﻿Limit of the Solution of a PDE in the Degenerate Case

Applied Mathematics
Vol.4 No.2(2013), Article ID:28206,5 pages DOI:10.4236/am.2013.42051

Limit of the Solution of a PDE in the Degenerate Case*

Alassane Diedhiou

Département de Mathématiques, Université de Ziguinchor, Ziguinchor, Senegal

Received May 21, 2012; revised January 10, 2013; accepted January 17, 2013

Keywords: Homogenization; Large Deviations Principle; Stochastic Differential Equations

ABSTRACT

In this paper we show that we can have the same conclusion for the limit of the solution if we suppose the case of hypoellipticity.

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).

REFERENCES

1. D. Nualart, “The Malliavin Calculus and Related Topics. Probability and Its Applications,” Springer-Verlag, New York, 1995.
2. A. Diédhiou and C. Manga, “Application of Homogeneization and Large Deviations to a Parabolic Semilinear Equation,” Journal of Mathematical Analysis and Applications, Vol. 342, No. 1, 2008, pp. 146-160.
3. M. I. Freidlin and R. B. Sowers, “A Comparison of Homogenization and Large Deviations, with Applications to Wavefront Propagation,” Stochastic Processes and Their Applications, Vol. 82, No. 1, 1999, pp. 23-32. doi:10.1016/S0304-4149(99)00003-4
4. É. Pardoux and S. Peng, “Backward Stochastic Differential Equations and Quasi-Linear Parabolic Differential Equations,” In: B. L. Rozovskii and R. B. Sowers, Eds., Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Information Sciences, Vol. 176, 1992, pp. 200-217. doi:10.1007/BFb0007334
5. A. Diédhiou and É. Pardoux, “Homogenization of Periodic Semilinear Hypoelliptic PDES,” Annales de la faculté des sciences de Toulouse Mathématiques, Vol. 16, No. 2, 2007, pp. 253-283.
6. É. Pardoux, “Homogenization of Linear and Semilinear Second Order Parabolic PDEs with Periodic Coefficients: A Probabilistic Approch,” Journal of Functional Analysis, Vol. 167, No. 2, 1999, pp. 498-520. doi:10.1006/jfan.1999.3441
7. É. Pardoux, “BSDEs, Weak Convergence and Homogenization of Semilinear PDEs,” In: F. H. Clarke and R. J. Stern, Eds., Nonlinear Analysis, Differential Equations and Control, Springer, Berlin, 1999, pp. 503-549.
8. A. Dembo and O. Zeitouni, “Large Deviations Techniques and Applications,” Jones and Bartlett, Boston, 2010.

NOTES

*This work was supported, in part, by grants from FIRST (Fonds d’Impulsion pour la Recherche Scientique et Technique).