Forward and backward generators for Feller process.

he notion of generator was introduced in the section
(
Forward and backward
generators
). If the process has a homogeneous transition function then the
generator does not depend on the time parameter.

Proposition

(Backward Kolmogorov for
Feller process) Let
be a Feller process in
,
is the associated backward generator and
.
We have

0.
.

1.
,
.

2. The function
is differentiable with respect to the strong topology in
and

3.

4. The set
is dense in
.

5.
is a closed operator in
.

Proof

(0),(1)
and the limit exists because
:

(2)

(3)

The claim (4) follows from the two
observations:
and

(5). We introduce the
notations
Let
,
and
.
We aim to show that
and
.
We
calculate
since
Note that
hence
and we conclude
and
.

Proposition

(Properties of Feller resolvent
3) For every
the map
from
to
is one-to-one and onto and its inverse is
.
(see the definition (
Resolvent of
Feller process
))

(Dissipation property of
Feller process) Let
be a generator of a Feller process in
,
and
is such that
.
Then

Proof

By
definition
and

Proposition

(Martingale property of
Feller process) Let
be a generator of the Feller process
in
,
,
and
is the
-generated
filtration. Then the
process
is an
-martingale.

Conversely, if
and there exists a
such
that
is an
-martingale
for every
then
and
.

Proof

To prove the direct statement we calculate for
:
By definition of
and Markovian property of
,
thus
It remains to note that according to the proposition
(
Backward Kolmogorov for
Feller
process
)-3,
Hence, we make a change
in the
integral:
and
conclude
To prove the converse statement we rewrite the
condition
as
Hence, we verify the conclusion of the converse statement as
follows
Hence,
and
.