Feller semigroups
Shiu-Tang Li
Finished: April 19, 2013
Last updated: November 2, 2013
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Feller semigroups
Definitions 1.1 Let E be a LCCB space, which means second countable
locally compact Hausdorff space. A complex function f is said to vanish at
infinity if for every > 0, there exists a compact set K ⊂ E s.t. |f (x)| < for all x ∈ K c . The class of all continuous function f on E which vanish at
infinity is called C0 (E). Notice that C0 (E) is a Banach space.
Definition 1.2 Let X be a Banach space. We define B(X) to be the
collection of all bounded linear maps from X to X.
Definition 1.3 Let X be a Banach space. A semigroup {S(t)}t≥0 is a
collection of mappings from [0, ∞) to B(X), which satisfies
(i) S(t1 + t2 ) = S(t1 ) ◦ S(t2 ) for all t1 , t2 ≥ 0.
(ii) S(0) = IX .
If in addition we have limt↓0 S(t)x = x for every x ∈ X, then {S(t)}t≥0 is
called a c0 -semigroup. If, furthermore, we have limt↓0 kS(t) − IX kB(X) = 0,
then {S(t)}t≥0 is called a uniformly continuous semigroup.
Definition 1.4 A Feller semigroup {S(t)}t≥0 is a c0 -semigroup on the
Banach space C0 (E).
Theorem 1.5 Let {Pt }t≥0 be transition function on (E, E ). Then {Pt }t≥0
is a Feller semigroup if and only if
(i) Pt (C0 (E)) ⊂ C0 (E) for all t ≥ 0.
(ii) For all f ∈ C0 (E), for all x ∈ E, limt↓0 (Pt f )(x) = f (x).
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Proof. If {Pt }t≥0 is Feller, then all the properties above obviously hold. Conversely, assume that we have all the properties above for {Pt }t≥0 . We have
for all x ∈ E, f ∈ C0 (E),
f (x) = lim(Pt f )(x) = lim((Pt P0 )f )(x) = lim(Pt (P0 f ))(x) = (P0 f )(x),
t↓0
t↓0
t↓0
so P0 is the identity map.
Now the goal is to prove (Pt f )(x) → f (x) uniformly in x ∈ E when t ↓ 0.
We divide the proof into several steps.
Step 1. (Defining resolvents)
Pn·2n −1Since (t, x) 7→ (Pt f )(x) is right continuf )(x), which is (R+ ×
ous in t, we define gn (t, x) := j=0 1{ jn ≤t< j+1
} (P j+1
2
2n
2n
E, B(R+ ) ⊗ E ) measurable, and gn (t, x) → (Pt f )(x) for all (t, x) ∈ (R+ × E).
It follows that the map (t, x) 7→ (Pt f )(x) is also (R+ × E, B(R+ ) ⊗ E ) measurable.
R∞
Fix f ∈ C0 (E). We define the map Up f : x 7→ 0 e−pt (Pt f )(x) dt, p > 0.
Pn·2n −1 −p· jn
Since the map is just the pointwise limit of x 7→ 21n · j=0
e 2 (P jn f )(x),
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which are all continuous maps, we have (Up f )(x) is measurable for all x ∈ E.
Step 2. (A key property for Feller resolvents) Let T be the topology of (E, E ) s.t. E = σ(T ). Assume that (E, E ) is not compact. Then,
by one-point compactifiction, we may define E ∗ = E ∪ {∞}, with the new
topology T ∗ := T ∪ {U ⊂ E ∗ : E ∗ \ U compact in (E, T )}. Since E is LCCB
and hence locally compact Hausdorff, (E ∗ , T ∗ ) is a compact Hausdorff space
[See Richard Bass, real analysis for graduate students, Sec 20.9]. Besides,
since (E, T ) is second countable, every base of (E, T ) has a countable subfamily which is still a base for (E, T ), and we may use this fact to construct
a countable base for (E ∗ , T ∗ ), which shows (E ∗ , T ∗ ) is also second countable.
In addition, every function f : E → R belongs to C0 (E) if and only if
the extension f˜ : E ∗ → R, f˜(∞) = 0, f˜(x) = f (x) for x ∈ E is a continuous
function on (E ∗ , T ∗ ).
We claim that (Up f )(x) ∈ C0 (E) for all f ∈ C0 (E). First, we fix
g
f ∈ C0 (E). Since Pt f ∈ C0 (E), we may denote its extension by P
t f , which
∗
belongs to C(E ) by the above arguments. We also denote the extension
R ∞ −pt
g
g
g
of Up f by U
(P
p f , and its not hard to see that Up f = 0 e
t f )(x) dt. To
∗
g
see Up f ∈ C(E ), by the sequence lemma for first countable spaces, it suf2
fices to check that for any xn → x ∈ E ∗ , (Up f )(xn ) → (Up f )(x) ((E ∗ , T ∗ ) is
second countable and hence first countable). But this is simply a fact from
dominated convergence theorem; note that since kf kC0 (E) = M for some
g
M > 0, we have kPt f kC0 (E) ≤ M for all t ≥ 0, and hence kP
t f kC(E ∗ ) ≤ M
for all t ≥ 0. Now, again by the arguments in the preceding paragraph,
Up f ∈ C0 (E) under the assumption that (E, E ) is not compact. When
(E, E ) is compact, it’s much easier to prove Up f ∈ C0 (E), by using the same
ideas as above, but avoiding all the compactification troubles.
Step 3. (More properties for Feller resolvents) We claim that
(pUp f )(x) → f (x) for all x ∈ E when p → ∞. Indeed, we first fix x ∈ E, for
any > 0, we may pick δ = δ(x) > 0 so that |(Pt f )(x) − f (x)| R< whenever
∞
0 ≤ t < δ. By dominated convergence theorem we have limp→∞ δ pe−pt (Pt f )(x) dt =
Rδ
Rδ
0. Since we also have 0 pe−pt |(Pt f )(x) − f (x)| dt ≤ 0 pe−pt dt ≤ and
R δ −pt
pe f (x) dt = f (x)(1 − epδ ), the claim follows.
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Next, we prove the resolvent equation Up f − Uq f = (q − p)Up Uq f =
(q − p)Uq Up f , for all f ∈ C0 (E). To see this,
Z ∞
(Up Uq f )(x) =
e−pt (Pt Uq f )(x) dt
Z ∞
Z0 ∞
Z
−pt
e−qs (Ps f )(y) ds dt
=
e
Pt (x, dy)
0
Z0 ∞
Z E∞
=
e−pt
e−qs (Ps+t f )(x) ds dt
Z0 ∞ Z ∞ 0
=
e−pt e−q(r−t) (Pr f )(x) dr dt
Z0 ∞ t
Z r
e−pt e−q(r−t) dt dr
(Pr f )(x)
=
0
Z0 ∞
1
e−pr − e−qr
dr =
(Up f − Uq f ).
=
(Pr f )(x)
q−p
q−p
0
From the resolvent equation, we know that for any p > 0, f ∈ C0 (E),
Up f = Uq (f + (q − p)Up f ) ∈ Uq (C0 (E)). This showsR Up (C0 (E)) is indepen∞
dent of p > 0. Besides, for all x ∈ E, |(pUp f )(x)| ≤ 0 pe−pt kPt f kC0 (E) dt =
kPt f kC0 (E) , so kpUp f kC0 (E) ≤ kPt f kC0 (E) ≤ kf kC0 (E) .
Step 4. (Up (C0 (E)) is dense in C0 (E)) Assume that Up (C0 (E)) is
not dense in C0 (E). Thus, Up (C0 (E)) is a proper closed linear subspace of
C0 (E). Now we recall an application of Hahn-Banach theorem [Rudin, real
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and complex analysis, Theorem 5.19]:
Let M be a linear subspace of a normed linear space X, and let x0 ∈ X.
Then x0 is in the closure M of M if and only if there is no bounded linear
functional f on X s.t. f (x) = 0 for all x ∈ M but f (x0 ) 6= 0.
Therefore, there is some bounded linear functional Λ : C0 (E) → R and
some g ∈ C0 (E) \ Up (C0 (E)), so that Λ(f ) = 0 for all f ∈ Up (C0 (E)), but
Λ(g) 6= 0.
Next, we state a version of C0 -Riesz representation theorem [Lawrence
Baggett, functional analysis, Theorem 1.4], as follows:
Let X be a second countable locally compact Hausdorff space, and let φ
be a bounded linear functional on C0 (X). Then φ is the difference φ1 − φ2 of
two positive linear functionals φ1 and φ2 , Rwhence there exists a unique finite
signed Borel measure µ such that φ(f ) = X f dµ for all f ∈ C0 (X).
R
We hence have 0 = pΛ(Up g) = Λ(pUp g) = E (pUp g)(x) dµ(x) for all p > 0,
where µ is a finite signed Borel measure, Λ, g is defined in preceding paragraphs. Since
R kpUp gkC0 (E) ≤ kgkC0 (E) , by bounded convergence theorem we
have 0 = E g(x) dµ(x) = Λ(g) 6= 0, a contradiction. Therefore, Up (C0 (E))
is dense in C0 (E).
Step 5 (The last approximation). For any f ∈ C0 (E), x ∈ E, we
have
|(Pt Up f )(x) − (Up f )(x)|
Z ∞
Z ∞
Z
−ps
e−ps (Ps f )(x) ds|
e (Ps f )(y) ds −
=| Pt (x, dy)
ZE∞
Z 0
Z0 ∞
=|
e−ps
Pt (x, dy)(Ps f )(y) ds −
e−ps (Ps f )(x) ds|
0
E
0
Z ∞
Z ∞
−ps
=|
e (Ps+t f )(x) ds −
e−ps (Ps f )(x) ds|
0
Z ∞
Z0 ∞
=|ept
e−ps (Ps f )(x) ds −
e−ps (Ps f )(x) ds|
Zt ∞
Z0 ∞
Z t
−ps
−ps
pt
≤|e
e (Ps f )(x) ds −
e (Ps f )(x) ds| + |
e−ps (Ps f )(x) ds|
t
0
Z ∞ t
≤|ept − 1| · kf kC0 (E) · |
e−ps ds| + tkf kC0 (E) .
t
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As a result, k(Pt Up f ) − (Up f )kC0 (E) ≤ |ept − 1| · kf kC0 (E) + tkf kC0 (E) → 0,
as t ↓ 0. Now, give > 0, and arbitrary g ∈ C0 (E), there exists some
h ∈ C0 (E) and p > 0 so that kg − Up hkC0 (E) < 3 . Therefore,
kPt g − gkC0 (E) ≤ kPt Up h − Pt gkC0 (E) + kPt Up h − Up hkC0 (E) + kg − Up hkC0 (E)
≤ 2kg − Up hkC0 (E) + kPt Up h − Up hkC0 (E)
2
<
+ kPt Up h − Up hkC0 (E)
3
< ,
where the last inequality holds for all t > 0 small enough.
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