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.
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