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 :
![](https://www.scirp.org/html/11-7400860\a8892c7c-1dc3-48d1-8643-38701c33e975.jpg)
where
and
are in
for every
and
![](https://www.scirp.org/html/11-7400860\9f806223-61f4-411c-bb30-42977626bf60.jpg)
where
be the collection of periodic continuous mappings from
into
.
The infinitesimal generator
is gigen by
![](https://www.scirp.org/html/11-7400860\b44a8ace-bf90-4b69-b5c8-06e0ec64d06a.jpg)
and where
is a bounded function and we set
![](https://www.scirp.org/html/11-7400860\b4fe9789-b2dd-441f-9889-438b5c2cb70f.jpg)
Let set
![](https://www.scirp.org/html/11-7400860\d4c96813-899e-4a0c-9216-785fcec6b771.jpg)
since
is continuous we have ![](https://www.scirp.org/html/11-7400860\2efd125e-59a5-4eba-87e9-b8f3a018f1a7.jpg)
We assume that
is periodic in each direction, with respect to the first argument, and it verifies:
• ![](https://www.scirp.org/html/11-7400860\9dc6767d-b09b-4966-9308-e5691bef49c2.jpg)
• There exists
bounded such that
![](https://www.scirp.org/html/11-7400860\da592b67-a730-477a-b0ef-4148b1954d09.jpg)
with
![](https://www.scirp.org/html/11-7400860\a4aec360-310c-4d78-b92e-a4a11607841c.jpg)
and we assume that
![](https://www.scirp.org/html/11-7400860\5737b0c5-b99e-4b06-83c3-887e5234041b.jpg)
Let us consider the progressive measurable process
solution of the BSDE:
![](https://www.scirp.org/html/11-7400860\0edfdb6d-9991-4359-94dd-c6a856ac7783.jpg)
By Pardoux and Peng [4], we have for all
,
![](https://www.scirp.org/html/11-7400860\c81b1138-feb9-44fe-b7cc-6b034d4bfc82.jpg)
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
![](https://www.scirp.org/html/11-7400860\ae81faa0-5715-4646-85e6-aa6f70f505d8.jpg)
where ![](https://www.scirp.org/html/11-7400860\f278df94-d7c9-4601-beff-68e9696b4376.jpg)
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
![](https://www.scirp.org/html/11-7400860\6c0e5854-6c3f-4fd0-ae42-5b64aeeffea5.jpg)
We say that the matrix
satisfies the strong Hörmander condition (called SHC) if for all
, there exists
such that
generates ![](https://www.scirp.org/html/11-7400860\88013692-91e0-4474-8283-6d89c3a41be8.jpg)
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
![](https://www.scirp.org/html/11-7400860\56fc741d-4c4d-4f42-a07f-2897e16a522e.jpg)
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
![](https://www.scirp.org/html/11-7400860\2c8ede0c-9773-4cc3-b414-9227c2a94471.jpg)
Let us define, for each
and ![](https://www.scirp.org/html/11-7400860\39cf93f2-1c76-4922-be66-730bbc1f90e1.jpg)
![](https://www.scirp.org/html/11-7400860\7e3a33e2-a1ee-464b-8c59-48ec6a237bd6.jpg)
We have
![](https://www.scirp.org/html/11-7400860\ca35df35-fce6-43a2-b94e-bc672e6168b5.jpg)
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
![](https://www.scirp.org/html/11-7400860\1cf4cba7-3bcb-48c8-951f-4d6fcc888f5a.jpg)
Then the random variables
satisfy a large deviations principle with rate function
defined by
![](https://www.scirp.org/html/11-7400860\176b7186-e208-41f3-a7bd-cb9555d019a8.jpg)
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
![](https://www.scirp.org/html/11-7400860\3570074c-0013-4bf1-b901-6b324a98a205.jpg)
this matrix satisfies the Hörmander condition, and
![](https://www.scirp.org/html/11-7400860\8eeae97b-ea8e-4dd1-9a43-1b6ef1c98469.jpg)
The invariant measure
has the density
.
Then we have
![](https://www.scirp.org/html/11-7400860\f04c6ec8-2a4e-454e-8981-2a4a203c305b.jpg)
Let us consider
![](https://www.scirp.org/html/11-7400860\e1ed56dd-65c6-490f-9933-1d60c11fefb1.jpg)
by the assumption on
, we get
![](https://www.scirp.org/html/11-7400860\e75b7178-c367-4d17-8ecf-8a7912a668c9.jpg)
Thus the form of
and the assumption on
imply that 3) is true. ![](https://www.scirp.org/html/11-7400860\3286ab39-bed6-4a39-a844-f198c749f0ea.jpg)
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
![](https://www.scirp.org/html/11-7400860\923f16b6-1415-431b-bc82-e513a0ef4879.jpg)
Furthermore, this LDP is uniform for all
and ![](https://www.scirp.org/html/11-7400860\4298b09a-5069-4d7e-b62f-2a97e474fc6a.jpg)
Proof: See Freidlin and Sowers [3]. ![](https://www.scirp.org/html/11-7400860\f1a3a310-74f3-4bb7-9c0a-32508cb72adb.jpg)
Let us consider some definitions:
![](https://www.scirp.org/html/11-7400860\179bcb7e-f6b5-4a82-b099-95a191c63869.jpg)
Since the function
is convex we can show that
![](https://www.scirp.org/html/11-7400860\5d6128de-15e7-40d1-b8b3-fbd873017c23.jpg)
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 ![](https://www.scirp.org/html/11-7400860\8f2ecbc6-71f9-4279-8eb6-777095e5bd60.jpg)
Proof: See Freidlin and Sowers [3].![](https://www.scirp.org/html/11-7400860\41b22318-d933-43b8-8f66-27628e7fe88d.jpg)
3. Asymptotic Behavior of ![](https://www.scirp.org/html/11-7400860\28ed0971-b160-44dd-b088-52b3f31de0be.jpg)
We want to apply the technics used by [6], so we consider now the BSDE:
![](https://www.scirp.org/html/11-7400860\5e1b7fb0-900a-4ad3-892d-119473636051.jpg)
We know that for all
, the solution
of the PDE is of the form:
![](https://www.scirp.org/html/11-7400860\b3ac979f-0f24-48fd-b876-ce3733ed152c.jpg)
and by the Feynman-Kac formula, we have
![](https://www.scirp.org/html/11-7400860\a9f4ee59-a85c-44c5-87b6-74d503004d74.jpg)
Our aim is to study the behavior of the
when
tends to zero.
• Remark 3.1
• If
, then
• ![](https://www.scirp.org/html/11-7400860\ca7484fb-e828-449e-a444-908dfe256de0.jpg)
• In the other cases, if
![](https://www.scirp.org/html/11-7400860\c54e8da6-c7f9-49eb-8a9b-c5fded7c4be4.jpg)
where
is Lipschitz continuous, then
![](https://www.scirp.org/html/11-7400860\78e9a349-e851-4256-b616-81b258d5bcb2.jpg)
uniformly in any compact set of ![](https://www.scirp.org/html/11-7400860\d273599c-46b8-4bcb-9a45-da6f22350ebd.jpg)
We give the Definition 3.2 A functional ![](https://www.scirp.org/html/11-7400860\3519e6b1-d0c1-4f4f-8908-324b06127ebe.jpg)
is a stopping time if for all
and all
for all
and
imply
![](https://www.scirp.org/html/11-7400860\ef31ae0c-7840-48cb-992b-66cd7c476bf1.jpg)
Let us set
the set of stopping times and
the set of elements
of
such that there exists
such that for all
![](https://www.scirp.org/html/11-7400860\b25b5697-0bfa-4ca9-afc7-81dd22add1d6.jpg)
with the convention
is hence a well defined element of
and
is the open set associated.
![](https://www.scirp.org/html/11-7400860\e05a481d-9c55-4a1d-81c3-899bb9986c48.jpg)
is an element of
(resp.
) if and only if, for all
(resp.
) where
![](https://www.scirp.org/html/11-7400860\29b62286-537b-4d72-9c12-f52a152c73fe.jpg)
Let us consider the function
defined in
by,
![](https://www.scirp.org/html/11-7400860\1eef984b-a688-4fb5-9cfc-eeecb7db3e15.jpg)
where
![](https://www.scirp.org/html/11-7400860\f1a33f1a-248d-41a8-8675-2b38143f6753.jpg)
Let
and
be a partition of
,
![](https://www.scirp.org/html/11-7400860\c45b7bc9-313c-447b-92f6-7d407a3427ec.jpg)
We have
![](https://www.scirp.org/html/11-7400860\bc482ec1-b7d6-43a4-bc50-74048bdc7463.jpg)
then we deduce that ![](https://www.scirp.org/html/11-7400860\7b62bd58-1efd-4127-8a8a-de482b77a8b5.jpg)
We have the Theorem 3.3 For
we have 1)
![](https://www.scirp.org/html/11-7400860\cbdc6dad-7e6f-490e-9089-f1985c6bc509.jpg)
2)
![](https://www.scirp.org/html/11-7400860\5cc1afbb-ce1a-4325-82de-b3717942eb6d.jpg)
uniformly in any compact set
of
.
3)
![](https://www.scirp.org/html/11-7400860\92502925-9004-4735-883d-9bea2d9d85b7.jpg)
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
![](https://www.scirp.org/html/11-7400860\8c8d2ebf-7410-4039-8df5-f08e125c86cf.jpg)
The third item is an immediate consequence of 1).
NOTES