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 in-
variant 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 theo- rem 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
pre- sented 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 mea- sure, 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
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 topo- logical 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
continu- ous 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 defini- tion 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-
6 1 Strong Markov processes
(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)
8 1 Strong Markov processes
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
contin- uous, 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) coin- cides 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 ).
10 1 Strong Markov processes
≤ ∞
f ∞
where u(x) = ∞
j=1 2−j1Kj (x) |µ| (E ).
2 We decompose the functional Λ into a combination of four positive
func- tionals: Λ = (Λ)+ − (Λ)− + i (Λ)+ − i (Λ)− where the linear
function- als (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
se- quence (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 de- creases 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 de- scribed 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,
14 1 Strong Markov processes
Proof. First suppose that M(E ) is relatively weakly compact.
Since the weak topology on M(E ) restricted to compact subsets
is metrizable and separa- ble, 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
mea- sure 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 fol- lows. 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 com- pact. 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
.
16 1 Strong Markov processes
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 col- lection 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 prob- ability 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 de- creases 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.
1.1 Strict topology 17
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 the- orem; 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 con- tained 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∗-conver- gence”, 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).
1.1 Strict topology 19
So the notion “equi-continuous for the strict topology” has a
functional ana- lytic 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} sat- isfies 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 oper- ators |T α|, α ∈ A.
20 1 Strong Markov processes
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 , de- fined 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 point- wise 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.
1.1 Strict topology 21
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 sub-
sets 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)
1.1 Strict topology 23
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 asser- tions 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 opera- tor
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 func- tional 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.
28 1 Strong Markov processes
(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 equiva- lent 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.
30 1 Strong Markov processes
[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