1. Introduction
Let us consider the parabolic PDE:
(1)
We study in this paper the behavior of 
when
tend to zero, and
. We suppose that the matrix of the second order coefficients of 
is degenerate, in fact we formulate here a hypoellipticity condition of Hörmander type (see e.g. David Nualart [1]). Diédhiou and Manga in [2] studied the limit of 
with a nondegenerate condition of the matrix. In Freidlin & Sowers[3], three cases are considered, with the assumption that the matrix is non, but we formulate here a hypoellipticity condition of. Since the parameter 
(homogeneization parameter) decreases quickly than 
(large deviations principle parameter) to zero we must homogenize first and apply the large deviations principle.
We use essentially probabilistic tools to solve our problem.
Let 
a probability filtered space. We consider the 
valued process 
solution of the SDE:
(2)
where 
and
is a d-dimensional standard Brownian motion.
We assume that 
and 
are smooth mappings from
, and periodic with period one in each direction.
The mapping 
is assumed to be of the form :
where 
and 
are in 
for every 
and
where 
be the collection of periodic continuous mappings from 
into
.
The infinitesimal generator 
is gigen by
and where 
is a bounded function and we set
Let set
since 
is continuous we have 
We assume that 
is periodic in each direction, with respect to the first argument, and it verifies:
• 
• There exists 
bounded such that
with
and we assume that
Let us consider the progressive measurable process 
solution of the BSDE:
By Pardoux and Peng [4], we have for all
,
The matrix 
(where 
is the symbol of transposition) is degenerate. Let us consider the Definition 1.1 The Lie bracket between the vector fields 
and 
is defined by
where 
We assume that the matrix 
of the column vectors 
verifies the strong Hörmander condition, defined by the Definition 1.2 Let 
be the set of Lie brackets of 
of order lower than 
at the point
We say that the matrix 
satisfies the strong Hörmander condition (called SHC) if for all
, there exists 
such that 
generates 
We organize this paper as follows. Section 2 contains the results of large deviations principle. In Section 3 we study the behavior of the solution of the PDE (1).
2. Large Deviations Principle
Since, 
(when we set
) we have a problem of homogenization because the matrix
is not elliptic.
Since 
tends to zero faster than
, the homogenization dominates, and the large deviations principle will be applied to the problem with constant coefficients.
For the homogeneization in the hypoellipticy case, we use the results of Diédhiou and Pardoux [5] and Pardoux [4,6,7].
Setting:
, we have
where 
is a standard Brownian motion.
The 
-valued process
, is a Feller process, then has a unique invariant measure
, and we have 
when 
see [5].
We assume that
(3)
and the homogenized coefficients see [3] are
Let us define, for each 
and 
We have
The details of the calculation of this limit are the same as in Freidlin and Sowers [3].
In order to establish a large deviations principle, we will consider the Theorem 2.1 ([8]) Fix 
and 
Assume that 1) For each 
is well-defined in
.
2) The origin is in the interior of the set
.
3) The set 
has a nonempty interior 
is well-defined for all
and
Then the random variables 
satisfy a large deviations principle with rate function 
defined by
The limit 
satisfies the conditions 1) and 2). For the condition 3) we may assume more that the matrix 
is strictly positive-definite. In fact it is not a strong assumption, for Example 2.2 If we choose 
and
this matrix satisfies the Hörmander condition, and
The invariant measure 
has the density
.
Then we have
Let us consider
by the assumption on
, we get
Thus the form of 
and the assumption on 
imply that 3) is true. 
We have the Theorem 2.3 (Freidlin and Sowers [3]) Fix 
and assume that the assumption (3) is true. For every 
and 
the family 
of 
valued random variables satisfies a large deviations principle (LDP) with rate function
Furthermore, this LDP is uniform for all 
and 
Proof: See Freidlin and Sowers [3]. 
Let us consider some definitions:
Since the function 
is convex we can show that
So we have the Theorem 2.4 For all
, we assume that the assumption (3) holds. The family
of 
-valued random variables satisfies a Large Deviations Principle (LDP) with rate function 
for all 
Proof: See Freidlin and Sowers [3].
3. Asymptotic Behavior of 
We want to apply the technics used by [6], so we consider now the BSDE:
We know that for all
, the solution 
of the PDE is of the form:
and by the Feynman-Kac formula, we have
Our aim is to study the behavior of the 
when 
tends to zero.
• Remark 3.1
• If
, then
• 
• In the other cases, if
where 
is Lipschitz continuous, then
uniformly in any compact set of 
We give the Definition 3.2 A functional 
is a stopping time if for all 
and all
for all 
and 
imply
Let us set 
the set of stopping times and 
the set of elements 
of 
such that there exists 
such that for all
with the convention 
is hence a well defined element of 
and 
is the open set associated.
is an element of 
(resp.
) if and only if, for all 
(resp.
) where
Let us consider the function 
defined in 
by,
where
Let 
and 
be a partition of
,
We have
then we deduce that 
We have the Theorem 3.3 For 
we have 1)
2)
uniformly in any compact set 
of
.
3)
in all compact set 
of
.
Proof: For first item, the proof is the same as in [2].
For the second point we can see that there exists 
such that
The third item is an immediate consequence of 1).
NOTES