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 1If is a smooth bounded solution of (1), and and are all bounded measurable functions. Then