(Forward and backward
generators) Let
be a Markov process in
and
are the associated propagators. We define the backward and forward
"generators"
and
as follows:

1.
,
(Backward Kolmogorov generator)
.

2.
,
(Forward Kolmogorov generator) .

A function
is included in the domain
of
if the limit 1 exists.

A measure
is included in the domain
of
if the limit 2 exists.

Proposition

(Kolmogorov equations in general setting).
Let
be a Markov process in
and
is the associated transition function. For any
,
and
we have