More generally, by using the same methods as in the previous Lecture, we can introduce iterated derivatives. If , we set

.

We may then consider as a square integrable random process indexed by and valued in . By using the integration by parts formula, it is possible to prove, as we did it in the previous Lecture, that for any , the operator is closable on . We denote by the domain of in , it is the closure of the class of cylindric random variables with respect to the norm

,

and

We have the following key result which makes Malliavin calculus so useful when one wants to study the existence of densities for random variables.

**Theorem.***(P. Malliavin) Let be a measurable random vector such that:*

- For every , ;
- The matrix

is invertible.

*Then has a density with respect to the Lebesgue measure. If moreover, for every ,
then this density is .
*

The matrix is often called the Malliavin matrix of the random vector .

This theorem relies on the following lemma of Fourier analysis for which we shall use the following notation: If is a smooth function then for , we denote

**Lemma.*** Let be a probability measure on such that for every smooth and compactly supported function ,
where , , . Then is absolutely continuous with respect to the Lebesgue measure with a smooth density.*

**Proof.** The idea is to show that we may assume that is compactly supported and then use Fourier transforms techniques. Let , and . Let be a smooth function on such that on the ball and outside the ball . Let be the measure on that has a density with respect to . It is easily seen, by induction and integrating by parts that for every smooth and compactly supported function ,

where , , . Now, if we can prove that under the above assumption has a smooth density, then we will able to conclude that has a smooth density because and are arbitrary. Let

be the Fourier transform of the measure . The assumption implies that is rapidly decreasing (apply the inequality with ). We conclude that has a smooth density with respect to the Lebesgue measure and that this density is given by the inverse Fourier transform formula:

We may now turn to the proof of the Theorem.

The proof relies on the integration by parts formula for the Malliavin derivative. Let be a smooth and compactly supported function on . Since , we easily deduce that and that

Therefore we obtain

We conclude that

As a consequence, we obtain

By using inductively this integration by parts formula, it is seen that for every , , there exists an integrable random variable such that,