feller book

Preface

VIII Preface

Preface IX
phrased in terms of hitting times. Theorem 9.36 is the most important one for readers interested in an existence proof of a σ-additive invariant measure which is unique up to a multiplicative constant. Assertion (e) of Proposition 9.40 together with Orey’s theorem for Markov chains (see Theorem 9.4) yields the interesting consequence that, up to multiplicative constants, σ-finite invariant measures are unique. In §9.4 Orey’s theorem is proved for recurrent Markov chains. In the proof we use a version of the bivariate linked forward recurrence time chain as explained in Lemma 9.50. We also use Nummelin’s splitting technique: see [162], §5.1 (and §17.3.1). The proof of Orey’s theorem is based on Theorems 9.53 and 9.62. Results Chapter 9 go back to Meyn and Tweedie [162] for time-homogeneous Markov chains and Seidler [207] for time-homogeneous Markov processes.
Interdependence
From the above discussion it is clear how the chapters in this book are related. Chapter 1 is a prerequisite for all the others except Chapter 7. Chapter 2 contains the proofs of the main results in Chapter 1; it can be skipped at a first reading. Chapter 3 contains material very much related to the contents of the first chapter. Chapter 5 is a direct continuation of 4, and is somewhat difficult to read and comprehend without the knowledge of the contents of Chapter 4. Chapter 6 is more or less independent of the other chapters in Part 2. For a big part Chapter 7 is independent of the other chapters: most of the results are phrased and proved for a finite-dimensional state space. The chapters 8 and 9 are very much interrelated. Some results in Chapter 8 are based on results in Chapter 9. In particular this is true in those results which use the existence of an invariant measure. A complete proof of existence and uniqueness is given in Chapter 9 Theorem 9.36. As a general prerequisite for understanding and appreciating this book a thorough knowledge of probability theory, in particular the concept of the Markov property, combined with a comprehensive notion of functional analysis is very helpful. On the other hand most topics are explained from scratch.
Acknowledgement

X Preface
Nebraska, May 12–14, 2006. Finally, another preliminary version was presented during a Conference on Evolution Equations, in memory of G. Lumer, at the Universities of Mons and Valenciennes, August 28–September 1, 2006. The author also has presented some of this material during a colloquium at the University of Amsterdam (December 21, 2007), and at the AMS Special Session on the Feynman Integral in Mathematics and Physics, II, on January 9, 2008, in the Convention Center in San Diego, CA.
The author is obliged to the University of Antwerp (UA) and FWO Flan- ders (Grant number 1.5051.04N) for their financial and material support. He was also very fortunate to have discussed part of this material with Karel in’t Hout (University of Antwerp), who provided some references with a cru- cial result about a surjectivity property of one-sided Lipschitz mappings: see Theorem 1 in Croezeix et al [63]. Some aspects concerning this work, like backward stochastic differential equations, were at issue during a conservation with Etienne Pardoux (CMI, Universite de Provence, Marseille); the author is grateful for his comments and advice. The author is indebted to J.-C. Zambrini (Lisboa) for interesting discussions on the subject and for some references. In addition, the information and explanation given by Willem Stannat (Technical University Darmstadt) while he visited Antwerp are gratefully acknowledged. In particular this is true for topics related to asymptotic stability: see Chap- ter 8. The author is very much obliged to Natalia Katilova who has given the ideas of Chapter 7; she is to be considered as a co-author of this chapter. Finally, this work was part of the ESF program “Global”.
Key words and phrases, subject classification
Some key words and phrases are: backward stochastic differential equation, parabolic equations of second order, Markov processes, Markov chains, ergod- icity conditions, Orey’s theorem, theorem of Chacon-Ornstein, invariant measure, Korovkin properties, maximum principle, Kolmogorov operator, squared gradient operator, martingale theory. AMS Subject classification [2000]: 60H99, 35K20, 46E10, 60G46, 60J25.
Antwerp, Jan A. Van Casteren February 2008

Part I Strong Markov processes
1 Strong Markov processes on polish spaces . . . . . . . . . . . . . . . . . 3 1.1 Strict topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Theorem of Daniell-Stone . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Measures on p olish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.3 Integral operators on the space of bounded continuous
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 Strong Markov processes and Feller evolutions . . . . . . . . . . . . . . 27
1.2.1 Generators of Markov processes and maximum principles 31 1.3 Strong Markov processes: main result . . . . . . . . . . . . . . . . . . . . . . 35
1.3.1 Some historical remarks and references . . . . . . . . . . . . . . . 40 1.4 Dini’s lemma, Scheffe’s theorem, and the monotone class
theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.4.1 Dini’s lemma and Scheffe’s theorem . . . . . . . . . . . . . . . . . . 42 1.4.2 Monotone class theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2 Strong Markov processes: proof of main result . . . . . . . . . . . . . 47 2.1 Proof of the main result: Theorem 1.39 . . . . . . . . . . . . . . . . . . . . . 47
2.1.1 Proof of item (a) of Theorem 1.39 . . . . . . . . . . . . . . . . . . . 47 2.1.2 Proof of item (b) of Theorem 1.39 . . . . . . . . . . . . . . . . . . . 69 2.1.3 Proof of item (c) of Theorem 1.39 . . . . . . . . . . . . . . . . . . . 72 2.1.4 Proof of item (d) of Theorem 1.39 . . . . . . . . . . . . . . . . . . . 76 2.1.5 Proof of item (e) of Theorem 1.39 . . . . . . . . . . . . . . . . . . . 94 2.1.6 Some historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

XII Contents
3.5 Measurability properties of hitting times . . . . . . . . . . . . . . . . . . . 150 3.5.1 Some side remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3.5.2 Some related remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Part II Backward Stochastic Differential Equations
4 Feynman-Kac formulas, backward stochastic differential equations and Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.2 A probabilistic approach: weak solutions. . . . . . . . . . . . . . . . . . . . 191 4.3 Existence and Uniqueness of solutions to BSDE’s . . . . . . . . . . . . 194 4.4 Backward stochastic differential equations and Markov
processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5 Viscosity solutions, backward stochastic differential equations and Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 5.1 Comparison theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 5.2 Viscosity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 5.3 Backward stochastic differential equations in finance . . . . . . . . . 248 5.4 Some related remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6 The Hamilton-Jacobi-Bellman equation and the stochastic Noether theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.2 The Hamilton-Jacobi-Bellman equation and its solution . . . . . . 258 6.3 The Hamilton-Jacobi-Bellman equation and viscosity solutions 267 6.4 A stochastic Noether theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
6.4.1 Classical Noether theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.4.2 Some problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
Part III Long time behavior

Contents XIII
8 Coupling methods and Sobolev type inequalities . . . . . . . . . . . 383 8.1 Coupling methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 8.2 Some related stability results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 8.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
9 Miscellaneous topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 9.1 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 9.2 Stopping times and time-homogeneous Markov processes . . . . . 463 9.3 Markov Chains: invariant measure . . . . . . . . . . . . . . . . . . . . . . . . . 465
9.3.1 Some definitions and results . . . . . . . . . . . . . . . . . . . . . . . . 465 9.3.2 Construction of an invariant measure . . . . . . . . . . . . . . . . 479
9.4 A proof of Orey’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 9.5 About invariant (or stationary) measures . . . . . . . . . . . . . . . . . . . 552 9.6 Weak and strong solutions to stochastic differential equations . 553
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
1.1 Strict topology
Throughout this book E stands for a complete metrizable separable topological space, i.e. E is a polish space. The Borel field of E is denoted by E. We write C b(E ) for the space of all complex valued bounded continuous functions on E . The space C b(E ) is equipped with the supremum norm: f ∞ = supx∈E |f (x)|, f ∈ C b(E ). The space C b(E ) will be endowed with a second topology which will be used to describe the continuity properties. This second topology, which is called the strict topology, is denoted as T β- topology. The strict topology is generated by the semi-norms of the form pu, where u varies over H (E ), and where pu(f ) = supx∈E |u(x)f (x)| = uf ∞, f ∈ C b(E ). Here a function u belongs to H (E ) if u is bounded and if for every real number α > 0 the set {|u| ≥ α} = {x ∈ E : |u(x)| ≥ α} is contained in a compact subset of E . It is noticed that Buck [44] was the first author who introduced the notion of strict topology (in the locally compact setting). He used the notation β instead of T β .

4 1 Strong Markov processes
Observe that continuity properties of functions f ∈ C b(E ) can be formulated in terms of convergent sequences in E which are contained in compact subsets of E . The topology of uniform convergence on C b(E ) is denoted by T u.
1.1.1 Theorem of Daniell-Stone
In Proposition 1.3 below we need the following theorem. It says that an abstract integral is a concrete integral. Theorem 1.2 will be applied with S = E , H = C +b , the collection of non-negative functions in C b(E ), and for I : C +B → [0,∞) we take the restriction to C +b of a non-negative linear functional defined on C b(E ) which is continuous with respect to the strict topology.
Theorem 1.2 (Theorem of Daniell-Stone). Let S be any set, and let H be a non-empty collection of functions on S with the following properties:
(1) If f and g belong to H , then the functions f + g, f ∨ g and f ∧ g belong to H as well;
(2) If f ∈ H and α is a non-negative real number, then αf , f ∧ α, and (f − α)+ = (f − α) ∨ 0 belong to H ;
(3) If f , g ∈ H are such that f ≤ g ≤ 1, then g − f belongs to H .
Let I : H → [0,∞] be an abstract integral in the sense that I is a mapping which possesses the following properties:
(4) If f and g belong to H , then I (f + g) = I (f ) + I (g); (5) If f ∈ H and α ≥ 0, then I (αf ) = αI (f ); (6) If (f n)n∈N is a sequence in H which increases pointwise to f ∈ H , then
I (f n) increases to I (f ).
Then there exists a non-negative σ-additive measure µ on the σ-field generated by H , which is denoted by σ(H ), such that I (f ) =
f dµ, for f ∈ H . If there
exists a countable family of functions (f n)n∈N ⊂ H such that I (f n) < ∞ for all n ∈ N, and such that S =
∞ n=1 {f n > 0}, then the measure µ is unique.
Proof. Define the collection H ∗ of functions on S as follows. A function f : S → [0,∞] belongs to H ∗ provided there exists a sequence (f n)n∈N ⊂ H which increases pointwise to f . Then the subset H ∗ has the properties (1) and (2) with H ∗ instead of H . Define the mapping I ∗ : H ∗ → [0,∞] by
I ∗(f ) = lim n→∞
I (f n) , f ∈ H ∗,
where (f n)n∈N ⊂ H is a sequence which pointwise increases to f . The definition does not depend on the choice of the increasing sequence ( f n)n∈N ⊂ H . In fact let (f n)n∈N and (gn)n∈N be sequences in H which both increase to f ∈ H ∗. Then by (6) we have
lim n→∞
sup m∈N
sup n∈N
= sup m∈N
I (gm) . (1.1)
From (1.1) it follows that I ∗ is well-defined. The functional I ∗ : H ∗ → [0,∞] has the properties (4), (5), and (6) (somewhat modified) with H ∗ instead of H and I replaced by I ∗. In fact the correct version of (6) for H ∗ reads as follows:
(6∗) Let (f n)n∈N be a sequence H ∗ which increases pointwise to a function f . Then f ∈ H ∗, and I ∗ (f n) increases to I ∗ (f ).
We also have the following assertion:
(3∗) Let f and g ∈ H ∗ be such that f ≤ g. Then I ∗(f ) ≤ I ∗(g).
We first prove (3∗) if f and g belong to H and f ≤ g. From (6), (3) and (4) we get
I (g) = sup m∈N
I (g ∧m) = sup m∈N
(I (g ∧m− f ∧m) + I (f ∧m))
≥ sup m∈N
I (f ∧m) = I (f ) . (1.2)
Here we used the fact that by (3) the functions g ∧ m − f ∧ m, m ∈ N, belong to H . Next let f and g be functions in H such that f ≤ g. Then there exist increasing sequences (f n)n∈N and (gn)n∈N in H such that f n converges pointwise to f ∈ H ∗ and gn to g ∈ H ∗. Then
I ∗ (f ) = sup n∈N
I (f n) ≤ sup n∈N
I (f n ∨ gn) = I ∗ (g) . (1.3)
Next we prove (6∗). Let (f n)n∈N be a pointwise increasing sequence in H ∗, and put f = supn∈N f n. Choose for every n ∈ N an increasing sequence (f n,m)m∈N ⊂ H such that supm∈N f n,m = f n. Define the functions gm, m ∈ N, by
gm = f 1,m ∨ f 2,m ∨ · · ·∨ f m,m.
Then gm+1 ≥ gm and gm ∈ H for all m ∈ N. In addition, we have
sup m∈N
sup m≥n
f n = f. (1.4)
Hence f ∈ H ∗. For 1 ≤ n ≤ m the inequalities f n,m ≤ f n ≤ f m hold pointwise, and hence gm ≤ f m. From (3∗) we infer
I ∗ (f ) = sup m∈N
I (gm) = sup m∈N
I ∗ (gm) ≤ sup m∈N
I ∗ (f m) ≤ I ∗ (f ) , (1.5)
and thus supm∈N I ∗ (f m) = I ∗ (f ). Next we will get closer to measure theory. Therefore we define the col-

(1) If the subsets G1 and G2 belong to G, then the same is true for the subsets G1 ∩G2 and G1 ∪G2;
(2) ∅ ∈ G; (3) If the subsets G1 and G2 belong to G and if G1 ⊂ G2, then µ (G1) ≤
µ (G2); (4) If the subsets G1 and G2 belong to G, then the following strong additivity
holds: µ (G1 ∩G2) + µ (G1 ∪G2) = µ (G1) + µ (G2); (5) µ (∅) = 0; (6) If (Gn)n∈N is a sequence in G such that Gn+1 ⊃ Gn, n ∈ N, then
n∈N Gn belongs to G and µ
n∈N Gn
= supn∈N µ (Gn).
These properties are more or less direct consequences of the corresponding properties of I ∗: (1∗)–(6∗).
Using the mapping µ we will define an exterior or outer measure µ∗
on the collection of all subsets of S . Let A be any subset of S . Then we put µ∗ (A) = ∞ if for no G ∈ G we have A ⊂ G, and we write µ∗ (A) = inf {µ (G) : G ∈ G, G ⊃ A}, if A ⊂ G0 for some G0 ∈ G. Then µ∗ has the following properties:
(i) µ∗(∅) = 0; (ii) µ∗ (A) ≥ 0, for all subsets A of S ; (iii)µ∗ (A) ≤ µ∗ (B), whenever A and B are subsets of S for which A ⊂ B; (iv)µ∗ (
∞ n=1 An) ≤∞
n=1 µ∗ (An) for any sequence (An)n∈N of subsets of S .
The assertions (i), (ii) and (iii) follow directly from the definition of µ∗. In order to prove (iv) we choose a sequence ( An)n∈N, An ⊂ S , such that µ∗ (An) < ∞ for all n ∈ N. Fix ε > 0, and choose for every n ∈ N an subset Gn of S which belongs to G and which has the following properties: An ⊂ Gn
and µ (Gn) ≤ µ∗ (An) + ε2−n. By the equality n∈N Gn =
m∈N
m n=1 Gn we
see that n∈N Gn belongs to G. From the properties of an exterior measure
we infer the following sequence of inequalities:
µ∗
n∈N An
n=1 µ∗ (An). Hence assertion (iv) follows.

1.1 Strict top ology 7
D = {A ⊂ S : µ∗ (D) ≥ µ∗ (A ∩D) + µ∗ (Ac ∩D) for all D ⊂ S } = {A ⊂ S : µ (D) ≥ µ∗ (A ∩D) + µ∗ (Ac ∩D)
for all D ∈ G with µ(D) <∞} . (1.7)
Here we wrote Ac = S \ A for the complement of A in S . The reader is invited to check the equality in (1.7). According to Caratheodory’s theorem the exterior measure µ∗ restricted to the σ-field D is a σ-additive measure. We will prove that D contains G. Therefore pick G ∈ G, and consider for D ∈ G for which µ(D) <∞ the equality
µ∗ (G ∩D) + µ∗ (Gc ∩D) = µ (G ∩D) + inf {µ (U ) : U ∈ G, U ⊃ Gc ∩D} . (1.8)
Choose h ∈ H ∗ in such that h ≥ 1Gc∩D. For 0 < α < 1 we have
1Gc∩D ≤ 1{h>α} ≤ 1
α h.
Since 1{h>α} = supm∈N 1 ∧ (m(h− α)+) we see that the set {h > α} is a member of G. It follows that T ∗(h) ≥ αµ ({h > α}) ≥ αµ∗ (Gc ∩D), and hence
µ∗ (Gc ∩D) ≤ inf {I ∗(h) : h ≥ 1Gc∩D, h ∈ H ∗} ≤ inf {I ∗(1U ) : U ⊃ Gc ∩D, U ∈ G} = µ∗ (Gc ∩D) . (1.9)
From (1.9) the equality
µ∗ (Gc ∩D) = inf {I ∗(h) : h ≥ 1Gc∩D, h ∈ H ∗}
follows. Next choose the increasing sequences (f n)n∈N and (gn)n∈N in such a way that the sequence f n increases to 1D and gn increases to 1G. Define the functions hn, n ∈ N, by
hn = 1D − f n ∧ gn = sup m≥n
{(f m − f n) + (f n − f n ∧ gn)} .
Since the functions f m−f n, m ≥ n, and f n−f n∧ gn belong to H we see that hn belongs to H ∗. Hence we get:
∞ > µ(D) = I ∗ (1D) = I ∗ (hn) + I ∗ (f n ∧ gn) = I ∗ (hn) + I (f n ∧ gn) . (1.10)
In addition we have hn ≥ 1Gc∩D. Consequently,
µ∗ (G ∩D) + µ∗ (Gc ∩D)
≤ µ (G ∩D) + inf n∈N
I ∗ (hn)
I (f n ∧ gn)

The equality in (1.11) proves that the σ-field D contains the collection G, and hence that the mapping µ, which originally was defined on G in fact the restriction is of a genuine measure defined on the σ-field generated by H , which is again called µ, to G.
We will show the equality I (f ) =
f dµ for all f ∈ H . For f ∈ H we have
f dµ =
= I ∗ (f ) = I (f ) . (1.12)
Finally we will prove the uniqueness of the measure µ. Let µ1 and µ2 be two measures on σ(H ) with the property that I (f ) =
f dµ1 =
f dµ2 for all
f ∈ H . Under the extra condition in Theorem 1.2 that there exist countable many functions (f n)n∈N such that I (f n) < ∞ for all n ∈ N and such that S =
∞ n=1 {f n > 0} we shall show that µ1(B) = µ2(B) for all B ∈ σ(H ).
Therefore we fix a function f ∈ H for which I (f ) < ∞. Then the collection B ∈ σ(H ) :
B
is a Dynkin system containing all sets of
the form {g > β } with g ∈ H and β > 0. Fix ξ > 0, β > 0 and g ∈ H . Then
the functions gm,n := min
+ ∧ 1
µ1 [{g > β } ∩ {f > ξ}] = lim m→∞
lim n→∞
gm,ndµ1 = lim
gm,ndµ2 = µ2 [{g > β } ∩ {f > ξ}] . (1.13)
Upon integration the extreme terms in (1.13) with respect to the Lebesgue measure dξ shows the equality
{g>β}
the collection
contains all sets of the form
{g > β } where g ∈ H and β > 0. Such collection of sets is closed under finite intersection. Hence, by a Dynkin argument, we infer the equality
B ∈ σ(H ) :
= σ(H ).
The same argument applies with (nf ) ∧ 1 replacing f . By letting n tend to ∞ this shows the equality
σ(H ) = {B ∈ σ(H ) : µ1 [B ∩ {f > 0}] = µ2 [B ∩ {f > 0}]} . (1.14)
Since the set H is closed under taking finite maxima, I (f ∨ g) ≤ I (f )+I (g) < ∞ whenever I (f ) and I (g) are finite, and S =
∞ n=1 {f n > 0} with I (f n) <∞,
n ∈ N, we see that

µ1(B) = lim n→∞

= µ2(B) (1.15)
for B ∈ σ(H ). This finishes the proof of Theorem 1.2.
1.1.2 Measures on polish spaces
Our first proposition says that the identity mapping f → f sends T β-bounded subsets of C b(E ) to ·∞-bounded subsets.
Proposition 1.3. Every T β-bounded subset of C b(E ) is ·∞-bounded. On the other hand the identity is not a continuous operator from (C b(E ),T β) to (C b(E ), ·∞), provided that E itself is not compact.
Proof. Let B ⊂ C b(E ) be T β-bounded. If B were not uniformly bounded, then there exists sequences (f n)n∈N ⊂ B and (xn)n∈N ⊂ E such that |f n (xn)| ≥ n2,
n ∈ N. Put u(x) = ∞
1
n 1xn . Then the function u belongs to H (E ), but
sup f ∈B
u (xn) |f (xn)| ≥ sup n∈N
n =∞. The latter shows
that the set B is not T β-bounded. By contra-position it follows that T β- bounded subsets are uniformly bounded.
Next suppose that E is not compact. Let u be any function in H (E ). Then limn→∞ u (xn) = 0. If the imbedding (C b(E ),T β)→ (C b(E ),T u) were continuous, then there would exist a function u ∈ H +(E ) such that f ∞ ≤ uf ∞ for all f ∈ C b(E ). Let K be a compact subset of E such that 0 ≤ u(x) ≤ 1
2 for x /∈ K . Since 1 ≤ u∞ = u (x0) for some x0 ∈ E , and since by assumption E is not compact we see that K = E . Choose an open neighborhood O of K , O = E , and a function f ∈ C b(E ) such that 1 − 1O ≤ f ≤ 1 − 1K . In particular, it follows that f = 1 outside of O, and f = 0 on K . Then 1 = f ∞ ≤ uf ∞ ≤ supx/∈K |u(x)f (x)| ≤ 1
2 f ∞ ≤ 1 2 . Clearly, this is a
contradiction. This concludes the proof of Proposition 1.3.
The following proposition shows that the dual of the space ( C b(E ),T β) coincides with the space of all complex Borel measures on E.
Proposition 1.4. 1. Let µ be a complex Borel measure on E . Then there exists a function u ∈ H (E ) such that
f dµ ≤ pu(f ) for all f ∈ C b(E ).
2. Let Λ : C b(E ) → C be a linear functional on C b(E ) which is continuous with respect to the strict topology. Then there exists a unique complex measure µ on E such that Λ(f ) =
f dµ, f ∈ C b(E ).


≤ ∞
f ∞
where u(x) = ∞
j=1 2−j1Kj (x) |µ| (E ).
2 We decompose the functional Λ into a combination of four positive functionals: Λ = (Λ)+ − (Λ)− + i (Λ)+ − i (Λ)− where the linear functionals (Λ)+ and (Λ)− are determined by their action on positive functions f ∈ C b(E ):
(Λ)+ (f ) = sup { (Λ(g)) : 0 ≤ g ≤ f, g ∈ C b(E )} , and
(Λ)− (f ) = sup { (−Λ(g)) : 0 ≤ g ≤ f, g ∈ C b(E )} .
Similar expressions cam be employed for the action of (Λ)+ and (Λ)− on functions f ∈ C +b . Since the complex linear functional Λ : C b(E ) → C is T β- continuous there exists a function u ∈ H +(E ) such that |Λ(f )| ≤ uf ∞ for
all f ∈ C b(E ). Then it easily follows that (Λ)+ (f )
≤ uf ∞ for all real-
+ (f )
2 uf ∞ for all f ∈ C b(E ),
which in general take complex values. Similar inequalities hold for (Λ)− (f ), (Λ)
+ (f ), and (Λ)
− (f ). Let (f n)n∈N be a sequence of functions in C +b (E )
which pointwise increases to a function f ∈ C +b (E ). Then limn→∞ Λ (f n) = Λ(f ). This can be seen as follows. Put gn = f − f n, and fix ε > 0. Then the sequence (gn)n∈N decreases pointwise to 0. Moreover it is dominated by f . Choose a strictly positive real number α in such a way that α f ∞ ≤ ε. Then it follows that
|Λ (gn)| ≤ ugn∞ = max u1{u≥α}gn
∞
1.1 Strict topology 11
where N chosen so large that u∞ 1{u≥α}gn
∞ ≤ ε for n ≥ N . By Dini’s
lemma such a choice of N is possible. An application of Theorem 1.2 then yields the existence of measures µj , 1 ≤ j ≤ 4, defined on the Baire field of E
such that (Λ)+ (f ) =
f dµ4 for f ∈ C b(E ). It follows that Λ(f ) =
f dµ1−
f dµ4 =
f dµ for f ∈ C b(E ). Here µ = µ1 − µ2 + iµ3 − iµ4
and each measure µj , 1 ≤ j ≤ 4, is finite and positive. Since the space E is polish it follows that Baire field coincides with the Borel field, and hence the measure µ is a complex Borel measure.
This concludes the proof of Proposition 1.4.
The next corollary gives a sequential continuity characterization of linear functionals which b elong to the space (C b(E ),T β)∗, the topological dual of the space C b(E ) endowed with the strict topology. We say that a sequence (f n)n∈N ⊂ C b(E ) converges for the strict topology to f ∈ C b(E ) if limn→∞ u (f − f n)∞ = 0 for all functions u ∈ H +(E ). It follows that a sequence (f n)n∈N ⊂ C b(E ) converges to a function f ∈ C b(E ) with respect to the strict topology if and only if this sequence is uniformly bounded and limn→∞ 1K (f − f n)∞ = 0 for all compact subsets K of E .
Corollary 1.5. Let Λ : C b(E )→ C be a linear functional. Then the following assertions are equivalent:
(1) The functional Λ belongs to (C b(E ),T β)∗; (2) limn→∞ Λ (f n) = 0 whenever (f n)n∈N is a sequence in C +b (E ) which con-
verges to the zero-function for the strict topology; (3) There exists a finite constant C ≥ 0 such that |Λ(f )| ≤ C f ∞ for all
f ∈ C b(E ), and limn→∞Λ (gn) = 0 whenever (gn)n∈N is a sequence in
C +b (E ) which is dominated by a sequence (f n)n∈N in C +b (E ) which decreases pointwise to 0;
(4) There exists a finite constant C ≥ 0 such that |Λ(f )| ≤ C f ∞ for all f ∈ C b(E ), and limn→∞ Λ (f n) = 0 whenever (f n)n∈N is a sequence in
C +b (E ) which decreases pointwise to 0; (5) There exists a complex Borel measure µ on E such that Λ(f ) =
f dµ for
all f ∈ C b(E ).
In (3) we say that a sequence (gn)n∈N in C +b (E ) is dominated by a sequence (f n)n∈N if gn ≤ f n for all n ∈ N.
Proof. (1) =⇒ (2). First suppose that Λ belongs to (C b(E ),T β) ∗
. Then there exists a function u ∈ H +(E ) such that |Λ(f )| ≤ uf ∞ for all f ∈ C b(E ). Hence, if the sequence (f n)n∈N ⊂ C +b (E ) converges to zero for the strict topology, then limn→∞ uf n∞ = 0, and so limn→∞Λ (f n) = 0. This proves the implication (1) =⇒ (2).
(2) =⇒ (3). Let (f n)n∈N be a sequence in C b(E ) which converges to 0 for
the uniform topology. From (2) it follows that the sequences
(f n)+ n∈N
(f n)−
n∈N
converge to 0 for the strict
topology T β, and hence limn→∞ Λ (f n) = 0. Consequently, the functional Λ : C b(E ) → C is continuous if C b(E ) is equipped with the uniform topology, and hence there exists a finite constant C ≥ 0 such that |Λ(f )| ≤ C f ∞ for all f ∈ C b(E ). If (f n)n∈N is a sequence in C +b (E ) which decreases to 0, then by Dini’s lemma it converges uniformly on compact subsets of E to 0. Moreover, it is uniformly bounded, and hence it converges to 0 for the strict topology. If the sequence (gn)n∈N ⊂ C +b (E ) is such that gn ≤ f n. Then the sequence (gn)n∈N converges to 0 for the strict topology. Assertion (2) implies that limn→∞ Λ (gn) = 0.
(3) =⇒ (4). This implication is trivial. (3) =⇒ (5). The boundedness of the functional Λ, i.e. the inequality
|Λ(f )| ≤ C f ∞, f ∈ C b(E ), enables us to write Λ in the form Λ =
Λ1−Λ2 +iΛ3−iΛ4 in such a way that Λ1 = (Λ) +
, Λ2 = (Λ) −
, Λ3 = (Λ) +
, and Λ3 = (Λ)−. From the definitions of these functionals (see the proof of assertion 2 in Proposition 1.4) assertion (3) implies that limn→∞ Λj (f n) = 0, 1 ≤ j ≤ 4, whenever the sequence (f n)n∈N ⊂ C +b (E ) decreases to 0. From Theorem 1.2 we infer that each functional Λj , 1 ≤ j ≤ 4, can be represented by a Borel measure µj: Λj(f ) =
f dµj, 1 ≤ j ≤ 4, f ∈ C B(E ). It follows
that Λ(f ) =
f dµ, f ∈ C b(E ), where µ = µ1 − µ2 + iµ3 − iµ4. (4) =⇒ (5). From the apparently weaker hypotheses in assertion (4) com-
pared to (3) we still have to prove that the functionals Λj , 1 ≤ j ≤ 4, as described in the implication (3) =⇒ (5) have the property that limn→∞ Λj (f n) = 0 whenever the sequence (f n)n∈N ⊂ C +b (E ) decreases pointwise to 0. We
will give the details for the functional Λ1 = (Λ)+. This suffices because Λ2 = ( (−Λ))+, Λ3 = ( (−iΛ))+, and Λ4 = ( (iΛ))+. So let the sequence (f n)n∈N ⊂ C +b (E ) decreases pointwise to 0. Fix ε > 0, and choose 0 ≤ g1 ≤ f 1, g1 ∈ C b(E ), in such a way that
Λ1 (f 1) = (Λ)+ (f 1) ≤ (Λ (g1)) + 1
2 ε. (1.18)
Then we choose a sequence of functions (uk)k∈N ⊂ C +b (E ) such that g1 = supn∈N
n k=1 uk =
∞ k=1 uk (which is a pointwise increasing limit), and such
that uk ≤ f k − f k+1, k ∈ N. In Lemma 1.6 below we will show that such a decomposition is possible. Then g1 −
n k=1 uk decreases pointwise to 0, and
hence by (4) we have
Λ (g1) ≤ Λ
2 ε, for n ≥ nε. (1.19)

2 ε
= n
k=1
Λ1 (f k − f k+1) + ε = Λ1 (f 1)− Λ1 (f n+1) + ε. (1.20)
From (1.20) we deduce Λ1 (f n) ≤ ε for n ≥ nε + 1. Since ε > 0 was arbitrary, this shows limn→∞ Λ1 (f n) = 0. This is true for the other linear functionals Λ2, Λ3 and Λ4 as well. As in the proof of the implication (3) =⇒ (5) from Theorem 1.2 it follows that each functional Λj , 1 ≤ j ≤ 4, can be represented by a Borel measure µj : Λj(f ) =
f dµj , 1 ≤ j ≤ 4, f ∈ C b(E ). It follows that
Λ(f ) =
f dµ, f ∈ C b(E ), where µ = µ1 − µ2 + iµ3 − iµ4. (5) =⇒ (1). The proof of assertion 1 in Proposition 1.4 then shows that
the functional Λ belongs to (C b(E ),T β) ∗
.
Lemma 1.6. Let the sequence (f n)n∈N ⊂ C +b (E ) decrease pointwise to 0, and 0 ≤ g ≤ f 1 be a continuous function. Then there exists a sequence of continuous functions (uk)k∈N such that 0 ≤ uk ≤ f k − f k+1, k ∈ N, and such that g = supn∈N
n k=1 uk =
increasing limit.
Proof. We write g = v1 = u1 + v2 = n
k=1 uk + vn+1, and vn+1 = un+1 + vn+2
where u1 = g ∧ (f 1 − f 2), un+1 = vn+1 ∧ (f n+1 − f n+2), and vn+2 = vn+1 − un+1. Then 0 ≤ vn+1 ≤ vn ≤ f n. Since the sequence (f n)n∈N decreases to 0, the sequence (vn)n∈N also decreases to 0, and thus g = supn∈N
n k=1 uk.
The latter shows Lemma 1.6.
In the sequel we write M(E ) for the complex vector space of all complex Borel measures on the polish space E . The space is supplied with the weak topology σ (E, C b(E )). We also write M+(E ) for the convex cone of all positive (= non- negative) Borel measures in M(E ). The notation M+
1 (E ) is employed for all probability measures in M+(E ), and M+
≤1(E ) stands for all sub-probability
measures in M+(E ). We identify the space M(E ) and the space (C b(E ),T β)∗.
Theorem 1.7. Let M be a subset of M(E ) with the property that for every sequence (Λn)n∈N in M there exists a subsequence (Λnk)k∈N such that
lim k→∞
iΛ(f )
, 0 ≤ ≤ 3,

Proof. First suppose that M(E ) is relatively weakly compact. Since the weak topology on M(E ) restricted to compact subsets is metrizable and separable, the weak closure of M is bounded for the variation norm. Without loss of generality we may and do assume that M itself is weakly compact. Fix f ∈ C b(E ), f ≥ 0. Consider the mapping Λ → (Λ)+ (f ), Λ ∈M(E ). Here we identify Λ = Λµ ∈ (C b(E ),T β)∗ and the corresponding complex Borel measure µ = µΛ given by the equality Λ(g) =
gdµ, g ∈ C b(E ). The mapping
Λ → (Λ) +
(f ), Λ ∈ M(E ), is weakly continuous. This can be seen as follows. Suppose Λn(g) → Λ(g) for all g ∈ C b(E ). Then (Λn)+ (f ) ≥ Λn (g) for all 0 ≤ g ≤ f , g ∈ C b(E ), and hence lim inf n→∞ (Λn)+ (f ) ≥ liminf n→∞Λn (g) = (Λ) (g). It follows that lim inf n→∞ (Λn)
+ (f ) ≥
sup0≤g≤f (Λ) (g) = (Λ)+ (f ). Since limn→∞ (Λn)+ (1) = (Λ)+ (1) we
also have lim inf n→∞ (Λn)+ (1−f ) ≥ sup0≤g≤1−f (Λ) (g) = (Λ)+ (1−f ).
Hence we see lim supn→∞ (Λn)+ (f ) ≤ (Λ)+ (f ).
In what follows we write K(E ) for the collection of compact subsets of E .
Theorem 1.8. Let M be a subset of M(E ). Then the following assertions are equivalent:
(a) For every sequence (f n)n∈N ⊂ C b(E ) which decreases pointwise to the zero
function the equality inf n∈N
sup µ∈M
(b) The equality inf K⊂E, K∈K(E)
sup µ∈M
|µ| (E ) <∞;
(c) There exists a function u ∈ H +(E ) such that for all f ∈ C b(E ) and for all µ ∈M the inequality
f dµ ≤ uf ∞ holds.
Moreover, if M ⊂M(E ) satisfies one of the equivalent conditions (a), (b) or (c), then M is relatively weakly compact.
Let Λ : C b(E ) → C be a linear functional such that inf n∈N |Λ| (f n) = 0 for every sequence (f n)n∈N ⊂ C b(E ) which decreases pointwise to zero. Here the linear functional |Λ| is defined in such a way that |Λ| (f ) = sup {|Λ(v)| : |v| ≤ f, v ∈ C b(E )} for all f ∈ C +b (E ). Then by Corollary 1.5 there exists a complex Borel measure µ such that Λ(f ) =
f dµ for all
f ∈ C b(E ). The positive Borel measure |µ| is such that |Λ| (f ) =
f d |µ| for all f ∈ C b(E ).
Proof. (a) =⇒ (b). By choosing the sequence f n = n−11 we see that supµ∈M |µ| (E ) <∞. Next let ρ be a metric on E which it a polish space, let (xn)n∈N be a dense sequence in E , and put Bk,n = {x ∈ E : ρ (x, xk) ≤ 2−n}. Choose continuous functions wk,n ∈ C b(E ) such that 1Bck,n ≤ wk,n ≤ 1Bck,n+1 . Put v,n = min1≤k≤ wk,n. Then for every n ∈ N the sequence → w,n
decreases pointwise to zero. So for given ε > 0 and for given n ∈ N there exists n(ε) such that
wn(ε),nd |µ| ≤ ε2−n for all µ ∈ M . It follows that
|µ| ∩n(ε) k=1 Bc
k,n
≤ ε, µ ∈M.
Put K (ε) = ∩∞n=1 ∪ n(ε) k=1 Bk,n. Then K (ε) is closed, and thus complete, and
completely ρ-bounded. Hence it is compact. Moreover, |µ| (E \ K (ε)) ≤ ε for all µ ∈M . Hence (b) follows from (a).
(b) =⇒ (c). This proof follows the lines of proof of assertion 1 of Proposition 1.4. Instead of considering just one measure we now have a family of measures M .
(c) =⇒ (a). Essentially speaking this is a consequence of Dini’s lemma. Here we use the following fact. If for some µ ∈ M(E ) the inequality
f dµ ≤
uf ∞ holds for all f ∈ C b(E ), then we also have f d |µ|
≤ uf ∞ for all
f ∈ C b(E ). Fix α > 0. If (f n)n∈N is any sequence in C +b (E ) which decreases pointwise to zero, then for µ ∈M we have the following estimate
f nd |µ| ≤max
∞

x∈E f n(x)
x∈E f 1(x)
. (1.21)
From the fact that the set {u ≥ α} is contained in a compact subset of E from (1.21) and Dini’s lemma we deduce that inf n∈N supµ∈M
f nd |µ| ≤
α supx∈E f 1(x) for all α > 0. Consequently, (a) follows.
Finally we prove that if M satisfies (c), then M is relatively weakly compact. First observe that µ ∈ M implies |µ| (E ) ≤ u∞. So the subset M is uniformly bounded, and since E is a polish space, the same is true for the ball {µ ∈M(E ) : |µ| (E ) ≤ u∞} endowed with the weak topology. There- fore, if (µn)n∈N is a sequence in M it contains a subsequence (µnk)k∈N such that Λ(f ) := limk→∞
f dµnk exists for all f ∈ C b(E ). Then it follows that
|Λ(f )| ≤ uf ∞ for all f ∈ C b(E ). Consequently, the linear functional Λ can be represented as a measure: Λ(f ) =
f dµ, f ∈ C b(E ). It follows that the
weak closure of the set M is weakly compact.
Definition 1.9. A family of complex measures M ⊂ M(E ) is called tight
if it satisfies one of the equivalent conditions in Theorem 1.8. Let M be a collection of linear functionals on C b(E ) which are continuous for the strict
topology. Then each Λ ∈ M can be represented by a measure: Λ(f ) = inf f dµΛ,
f ∈ C b(E ). Then the collection M of linear functionals is called tight, provided
the same is true for the family M =
µΛ : Λ ∈ M
.

to zero in fact converges uniformly on M . Assertion (b) says that the family M is tight in the usual sense as it can be found in the standard literature. Assertion (c) says that the family M is equi-continuous for the strict topology.
The following corollary says that if for M in Theorem 1.8 we choose a collection of positive measures, then the family M is tight if and only if it is relatively weakly compact. Compare these results with Stroock [224].
Corollary 1.11. Let M be a collection of positive Borel measures. Then the following assertions are equivalent:
(a) The collection M is relatively weakly compact. (b) The collection M is tight in the sense that supµ∈M µ(E ) < ∞ and
inf K∈K(E) supµ∈M µ (E \ K ) = 0.
(c) There exists a function u ∈ H +(E ) such that f dµ
≤ uf ∞ for all µ ∈M and for all f ∈ C b(E ).
Remark 1.12. Suppose that the collection M in Corollary 1.11 consists of probability measures and is closed with respect to the Levy metric. If M satisfies one of the equivalent conditions in 1.11, then it is a weakly compact subset of P (E ), the collection of Borel probability measures on E .
Proof. Corollary 1.11 follows more or less directly from Theorem 1.8. Let M be as in Corollary 1.11, and (f n)n∈N be a sequence in C b(E ) which decreases to the zero function. Then observe that the sequence of functions µ →
f nd |µ| =
f ndµ, µ ∈ M , decreases pointwise to zero. Each of these
functions is weakly continuous. Hence, if M is relatively weakly compact, then Dini’s lemma implies that this sequence converges uniformly on M to zero. It follows that assertion (a) in Corollary 1.11 implies assertion (a) in Theorem 1.8. So we see that in Corollary 1.11 the following implications are valid: (a) =⇒ (b) =⇒ (c). If M ⊂ M+(E ) satisfies (c), then Theorem 1.8 implies that M is relatively weakly compact. This means that the assertions (a), (b) and (c) in Corollary 1.11 are equivalent.
We will also need the following theorem.
Theorem 1.13. Let (µn)n∈N ⊂ M(E ) be a tight sequence (see Definition 1.9) with the property that Λ(f ) := limn→∞
f dµn exists for all f ∈
C b(E ). Let Φ ⊂ C b(E ) be a family of functions which is equi-continuous and bounded. Then Λ can be represented as a complex Borel measure µ, and
lim n→∞
dµn −
dµ
= 0.

Proof. The fact that the linear functional Λ can be represented by a Borel measure follows from Corollary 1.5 and Theorem 1.8. Assume to arrive at a contradiction that
limsup n→∞
dµn −
dµ
> 0.
Then there exist ε > 0, a subsequence (µnk)k∈N, and a sequence (k)k∈N ⊂ Φ such that
kdµnk −
kdµ
> ε, k ∈ N. (1.22)
Choose a compact subset of E in such a way that
sup ∈Φ ∞ × sup
n∈N |µn| (E \ K ) ≤ ε
16 . (1.23)
By the Bolzano-Weierstrass theorem for bounded equi-continuous families of functions, there exists a continuous function K ∈ C (K ) and a subsequence of the sequence (k)k∈N, which we call again (k)k∈N, such that
lim k→∞
k(x)− K(x) = 0. (1.24)
By Tietze’s extension theorem there exists a continuous function ∈ C b(E ) such that restricted to K coincides with K and such that || ≤ 2 sup
ψ∈Φ ψ∞.
From (1.24) it follows there exists kε ∈ N such that for k ≥ kε the inequality
sup n∈N
8 . (1.25)

≤ 1K (k − )∞ (|µnk | (K ) + |µ| (K ))

|µnk | (K )
k∈N |µnk | (E \ K ) +

≤ 3

< ε (1.27)
for k large enough. The conclusion in (1.27) contradicts our assumption in (1.22).
This proves Theorem 1.13.
Occasionally we will need the following version of the Banach-Alaoglu theorem; see e.g. Theorem 7.18. We use the notation f, µ =
E f (x) dµ(x),
f ∈ C b(E ), µ ∈ M (E ). For a proof of the following theorem we refer to e.g. Rudin [205]. Notice that any T β-equi-continuous family of measures is contained in Bu for some u ∈ H (E ). Here Bu is the collection defined in (1.28) below.
Theorem 1.15. (Banach-Alaoglu) Let u be a function in H (E ), and define the subset Bu of M (E ) by
Bu = {µ ∈M (E ) : |f, µ| ≤ uf ∞ for all f ∈ C b(E )} . (1.28)
Then Bu is σ (M (E ), C b(E ))-compact.
Since the space (C b(E ),T β) is separable, it follows that for every sequence (µn)n∈N in Bu there exists a measure µ ∈M (E ) and a subsequence (µnk)k∈N such that lim
k→∞ f, µnk = f, µ for all f ∈ C b(E ).
Instead of “σ (M (E ), C b(E ))”-convergence we often write “weak∗-convergence”, which is a functional analytic term. In a probabilistic context people usually write “weak convergence”.
1.1.3 Integral operators on the space of bounded continuous functions
We insert a short digression to operator theory. Let E 1 and E 2 be two polish spaces, and let T : C b (E 1)→ C b (E 2) be a linear operator with the property that its absolute value |T | : C b (E 1)→ C b (E 2) determined by the equality
|T | (f ) = sup {|T g| : |g| ≤ f } , f ∈ C b (E 1) , f ≥ 0,
is well-defined and acts as a linear operator from C b (E 1) to C b (E 2).

So the notion “equi-continuous for the strict topology” has a functional analytic flavor.
Definition 1.17. A family of linear operators {T α : α ∈ A}, where every T α is a continuous linear operator from C b (E 1) to C b (E 2) is called tight if for every compact subset K of E 2 the family of functionals {Λα,x : α ∈ A, x ∈ K } is tight in the sense of Definition 1.9. Here the functional Λα,x : C b (E 1)→ C is defined by Λα,x(f ) = T αf (x), f ∈ C b (E 1). Its absolute value |Λα,x| has then the property that |Λα,x| (f ) = |T α| f (x), f ∈ C b (E 1).
The following theorem says that a tight family of operators {T α : α ∈ A} is equi-continuous for the strict topology and vice versa. Both spaces E 1 and E 2 are supposed to be polish.
Theorem 1.18. Let A be some index set, and let for every α ∈ A the mapping T α : C b (E 1) → C b (E 2) be a linear operator, which is continuous for the uniform topology. Suppose that the family {T α : α ∈ A} is tight. Then for every v ∈ H (E 2) there exists u ∈ H (E 1) such that
vT αf ∞ ≤ uf ∞ , for every α ∈ A and for all f ∈ C b (E 1). (1.29)
Conversely, if the family {T α : α ∈ A} is equi-continuous in the sense that for every v ∈ H (E 2) there exists u ∈ H (E 1) such that (1.29) is satisfied. Then the family {T α : α ∈ A} is tight.
If the family {T α : α ∈ A} satisfies (1.29), then the family {|T α| : α ∈ A} satisfies the same inequality with |T α| instead of T α. The argument to see this goes in more or less the same way as we will prove the first part of Proposi- tion 1.28 below. Fix f ∈ C b (E 1), α ∈ A, and x ∈ E 1, and let the functions u ∈ H (E 1) and v ∈ H (E 2) be such that (1.29) is satisfied. Choose ϑ ∈ [−π, π] in such a way that
|v(x) |T α| (f )(x)| = |v(x)| |T α|
eiϑf
+ ≤ uf ∞ . (1.30)
From (1.30) we see that the inequality in (1.29) is also satisfied for the operators |T α|, α ∈ A.

Proof (Proof of Corollary 1.19.). Choose v ∈ H +(E ). The proof follows by considering the family of functionals Λα,x : C b(E ) → C, α ∈ A, x ∈ E , defined by Λα,xf (x) = u(x)T αf (x), f ∈ C b(E ). If the family {T α : α ∈ A} is T β-equi-continuous, then the family {Λα,x : α ∈ A, x ∈ E } is tight. For exam- ple, it then easily follows that {Λu,α,xf m : α ∈ A, x ∈ E } converges uniformly in α ∈ A, x ∈ E , to 0, provided that the sequence (f m)m∈N decreases pointwise to 0. Conversely, suppose that for any given v ∈ H + (E ), and for any sequence of functions (f m)m∈N ⊂ C b(E ) which decreases pointwise to 0, the sequence {Λv,α,xf m : α ∈ A, x ∈ E }m∈N converges uniformly to 0. Then the family {Λα,x : α ∈ A, x ∈ E } is tight: see 1.19.
Proof (Proof of Theorem 1.18.). Like in Definition 1.17 the functionals Λα,x, α ∈ A, x ∈ E 1, are defined by Λα,x(f ) = [T αf ] (x), f ∈ C b (E 1). First we suppose that the family {T α : α ∈ A} is tight. Let (f n)n∈N ⊂ C +b (E 1) be sequence of continuous functions which decreases pointwise to zero, and let v ∈ H (E 2) be arbitrary. Since the family {T α : α ∈ A} is tight, it follows that, for every compact subset K the collection of functionals {Λα,x : α ∈ A, x ∈ K } is tight. Then, since the sequence (f n)n∈N ⊂ C +b (E 1) decreases pointwise to zero, we have
lim n→∞
sup α∈A, x∈K
|Λα,x| (f n) = 0 for every compact subset K of E 1. (1.31)
From (1.31) it follows that limn→∞ supα∈A,x∈K |v(x)| |Λα,x| (f n) = 0. Hence the family of functionals {|v(x)|Λα,x : α ∈ A, x ∈ E 1} is tight. By Theorem 1.8 (see Definition 1.9 as well) it follows that there exists a function u ∈ H (E 1) such that
|v(x) [T αf ] (x)| = |v(x)Λα,x(f )| ≤ uf ∞ (1.32)
for all f ∈ C b (E 1), for all x ∈ E and for all α ∈ A. The inequality in (1.32) implies the equi-continuity property (1.29).
Next let the family {T α : α ∈ A} be equi-continuous in the sense that it satisfies inequality (1.29). Then the same inequality holds for the family {|T α| : α ∈ A}; the argument was given just prior to the proof of Theorem 1.18. Let K be any compact subset of E 1 and let (f n)n∈N ⊂ C +b (E 1) be a sequence which decreases to zero. Then there exists a function u ∈ H (E 1) such that
sup α∈A, x∈K
[|T α| f n] (x) = 1K |T α| f n∞ ≤ uf n∞ . (1.33)
From (1.33) it readily follows that limn→∞ supα∈A, x∈K [|T α| f n] (x) = 0. By Definition 1.17 it follows that the family {T α : α ∈ A} is tight.
This completes the proof of Theorem 1.18.

1. If f 1 and f 2 ∈ C b (E 1) are such that f 1 ≤ f 2, then U (f 1) ≤ U (f 2). In other words the mapping f → U f , f ∈ C b (E 1,R) is monotone.
2. If f 1 and f 2 belong to C b (E 1,R), and if α ≥ 0, then U (f 1 + f 2) ≤ U (f 1)+ U (f 2), and U (αf 1) = αU (f 1).
3. U is unit preserving: U (1E1) = 1E2 . 4. If (f n)n∈N ⊂ C b (E 1,R) is a sequence which decreases pointwise to zero,
then so does the sequence (U (f n))n∈N.
Then for every v ∈ H + (E 2) there exists u ∈ H + (E 1) such that
sup y∈E2
v(y)U (f ) (y) ≤ sup x∈E1
u(x)f (x), for all f ∈ C b (E 1) and hence
sup y∈E2
v(y)U |f | (y) ≤ sup x∈E1
u(x) |f (x)| , for all f ∈ C b (E 1). (1.34)
If the mapping U maps C b (E 1) to L∞ (E, R,E), then the conclusion about its continuity as described in (1.34) is still true provided it possesses the properties (1), (2), (3), and (4) is replaced by
4 If (f n)n∈N ⊂ C b (E 1,R) is a sequence which decreases pointwise to zero, then the sequence (U (f n))n∈N decreases to zero uniformly on compact subsets of E 2.
Proof. Put
M vU =
y∈E2
v(y) (U g) (y)
and
y∈E2
v(y) (U |g|) (y)
. (1.35)
A combination of Theorem 1.8 and its Corollary 1.11 shows that the collections
M vU and M |·| vU are tight. Here we use hypothesis 4. We also observe that
M vU = M |·| vU . This can be seen as follows. First suppose that ν ∈ M
|·| vU and
g + g∞ , ν ≤ sup y∈E2
v(y) (U |g + g∞|) (y)
≤ sup y∈E2
v(y) g∞
(v(y)U (g) (y)) + ν (E 1) g∞ . (1.36)
From (1.36) we deduce g, ν ≤ supy∈E1 (v(y)U (g) (y)), and hence M
|·| vU ⊂

v(y)U (|g|) (y). (1.37)
From (1.37) the inclusion M vU ⊂M |·| vU follows. So from now on we will write
M vU = M vU = M |·| vU . There exists a function u ∈ H +(E ) such that for all
f ∈ C b(E ) and for all µ ∈ M the inequality
f dµ ≤ supx∈E (u(x)f (x)) holds. The result in Theorem 1.20 follows from assertion in the following equalities
sup y∈E2
v(y)U f (y) = sup {f, ν : ν ∈M vU } , and (1.38)
sup y∈E2
v(y)U |f | (y) = sup {|f, ν | : ν ∈M vU } . (1.39)
The equality in (1.38) follows from the Theorem of Hahn-Banach. In the present situation it says that there exists a linear functional Λ : C b (E 1,R)→ R such that Λ(f ) ≤ sup
y∈E2
v(y)U f (y), for all f ∈ C b (E 1,R), and
Λ (1E1) = sup y∈E2
v(y)U (1E1) (y) = sup y∈E2
v(y)1E2(y) = sup y∈E2
v(y). (1.40)
Let f ∈ C b (E 1,R), f ≤ 0. Then Λ(f ) ≤ sup y∈E2
v(y)U f (y) ≤ 0. Again using
Hypothesis 4 shows that Λ can be identified with a positive Borel measure on E 1, which than belongs to M vU . Consequently, the left-hand side of (1.38) is less than or equal to its right-hand side. Since the reverse inequality is trivial, the equality in (1.38) follows. The equality in (1.39) easily follows from (1.38).
The assertion about a sub-additive mapping U which sends functions in C b (E 1) to functions in L∞ (E, R,E) can easily be adopted from the first part of the proof.
This concludes the proof of Theorem 1.20.
The results in Proposition 1.21 should be compared with Definition 3.6. We describe two operators to which the results of Theorem 1.20 are applicable. Let L be an operator with domain and range in C b(E ), with the property that for all µ > 0 and f ∈ D(L) with µf − Lf ≥ 0 implies f ≥ 0. There is a close connection between this positivity property (i.e. positive resolvent property) and the maximum principle: see Definition 3.4 and inequality (3.46). In addition, suppose that the constant functions belong to D(L), and that L1 = 0. Fix λ > 0, and define the operators U jλ : C b(E, R) → L∞ (E, R,E), j = 1, 2, by the equalities (f ∈ C b (E,R)):
U 1λf = sup K∈K(E)
inf g∈D(L)
U 2λf = inf g∈D(L)
{g ≥ f : λg − Lg ≥ 0} . (1.42)

Here the symbol K(E ) stands for the collection of all compact subsets of E . Observe that, if g ∈ D(L) is such that λg − Lg ≥ 0, then g ≥ 0. This follows from the maximum principle.
Proposition 1.21. Let the operator L be as above, and let the operators U 1λ and U 2λ be defined by (1.41) and (1.42) respectively. Then the following assertions hold true:
(a) Suppose that the operator U 1λ has the additional property that for every sequence (f n)n∈N ⊂ C b(E ) which decreases pointwise to zero the sequence
U 1λf n n∈N
does so uniformly on compact subsets of E . Then for every
u ∈ H +(E ) there exists a function v ∈ H +(E ) such that
sup x∈E
v(x)f (x), and
sup x∈E
u(x)U 1λ |f | (x) ≤ sup x∈E
v(x) |f (x)| for all f ∈ C b (E, R). (1.43)
(b) Suppose that the operator U 2λ has the additional property that for every sequence (f n)n∈N ⊂ C b(E ) which decreases pointwise to zero the sequence
U 2λf n n∈N
does so uniformly on compact subsets of E . Then for every
u ∈ H +(E ) there exists a function v ∈ H +(E ) such that the inequalities in (1.43) are satisfied with U 2λ instead of U 1λ. Moreover, for f ∈ D (Ln), µ ≥ 0, and n ∈ N, the following inequalities hold:
µnf ≤ U 2λ (((λ + µ) I − L) n
f ) , and (1.44)
f ∞ . (1.45)
In (1.45) the functions u and v are the same as in (1.43) with U 2λ replacing U 1λ.
The inequality in (1.45) could be used to say that the operator L is T β- dissipative: see inequality (3.14) in Definition 3.5. Also notice that U 1λ(f ) ≤ U λ2(f ), f ∈ C b (E, R). It is not clear, under what conditions U 1λ(f ) = U 2λ(f ). In Proposition 1.22 below we will return to this topic. The mapping U 1λ is heavily used in the proof of (iii) =⇒ (i) of Theorem 3.10. If the operator L in Proposition 1.21 satisfies the conditions spelled out in assertion (a), then it is called sequentially λ-dominant: see Definition 3.6.

U 2λ ((λ + µ) I − L) f
= inf g∈D(L)
= inf g∈D(L)
{g ≥ ((λ + µ) I − L) f :
(λ + µ) g − Lg ≥ µg ≥ ((λ + µ) I − L) (µf )} = inf
g∈D(L) {g ≥ ((λ + µ) I − L) f : λg − Lg ≥ 0, g ≥ µf } ≥ µf. (1.46)
Repeating the arguments which led to (1.46) will show the inequality in (1.44). From (1.46) and (1.43) with U 2λ instead of U 1λ we obtain
sup x∈E
U 2λ ((λ + µ) f − Lf ) (x)
≤ sup x∈E
f (x), (1.47)
for µ ≥ 0 and f ∈ D (Ln). The inequality in (1.45) is an easy consequence of (1.47). This concludes the proof of Proposition 1.21.
Proposition 1.22. Let the operator L with domain and range in C b(E ) have the following properties:
1. For every λ > 0 the range of λI −L coincides with C b(E ), and the inverse
R(λ) := (λI − L) −1
exists as a positivity preserving bounded linear operator from C b(E ) to C b(E ). Moreover, 0 ≤ f ≤ 1 implies 0 ≤ λR(λ)f ≤ 1.
2. The equality lim λ→∞
λR(λ)f (x) = f (x) holds for every x ∈ E , and f ∈ C b(E ).
3. If (f n)n∈N ⊂ C b(E ) is any sequence which decreases pointwise to zero, then for every λ > 0 the sequence (λR(λ)f n)n∈N decreases to zero as well.
Fix λ > 0, and define the mappings U 1λ and U 2λ as in (1.41) and (1.42) respectively. Then the (in-)equalities
sup
f ; µ > 0, k ∈ N ≤ U 1λ(f ) ≤ U 2λ(f ) (1.48)
hold for f ∈ C b (E, R). Suppose that f ≥ 0. If the function in the left extremity of (1.48) belongs to C b(E ), then the first two terms in (1.48) are equal. If it belongs to D(L), then all three quantities in (1.48) are equal.
Proof. First we observe that for every (λ, x) ∈ (0,∞) × E there exists a Borel measure B → r (λ ,x,B) such that λr (λ ,x,E ) ≤ 1, and R(λ)f (x) = E
f (y)r (λ,x,dy), f ∈ C b(E ). This result follows by considering the functional Λλ,x : C b(E )→ C, defined by Λλ,x(f ) = R(λ)f (x). In fact
r (λ ,x,B) = sup K∈K(E),K⊂B
inf {R(λ)f (x) : f ≥ 1K} , B ∈ E.
This result follows from Corollary 1.5. Often we write

f (y)r (λ,x,dy) , B ∈ E, f ∈ C b(E ).
Observe that the mapping B → R(λ) (f 1B) is a positive Borel measure on E . Moreover, by Dini’s lemma we see that
lim n→∞
λR(λ)f n(x) = 0, λ0 > 0. (1.49)
whenever the sequence (f n)n∈N ⊂ C b(E ) decreases pointwise to zero. From Theorem 1.18 and its Corollary 1.19 it then follows that the family of operators {λR(λ) : λ ≥ λ0} is equi-continuous for the strict topology T β , i.e. for every function u ∈ H +(E ) there exists a function v ∈ H +(E ) such that
λ uR(λ)f ∞ ≤ vf ∞ for all λ ≥ λ0 and all f ∈ C b(E ). (1.50)
Fix f ∈ C b (E, R) and λ > 0. Next we will prove the
U 1λ(f ) ≥ sup
. (1.51)
A version of this proof will be more or less retaken in (3.137) in the proof of the implication (iii) =⇒ (i) of Theorem 3.10 with D1 + L instead of L. First we observe that for g ∈ D(L) we have
λg(x)− Lg(x) = lim µ→∞
µ (g(x)− µR (λ + µ) g(x)) , x ∈ E. (1.52)
If g ∈ D(L) is such that λg − Lg ≥ 0, then (λ + µ) g − Lg ≥ µg, and hence g ≥ µR(λ+ µ)g for all µ > 0. If g ≥ µR(λ+µ)g, then µ (g − µR(λ + µ)g) ≥ 0, and by (1.52) we see λg − Lg ≥ 0. So that we have the following equality of subsets
{g ∈ D(L) : λg − Lg ≥ 0} = {g ∈ D(L) : g ≥ µR (λ + µ) g for all µ > 0} . (1.53)
From (1.53) we infer
g ∈ D(L) : g ≥ sup
k g
.
(1.54) Let g ∈ D(L) be such that g ≥ f 1K and such that λg − Lg ≥ 0, then (1.54)
implies g ≥ sup µ>0, k∈N
(µR (λ + µ)) k
,
µ > 0, k ∈ N, are integral operators, and bounded Borel measures are inner- regular (with respect to compact subsets), we obtain
g ≥ sup µ>0, k∈N
(µR (λ + µ)) k
sup K∈K(E)
inf g∈D(L)
{g ≥ f 1K : λg − Lg ≥ 0} ≥ sup µ>0, k∈N
(µ ((λ + µ) I − L)) k
f.
(1.55) The inequality in (1.55) implies (1.51) and hence, since the inequality U 1λ(f ) ≤ U 2λ(f ) is obvious, the inequalities in (1.48) follow. Here we employ the fact that λg − Lg ≥ 0 implies g ≥ 0. Fix a compact subset K of E , and f ≥ 0, f ∈ C b(E ). If the function g = sup
µ>0, k∈N (µ (λ + µ) I − L)
k f belongs to C b(E ),
then g ≥ f 1K , and g ≥ µR (λ + µ) g for all µ > 0. Hence it follows that
sup µ>0, k∈N
(µ (λ + µ) I − L) k
f ≥ inf {g ≥ f 1K : g ≥ µR (λ + µ) g, g ∈ C b(E )} .
(1.56) Next we show that τ β- lim
α→∞ αR(α)f = f . From the assumptions 2 and 3, and
from (1.50) it follows that D(L) = R
(βI − L)−1
is T β-dense in C b(E ).
Therefore let g be any function in D(L), and let u ∈ H +(E ). Consider, for α > λ0 the equalities
f − αR(α)f = f − g − αR(α) (f − g) + g − αR(α)g
= f − g − αR(α) (f − g)−R(α) (Lg) , (1.57)
and the corresponding inequalities
u (f − αR(α)f )∞ ≤ u (f − g)∞ + uαR(α) (f − g)∞ + uR(α) (Lg)∞ ≤ u (f − g)∞ + v (f − g)∞ +
u∞ α Lg∞ . (1.58)
So that for given ε > 0 we first choose g ∈ D(L) in such a way that
u (f − g)∞ + v (f − g)∞ ≤ 2
3 ε. (1.59)
Then we choose αε ≥ λ0 so large that u∞ α Lg∞ ≤
1
u (f − αR(α)f )∞ ≤ ε, for α ≥ αε. (1.60)
From (1.60) we see that T β- lim α→∞
αR(α)f = f . So that the inequality in (1.56)
implies:
(µ (λ + µ) I − L) k
f ≥ inf {g ≥ f 1K : g ≥ µR (λ + µ) g, g ∈ D((L)} ,
(1.61)
and consequently U 1λ(f ) ≤ f λ := sup µ>0, k∈N
(µ (λ + µ) I − L) k
f . It follows that
f λ = U 1λ(f ) provided that f and f λ both belong to C b(E ). If f λ ∈ D(L), then f λ = U 1λ(f ) and f λ ≥ µR(λ + µ)f λ, and consequently λf λ − Lf λ. The conclusion U 2λ (f ) = f λ is then obvious.
This finishes the proof of Proposition 1.22.

1.2 Strong Markov processes and Feller evolutions 27
In the following proposition we see that a multiplicative Borel measure is a point evaluation.
Proposition 1.23. Let µ be a non-zero Borel measure with the property that fgdµ =
f dµ
gdµ for all functions f and g ∈ C b(E ). Then there exists
x ∈ E such that
f dµ = f (x) for f ∈ C b(E ).
Proof. Since µ = 0 there exists f ∈ C b(E ) such that 0 =
f dµ =
1dµ = 1. Let
f and g be functions in C +b (E ). Then we have
f gd |µ| = sup

=
f d |µ|
gd |µ| . (1.62)
From (1.62) it follows that the variation measure |µ| is multiplicative as well. Since E is a polish space, the measure |µ| is inner regular. So there exists a compact subset K of E such that |µ| (E \ K ) ≤ 1/2, and hence |µ| (K ) > 1/2. Since |µ| is multiplicative it follows that |µ| (K ) = 1 = |µ| (E ). It follows that the multiplicative measure |µ| is concentrated on the compact subset K , and hence it can be considered as a multiplicative measure on C (K ). But then there exists a point x ∈ K such that |µ| = δx, the Dirac measure at x. So there exists a constant cx such that µ = cx |µ| = cxδx. Since µ(E ) = δx(E ) = 1 it follows that cx = 1. This proves Proposition 1.23.
1.2 Strong Markov processes and Feller evolutions
In the sequel E denotes a separable complete metrizable topological Hausdorff space. In other words E is a polish space. The space C b(E ) is the space of all complex valued bounded continuous functions. The space C b(E ) is not only equipped with the uniform norm: f ∞ := supx∈E |f (x)|, f ∈ C b(E ), but also with the strict topology T β . It is considered as a subspace of the bounded Borel measurable functions L∞(E ), also endowed with the supremum norm.

(i) It leaves C b(E ) invariant: P (s, t)C b(E ) ⊆ C b(E ) for 0 ≤ s ≤ t ≤ T ; (ii) It is an evolution: P (τ, t) = P (τ, s) P (s, t) for all τ , s, t for which
0 ≤ τ ≤ s ≤ t and P (t, t) = I , t ∈ [0, T ]; (iii) It consists of contraction operators: P (s, t)f ∞ ≤ f ∞ for all t ≥ 0
and for all f ∈ C b(E ); (iv) It is positivity preserving: f ≥ 0, f ∈ C b(E ), implies P (s, t)f ≥ 0; (v) For every f ∈ C b(E ) the function (s,t,x) → P (s, t)f (x) is continuous
on the diagonal of the set {(s,t,x) ∈ [0, T ] × [0, T ] × E : 0 ≤ s ≤ t ≤ T } in the sense that for every element (t, x) ∈ (0, T ] × E the equality
lim s↑t,y→x
P (s, t)f (y) = f (x) holds, and for every element (s, x) ∈ [0, T ) × E
the equality lim t↓s,y→x
P (s, t)f (y) = f (x) holds.
(vi) For every t ∈ [0, T ] and f ∈ C b(E ) the function (s, x) → P (s, t)f (x) is Borel measurable and if (sn, xn)n∈N is any sequence in [0, t] × E such that sn decreases to s ∈ [0, t], xn converges to x ∈ E , and lim
n→∞ P (sn, t) g (xn)
exists in C for all g ∈ C b(E ), then lim n→∞
P (sn, t) f (xn) = P (s, t)f (x).
(vii) For every (t, x) ∈ (0, T ] × E and f ∈ C b(E ) the following equality holds: lims↑t, s≥τ P (τ, s) f (x) = P (τ, t) f (x), τ ∈ [0, t).
Remark 1.25. Since the space E is polish, the continuity as described in (v) can also be described by sequences. So (v) is equivalent to the following condition: for all element (t, x) ∈ (0, T ] × E and (s, x) ∈ [0, T ) × E the equalities
lim n→∞
P (sn, t) f (yn) = f (x) and lim n→∞
P (s, tn) f (yn) = f (x) (1.63)
hold. Here (sn)n∈N ⊂ [0, t] is any sequence which increases to t, (tn)n∈N ⊂ [s, T ] is any sequence which decreases to s, and (yn)n∈N is any sequence in E which converges to x ∈ E . If for f ∈ C b(E ) and t ∈ [0, T ] the function (s, x) → P (s, t)f (x), (s, x) ∈ [0, t] × E , is continuous, then (vi) and (vii) are satisfied. If the function (s,t,x) → P (s, t) f (x) is continuous on the space {(s,t,x) ∈ [0, T ] × [0, T ] × E : s ≤ t}, then the propagator P (s, t) possesses property (v) through (vii). In Proposition 1.26 we will single out a closely related property. Its proof is part of the proof of part (b) in Theorem 1.39.
Proposition 1.26. Let the family {P (τ, t) : 0 ≤ τ ≤ t ≤ T } which possesses properties (i) through (iv) of Definition 1.24. Suppose that for every f ∈ C b(E ) the function (τ , t , x) → P (τ, t) f (x) is continuous on the space
{(τ ,t ,x) ∈ [0, T ] × [0, T ] × E : τ ≤ t} . (1.64)
Then for every f ∈ C b ([0, T ] × E ) the function (τ , t , x) → P (τ, t) f (t, ·) (x) is continuous on the space in (1.64).
It is noticed that assertions (iii) and (iv) together are equivalent to

1.2 Strong Markov processes and Feller evolutions 29
In the presence of (iii), (ii) and (i), property (v) is equivalent to:
(v) lim t↓s u (P (s, t)f − f )∞ = 0 and lim
s↑t u (P (s, t)f − f )∞ = 0 for all f ∈
C b(E ) and u ∈ H (E ). So that a Feller evolution is in fact T β-strongly
continuous in the sense that, for every f ∈ C b(E ) and u ∈ H (E ),
lim (s,t)→(s0,t0)
s≤s0≤t0≤t
u (P (s, t) f − P (s0, t0) f )∞ = 0, 0 ≤ s0 ≤ t0 ≤ T. (1.65)
Remark 1.27. Property (vi) is satisfied if for every t ∈ (0, T ] the function (s, x) → P (s, x; t, E ) = P (s, t) 1(x) is continuous on [0, t] × E , and if for every sequence (sn, xn)n∈N ⊂ [0, t] × E for which sn decreases to s and xn converges to x, the inequality lim supn→∞ P (sn, t) f (xn) ≥ P (s, t) f (x) holds for all f ∈ C +b (E ). Since functions of the form x → P (s, t)f (x), f ∈ C b(E ), belong to C b(E ), it is also satisfied provided for every f ∈ C b(E ) we have
lim n→∞
P (sn, t) f = P (s, t) f, uniformly on compact subsets of E.
This follows from the inequality:
|P (sn, t) f (xn)− P (s, t) f (x)| ≤ |P (sn, t) f (xn)− P (s, t) (xn)| + |P (s, t) f (xn)− P (s, t) f (x)|
where sn ↓ s, xn → as n→ ∞, and f ∈ C b(E ).
Proposition 1.28. Let {P (s, t) : 0 ≤ s ≤ t ≤ T } be a family of operators having property (i) and (ii) of Definition 1.24. Then property (iii ) is equivalent to the properties (iii) and (iv) together.
Moreover, if such a family {P (s, t) : 0 ≤ s ≤ t ≤ T } possesses property (i), (ii) and (iii), then it possesses property (v) if and only if it possesses (v ).
Proof. First suppose that the operator P (s, t) : L∞(E ) → L∞(E ) has the properties (iii) and (iv), and let f ∈ C b(E ) be such that 0 ≤ f ≤ 1. Then by (iii) and (iv) we have 0 ≤ P (s, t)f (x) ≤ supy∈E f (y) ≤ 1, and hence (iii) is satisfied. Conversely, let f ∈ C b(E ) and x ∈ E . Then by (iii) the operator P (s, t) satisfies
P (s, t)f (x) = [P (s, t)f ] (x) ≤ sup y∈E f (y) ≤ f ∞ . (1.66)
There exists ϑ ∈ [−π, π] such that by (1.66) we have
|P (s, t)f (x)| =
from which (iii) easily follows.

[0, T ] in such a way that s0 ≤ t0. For the converse implication we employ Theorem 1.18 with the families of operators
{P (sm, s0) : 0 ≤ sm ≤ sm+1 ≤ s0} and {P (t0, tm) : t0 ≤ tm+1 ≤ tm ≤ T } (1.67)
respectively. Let (f n)n∈N be a sequence functions in C +b (E ) which decreases pointwise to zero. Then by Dini’s lemma and assumption (v) we know that
lim n→∞
sup m∈N
sup x∈K
P (t0, tm) f n(x) = 0 (1.68)
for all compact subsets K of E . From (1.68) we see that the sequences of operators in (1.67) are tight. By Theorem 1.18 it follows that they are equi- continuous. If the pair (s, t) belongs to [0, s0] × [t0, T ], then we write
P (s, t) f − P (s0, t0) f = P (s, t0) (P (t0, t)− I ) f + (P (s, s0)− I ) P (s0, t0) f. (1.69)
Let v be a function in H (E ). Since the first sequence in (1.67) is equi- continuous and by invoking (1.69) there exists a function v ∈ H (E ) such that the following inequality holds for all m ∈ N and all f ∈ C b(E ):
u (P (sm, tm) f − P (s0, t0) f )∞ ≤ v (P (t0, tm)− I ) f ∞ + u (P (sm, s0)− I ) P (s0, t0) f . (1.70)
In order to prove the equality in (1.65) it suffices to show that the right-hand side of (1.70) tends to zero if m → ∞. By the properties of the functions u and v it suffices to prove that
lim m→∞
1K (P (sm, s0) f − f )∞ = lim m→∞
1K (P (t0, tm) f − f )∞ = 0 (1.71)
for every compact sub

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