We translate into the language of semi-group theory Bismut's Calculus on boundary processes (Bismut (1983), Lèandre (1989)) which gives regularity result on the heat kernel associated with fractional powers of degenerated Laplacian. We translate into the language of semi-group theory the marriage of Bismut (1983) between the Malliavin Calculus of Bismut type on the underlying diffusion process and the Malliavin Calculus of Bismut type on the subordinator which is a jump process.

Let

Let us introduce the local time

Classically [

Bismut [

The main remark of Bismut in [

Recently, we have translated into the language of semi-group theory the Malliavin Calculus of Bismut type for diffusion [

On the general problematic on this work, we refer to the review papers of Léandre [

Let us recall some basis on the study of fractional powers of operators [

On

We perform the same algebraic considerations on

We get a theorem which enters in the framework of the Malliavin Calculus for heat-kernel.

Let one suppose that the Malliavin condition in

If the quadratic form

We give simple statements to simplify the exposition. It should be possible by the method of this paper to translate the results of [

We consider the vector fields on

We consider the generator

Let us consider the Hoermander’s type generator associated with the smooth Lipschitz vector fields on

One has the relation

Let us consider the semi-group

For

For the integrability conditions, we refer to the appendix.

We remark that

By Lemma

Therefore,

We consider the Malliavin generator

If

We can perform the same improvements as in [

We can consider the generator

If

We refer to the appendix for the proof and the subsequent estimates.

Let us show from where come these identities, by using (

if

if

According to the fact that

Let us consider diffusion type generator of the previous part:

Therefore, we can write

According the line of stochastic analysis, we consider a generator

It is not clear that

For

By linearity,

Let us show from where this formula comes. In the previous part, we have done a perturbation of the leading Brownian motion

Bismut’s idea to state hypoellipticity result is to take the derivative in

First of all, let us compute

For

Let us introduce the vector fields on

If the Volterra expansion converges, then

Let us show from where this formula comes. Classically,

Let us introduce the generator on

We get the following.

For

We have

For

If the Volterra expansion converges, then

Analogous formula works for

Let us compute

Therefore,

Let

For

If the Volterra expansion converges, then

We can summarize the previous considerations in the next theorem.

If

Let one suppose that

It is enough to show that wecan approximate

if

if

if

We can consider vector fields at the manner of (

If

The proof of Theorem

There are two partial derivatives to treat:

the partial derivative in the time of the subordinator

the partial derivatives in the space of the underlying diffusion

Let us begin by the most original part of Bismut's Calculus on boundary process, that is, the integration by parts in the time

We look at (

For a conveniently enlarged semi-group in the manner of Theorem

If

We consider

By the same way, we deduce that if

We could do integration by parts to each order in order to show that the semi-group

Let

By the result of the appendix,

If

We consider a convex function decreasing from

We deduce from that that

We could improve (

We consider a very small

It remains to show the following.

For all

We remark that if we consider only functions of

Let

Following the idea of [

Let us consider an improvement of the Gronwall lemma: if

We remark that

By Gronwall lemma,

By the same procedure, we get the following.

Let be

and we get the following.

Let

We can show (