Ito’s formula provides probabilistic representation of PDE with Cauchy terminal data backward in time. By simple change of variable in time, it also provides a useful probabilistic representation of PDE with Cauchy initial data, which recovers Duhamel’s formula.
We will consider the following PDE with initial data:
where is a time-homogeneous Feller-Dynkin operator. As an example, if
is given, then the above PDE is inhomogeneous heat equation.
We fix . Suppose
is smooth bounded solution of (1), then the function
satisfies
Let be the Markov process with its generator
, denoted by
. It is straightforward to write
in the form of
Since is time-homogeneous, there exists time-homogeneous Feller-Dynkin semigroup
satisfying
for smooth test functions
. We can rewrite
in terms of
as
Finally, we could replace in the above by
and obtain the probabilistic representation of
.
Theorem 1 If
is a smooth bounded solution of (1), and
and
are all bounded measurable functions. Then