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such that for x>
0
and positive constants a
,
b
∞
e
−
xs
)μ(
d
s).
f(x)
=
a
+
bx
+
(
1
−
0
+
Proof
See [94, Theorem 3.9.4].
If we choose as Lévy process a subordinator, Bernstein functions allow a com-
plete characterization of the process. It is based on the observation that subordinators
can be described via their convolution semigroup.
Definition 16.3.4
A family
(η
t
)
t
≥
0
of bounded Borel measures on
R
is called con-
volution semigroup on
R
if the following conditions are fulfilled:
(i)
η
t
(
R
)
≤
1, for all
t
≥
0,
(ii)
η
s
∗
η
t
=
η
s
+
t
,
s,t
≥
0 and
μ
0
=
δ
0
,
(iii)
η
t
→
→
δ
0
vaguely as
t
0,
where
δ
0
denotes the Dirac measure at 0. By vague convergence of a sequence
(η
t
)
t>
0
of measures to
η
0
we mean that for all continuous functions with compact
support
u
∈
C
0
(
R
)
,wehave
lim
t
u(x)η
t
(
d
x)
=
u(x)η
0
(
d
x).
→
0
R
R
The relation between convolution semigroups and Bernstein functions is given in
the following theorem.
Theorem 16.3.5
be a Bernstein function
.
Then
,
there exists a
unique convolution semigroup (η
t
)
t
≥
0
supported on
Let f
:
(
0
,
∞
)
→ R
[
0
,
∞
) such that
e
−
tf (x)
,x>
0
and t>
0
,
L
(η
t
)(x)
=
(16.16)
:=
0
e
−
zx
η
t
(
d
z)
,
for
appropriate η
t
and x>
0.
Conversely
,
for any convolution semigroup (η
t
)
t
≥
0
sup-
ported by
L
L
holds
,
where
denotes the Laplace transform
,
i
.
e
.
(η
t
)(x)
[
0
,
∞
) there exists a unique Bernstein function f such that
(
16.16
)
holds
.
Proof
See, e.g. [94, Theorem 3.9.7].
We recall the correspondence between convolution semigroups and Lévy pro-
cesses.
Theorem 16.3.6
Let X be a Lévy process
,
where for each t
≥
0
X(t) has law η
t
,
then (η
t
)
t
≥
0
is convolution semigroup
.
Proof
See [3, Proposition 1.4.4].
The semigroup of subordinated Feller processes can now be characterized.
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