t)2 x) and set T , = T AT and T , = T in (2.4).
(2.1 1) E[X(T)]2 E [ X ( t A7'11 = ECX(T)Zlr<mJ + ECX(T)Xir=a)I*
and hence
which implies (2.9).The proof of (2.10)is similar. 0

67. 2. MARTINGALES 57
2.4 Corollary Let X be a submartingale and let F c [0, 00) be countable.
Then for each x > 0 and 7' > 0,
(2.13) .( sup x ( t ) 2 s x ~ E C X + ( T ) I
t a F n ( 0 . T I
and
(2.14)
Proot Let F,c F, c . . . be finite and F = UF,. Then, for 0 < y < x,
(2.15)
P{ inf X(r)S - x s; .Y '(E[Xt(7)1 - E[X(O)]).
p{ sup x ( t ) 2x s y - l ~ ~ ~ + ( r ) ~
r c F n ( 0 . TI
0Letting y-' x we obtain (2.13), and (2.14) follows similarly.
Let X be a real-valued process, and let F c LO, 00) be finite. For a < h
define rl = min { t E F:X(t) I a}, and for k = 1,2,. ..define ak= min { t > t k :
I E F, X(r)2 h} and r k t , = min { t > a*:t E F,X(r)< u}. Define
(2.16) V(a,h, F) = max {k:ak< ao}.
The quantity V(a,b, F) is called the number of upcrossings of the interval (a, h)
by X restricted to F.
I E F A 10. 7 1
2.5 lemma
then
Let X be a submartingale. If T > 0 and F c [0, 7'1 is finite,
(2.17)
Proof. Since ut A T I rk + I A 7 ,Lemma 2.2 implics
which gives (2.17).