Limit of the Solution of a PDE in the Degenerate Case

Limit of the Solution of a PDE in the Degenerate Case

*

Alassane Diedhiou

Département de Mathématiques, Université de Ziguinchor, Ziguinchor, Senegal Email: [email protected]

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

ABSTRACT

In this paper we show that we can have the same conclusion for the limit of the solution ,

u if we suppose the case of

hypoellipticity.

Keywords: Homogenization; Large Deviations Principle; Stochastic Differential Equations

1. Introduction

Let us consider the parabolic PDE:

 

 

 

 

 

,

, ,

,

, d

1

, , ,

0, , IR

u x

t x L u t x f u t x

t

u x g x x



 







 _{ } 

 _  

 



 _{ }

 

 





 ,



(1)

We study in this paper the behavior of ,

u when

,

 tend to zero, and lim_,0 0

 . We suppose that the matrix of the second order coefficients of L_,

u

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



, , 

 

t t₀ a probability filtered space. We

d







consider the IR



d1 valued process



Xtx, , 

 _solu-

tion of the SDE:

, , , ,

, , ,

, , 0

d d

x x

x t

t t

x

X X

d

t

X W B t

X x



 



 

    

 

    

    

 _



 

 









d



(2)

where _{: IR}d _IRd d_, , _{: IR}d _{IR ,} and

B 

 _  _



W tt; 0 is a d-dimensional standard Brownian mo-

tion.

We assume that  and _B, _{are smooth mappings} from , and periodic with period one in each direc- tion.

d IR

The mapping _B, _{is assumed to be of the form :}

, ,

0 1 2

B  B B B 



  

  

where B B0, 1 and B2, are in



for every

d d

IR , IR







0, 0



  and



d d



, 2

, 0 _{IR ,IR}

lim 0,

p

B 

   

 _

where



_{IR , IR}d d



p

C be the collection of periodic con-

tinuous mappings from _IRd into _IRd. The infinitesimal generator L, is gigen by

2

, ,

, 1 1

2

d d

ij i

i j i j i i

x x

L a B

x x x



 _{ }

 

 

   

 _{ }  _{ }

  

   











and where



_{IR , IR}d



g  is a bounded function and we set

 

IR

. sup

d

x

g x g



  

Let set

 



d



0 IR : 0 ,

G  x g x 

since g is continuous we have o

0 = 0.

G G

We assume that is periodic in each direction, with respect to the first argument, and it verifies:

f

 _{IR ,}d

 

_{,1 0}

x f x

  

 There exists



_IRd _{IR, IR bounded such that}



C 

 

,

 

,

f x y C x y y

with

 





 

, 0, IR , 0,1

, 0, IR , 1

d

d

C x y x y

C x y x y

    

    

*_{This work was supported, in part, by grants from FIRST (Fonds}

and we assume that

 

 

 

d

IR

max , ,0 0, IR .

y C x y C x C x   x

Let us consider the progressive measurable process solution of the BSDE:



, , ,t x_, , , ,t x

Y Z







, , ,

, , , , , , , , ,

, , , 2 , , , 0

1 _{, d}

1 _{d ,0 ,}

IE d .

t t x

t x t x r t x

s t r

s t

t x

r r

s t

t x r

X

Y g X f Y

Z W s t

Z r



  







  

 

  

 



_{  }

 

  

 _{ } 

  











  









r

, ,

By Pardoux and Peng [4], we have for all

 

_{t x, }



_{0, }



_IRd_,

 

, ,

0

, t x.

u t x Y

The matrix a (where is the symbol of

transposition) is degenerate. Let us consider the

 



Definition 1.1 The Lie bracket between the vector fields A and j Ak is defined by

, ,

j k j k k j

A A A A A A

   

 

where i .

j k j k i

A A A A

x

 _{ } 



 

We assume that the matrix  of the column vectors j

 verifies the strong Hörmander condition, defined by the

Definition 1.2 Let H n x



,



be the set of Lie brackets of



j

 

x



_{1 j d} of order lower than n at the point

d

IR .

x

We say that the matrix  satisfies the strong Hör- mander condition (called SHC) if for all , there exists such that

d

IR

x

IN

x

n  H n x



_x,



generates IRd.

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, lim_0 0

 _ (when we set   ) we have a

problem of homogenization because the matrix

 

 

a x  x is not elliptic.

Since  tends to zero faster than , the homogeni-

zation 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: 2

, ,

, , 1 x

x t

t

X  X 



  

   



  



 

 

, we have









, , , , , , , ,

, , 0

d d

,

x x x

t t t

x

X B X t X W

x X

d t

   





 _





 _{ }

 

 _



  



   



    





 

where



, _{: 0}



t

W t

d

is a standard Brownian motion. The T -valued process X_tx, ,_{, is a Feller process,}

then has a unique invariant measure , and we have

0,

_ when 0 see [5]. We assume that

   

d

0 0 d 0

B z  z 



T

, (3)

and the homogenized coefficients see [3] are





   



 



   

d

d

0 1 0

0 0 0

ˆ _{d ;}

ˆ ˆ _{d .}

B I B B x x

A I B I B x x



  

  

    





T

T

Let us define, for each T 0 and xIR ,d

 

, ,

,

d

1

log IE exp , ,

0, IR .

x

T x T

g   X 



  

 _{  }

 

 

 



  _

 

We have

 

 

, ₀ ,

d 1

lim

1

, , , ,

2

T x T x

g g

x T A B IR .

 

    





 

  _  _ 

 

 

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

0

T  xIR .d

1) For each d

 

,

IR ,gT x

  is well-defined in



 ,



.

2) The origin is in the interior of the set

 



d



, IR :g_{T x}

    .

3) The set



IR : ,

 



d T x

A  g    has a no-

nempty interior Ao, g_{T x,}

 

 is well-defined for all o

,, A  and

 

o ,

, ,

lim sup _{T x} .

A A

g

  



 

  

Then the random variables



x, , _{: 0}



satisfy a T

large deviations principle with rate function 1 ,

T x

I defin-

ed by 1 0 0

1 1

0 .

2 4π

1 1

0

4π 2

A

 

 

 

 

_  _

 

 _ 

 

 

 

_d



 



1

, ,

IR d

sup , ,

IR .

T x T x

I z z g

z



 



 

 ▲

The limit gT x,

 

 satisfies the conditions 1) and 2).

For the condition 3) we may assume more that the matrix

A is strictly positive-definite. In fact it is not a strong

assumption, for

Let us consider

 



 



d

1 1

IR

sup , , IR ,



    



 

  

  d

Example 2.2 If we choose d 3, B₀0 and

by the assumption on A, we get













1 0

, , 0 sin 2π , 0 cos 2π

x y z x

x



 

 

  

 

 

 





1 1 1 _{, , IR}

2 A B B

 _  _ _ _

 d_.

Thus the form of _1 and the assumption on

A

imply that 3) is true.



this matrix satisfies the Hörmander condition, and _{We have the}





















1 0 0

1 cos 4π 1

, , 0 sin 4π .

2 2

1 cos 4π

1 0 sin 4π

2 2

x

x y z x

x x



 

 

 

 

 

 



 

 

 

 

Theorem 2.3 (Freidlin and Sowers [3]) Fix 0

and assume that the assumption (3) is true. For every

and 0 0

0

T 

d

IR

x  T T, the family





of

valued random variables satisfies a large devia- tions principle (LDP) with rate function

, , _:

x T

X  0

d IR

- 

1 1

, , IR

T x

z x

I z T z

T



 

 _{ } 

 

 d_.

The invariant measure 0 has the density Furthermore, this LDP is uniform for all 0 T T₀

and xIR .d



, ,



2 I1d



, ,

p x y z  x x y z

T



.

Proof: See Freidlin and Sowers [3].



Then we have Let us consider some definitions:

 

1



 



 

1

0 0

d ,if is absolutely continuous and 0

, if not

T

T

s s x

S   



 

   



 

Since the function _1 is convex we can show that





, ,

, , , , , , ,

, , 2 , , 0

1 _{, d}

1 _{d , 0}

IE d

t x

x t x r x

s t r

s t

x

r r

s t

x r

X

Y g X f Y

Z W s t

Z r



  







  

 

  

 



_{  }

 

  

 _{ } 

  











  







  











r

 





   



1 d





 



1 1

0, ;IR , 0 , ₀

inf d .

T

T x T y

y x

s s T

T

     



 

 _{ }

 







 

So we have the

Theorem 2.4 For all , we assume that the

assumption (3) holds. The family 0

T 



x, , _{;0 ; 0}



t

X   t T _ _,

We know that for all

 

_,



_0,



_IRd

t x    , the solu-

tion ,

 

_,

u t x of the PDE is of the form: of 



 

0,T ; IRd

 

T



-valued random variables satisfies a Large Deviations Principle (LDP) with rate function

1 0

S  for all 



 

0,T ; IR .d



 

, 0 ,

, x, , , t

 

u  t x Y 

Proof: See Freidlin and Sowers [3].



_{and by the Feynman-Kac formula, we have}





, ,

, , , , , ,

0

0

1

IE exp , d .

x t

x x r x

t r

X

Y g X C Y



  

 r

    

     

 

    









  



  





3. Asymptotic Behavior of

u

,

We want to apply the technics used by [6], so we

consider now the BSDE: Our aim is to study the behavior of the ,

 

_,

when tends to zero. 

Remark 3.1

 If g 1, then  x IR ,d   0,

, , ,

0 x t 1, dIP

s

Y  s

    _{d as}

 In the other cases, if

 

_,

 

_{0, ,}

 

_IRd



₁



C x y  y  x y   , 

where  is Lipschitz continuous, then

, , , 0

limsup x t 1

Y  

uniformly in any compact set of



_{0, }



_{IR .}d

We give the

Definition 3.2 A functional :C



 

0, , IRt d





 

0,t is a stopping time if for all   , C



 

0, , IRt d



and all

 

0, , _r

s t  _r for all r

 

0,s and  

 

s imply

 

 

.

   

Let us set t the set of stopping times and t the set of elements  of such that there exists 

such that for all t



 



_{0, , IR :}d



 

_inf



_:



_,





s

C s t t

      s 

with the convention inf t.  is hence a well de- fined element of t and  is the open set associated.











d







: 0, C 0, , IR 0,

     

is an element of (resp.  ) if and only if, for all

 

0, t ,

t   t   t (resp. t) where

 

t, t

 

_{ }0,t .

   

Let us consider the function V



t x,



defined in



_{0, }



_IRd _by,

 

 



 





0 0 0

,

inf sup , , , 0,

t t

V t x

R_ x G

 _    

    t





where

 

 

   

d

0 0 , 0 d

R_  CS_  C



C x  x

T





; .

Let  and be a partition of  _IR_IRd,

 

 



 

 



d d

, IR IR ; , 0

, IR IR ; , 0 .

t x V t x

t x V t x

 

 

   

   





We have

 





,

, ,

, , , ,

0 ,

1

IE exp , d ,

t x

x r x

t r

u t x

X

g X C Y



 



   

    



   







 



 



 r

 _

  





.

then we deduce that ,

 

_{, 0}

u t x 

We have the

Theorem 3.3 For

 

t x, 



0, 



IRd, we have

1)

 

 

 



 







, 0

0 0 0

lim log , ,

inf inf , , , 0, .

t t

u t x V t x

C S x G t





    

 





    



  



2)

 

, 0

limu  t x, 0,

 

  

uniformly in any compact set  of . 3)

 

, 0

limu  t x, 1,

 

  

in all compact set  of o .

Proof: For first item, the proof is the same as in [2].

For the second point we can see that there exists such that

0

C

 

 

,

0 , e , ,

C

u  t x  t x

      _.

The third item is an immediate consequence of 1).

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