It is now time to give some applications of the theory of stochastic differential equations to parabolic second order partial differential equations. In particular we are going to prove that solutions of such equations can represented by using solutions of stochastic differential equations. This representation formula is called the Feynman–Kac formula.

As usual, we consider a filtered probability space which satisfies the usual conditions and on which is defined a -dimensional Brownian motion . Again, we consider two functions and and we assume that there exists such that

Let be the second order differential operator

where .

As we know, there exists a bicontinuous process such that for ,

Moreover, as it has been stressed before, for every , and

As a consequence, if is a Borel function with polynomial growth, we can consider the function

**Theorem.** *For every , is a Markov process with semigroup . More precisely, for every Borel function with polynomial growth and every ,
*

**Proof.** The key point, here, is to observe that solutions are actually adapted to the natural filtration of the Brownian motion . More precisely, there exists on the space of continuous functions a predictable functional such that for :

Indeed, let us first work on where is small enough. In that case, as seen previously, the process is the unique fixed point of the application defined by

Alternatively, one can interpret this by observing that is the limit of the sequence of processes inductively defined by

It is easily checked that for each there is a predictable functional such that

which proves the above claim when is small enough. To get the existence of for any , we can proceed

With this hands, we can now prove the Markov property. Let . For , we have

Consequently, from uniqueness of solutions,

We deduce that for a Borel function with polynomial growth,

because is a Brownian motion independent of

**Theorem*** Let be a Borel function with polynomial growth and assume that the function
is , that is once differentiable with respect to and twice differentiable with respect to . Then solves the Cauchy problem
in , with the initial condition .*

**Proof.** Let and consider the function . According the previous theorem, we have

As a consequence, the process is a martingale. But from Ito’s formula the bounded variation part of is which is therefore 0. We conclude

**Exercise*** Show that if is a function such that and have polynomial growth, then the function is . Here, we denote by the Hessian matrix of .*

**Theorem.** * Let be a Borel function with polynomial growth. Let be a solution of the Cauchy problem
with the initial condition .
If there exists a locally integrable function and , such that for every and ,
then .
*

**Proof.** Let and, as before, consider the function . As a consequence of Ito’s formula, we have

where is a local martingale with quadratic variation . The conditions on and $u$ imply that this quadratic variation is integrable. As a consequence, is a martingale and thus

The previous results may be extended to study parabolic equations with potential as well. More precisely, let be a bounded function. If is a Borel function with polynomial growth, we define

.

The same proofs as before will give the following theorems.

**Theorem.** *For every and every Borel function with polynomial growth and every ,
*

**Theorem.** * Let be a Borel function with polynomial growth and assume that the function
is , that is once differentiable with respect to and twice differentiable with respect to . Then solves the Cauchy problem
in , with the initial condition
.
*

**Theorem.** * Let be a Borel function with polynomial growth. Let be a solution of the Cauchy problem
with the initial condition . If there exists a locally integrable function and , such that for every and ,
,
then .
*