Proposition
Let
,
then
Note that when any of the operators
acts on the transitional probability
the following rules apply. The backward operator
acts on the backward space argument
and extends the backward time argument
backward in time. The forward *operator
acts on the forward space set
and extends the forward parameter
forward in time. The precise manner of the extension is natural in every case.
The next observation is on the symmetry of
and
.
Notation
We introduce the notation
for the natural scalar product of function and
measure:
Proof
By the
definitions,
