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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 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
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
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 .
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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