The goal of this paper is to give the proof of the two next theorems originally proved by [1] by using stochastic analysis and the Malliavin Calculus of Bismut type for jump processes of [28].
Let us consider functions positive with compact support on continuous except in 0 equal to near 0 with .
Let us introduce functions with bounded derivatives at each order, equal to 0 in 0 with values in .
We consider the Markov generator
We do the following hypothesis.
Hypothesis 2.1.
There exists an such that the family of vectors generates .
generates a convolution linear semigroup in probability measures acting on differentiable bounded functions . satisfies the parabolic equation
Under Hypothesis 2.1, [20, 29, 30] proved that has a smooth heat kernel :
We denote
where .
Theorem 2.2.
If , then there exists , such that such that,

(i)
(i),

(ii)
is a submersion in .
Remark 2.3.
Let us explain heuristically the theorem. Let be the process with independent increments associated to the generator
where . The processes are independents, and the time of their jumps are disjoints. We put
Then,
The theorem explains that we have to jump in a finite numbers of jumps in a submersive way from to if we want . Let us give some explanations what we mean about this fact, because the jump process has in fact an infinite number of jumps because the Lévy measure is of infinite mass. We take
has generator
The jump process
has only a finite number of jumps because its Lévy measure is of finite mass and its law gives a good approximation of the law of if is small enough!
We consider some vectors and a smooth vector fields with bounded derivatives at each order. We consider the generator
It generates a Markov semigroup ,
if is bounded differentiable. If , the is classically related to fractional powers of the Laplacian [31].
We do the following Hypothesis.
Hypothesis 2.4.
Consider .
In such a case, [19, 29, 30] has proven that there exists a smooth heat kernel :
We consider and we denote by . We introduce the differential impulsive equation starting from :
We denote
Theorem 2.5.
The condition implies that there exists , , , and , such that:
(i),
(ii) is a submersion in .
Remark 2.6.
Let us explain heuristically this theorem. We consider the processes with independent increments . We consider the stochastic differential equation
Then,
It has since an infinite number of jumps. We take
has a finite number of jumps and has generator
We consider the stochastic differential equation
The law of is a good approximation of the law of if is small enough. This express the fact that by a finite number of jumps, has to pass from to in a submersive way if .