(Generator of Levy process) Let
be a Levy process in
with generator
.
The space
is included in
and there are
,
and a measure
on
with
such that for any
we
have

Proof

According to the definition (
Levy process
), the
increment
is infinitely divisible for any
.
Hence, if
is the characteristic exponent of
:
then
According to the proposition
(
Levy-Khintchine formula
2
)
thus
Therefore, there are
,
and a measure
such that for any
,
We are aiming to
calculate
in terms of
and
.
The
is connected to the
via the
relationships
By the stationary property of the process, the matrix
is only dependent on the distance between
and
:
We introduce the
notation
thus
For any
let
be a function such
that
Then
Therefore
We note
that
and
conclude