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Homogenization of divergence-form operators with lower
order terms in random media
Antoine Lejay
To cite this version:
Antoine Lejay. Homogenization of divergence-form operators with lower order terms in ran-
dom media. Probability Theory and Related Fields, Springer Verlag, 2001, 120 (2), pp.255-276.
�10.1007/s004400100135�. �inria-00001220�
Homogenization of divergence-form operators with
lower order terms in random media
Antoine Lejay
1
— Projet SYSDYS (INRIA/LATP)
Abstract: The probabilistic machinery (Central Limit Theorem,
Feynman-Kac formula and Girsanov Theorem) is used to study
the homogenization property for PDE with second-order partial
differential operator in divergence-form whose coefficients are sta-
tionary, ergodic random fields. Furthermore, we use the theory
of Dirichlet forms, so that the only conditions required on the
coefficients are non degeneracy and boundedness.
Keywords: random media, random potential, homogenization,
Dirichlet form, divergence-form operators
AMS Classification: 35B27 (31C25 35R60 60H30 60J60)
Published in Probability Theory and Related Fields. 120:2,
pp. 255–276, 2001
Archives, links & reviews:
◦ doi: 10.1007/s004400100135
◦ MR Number: 1841330
1
Current address: Projet OMEGA (INRIA Lorraine/IECN)
IECN, Campus scientifique
BP 239
54506 Vandœuvre-l`es-Nancy CEDEX France
E-mail: uji - c ..n et nae ryo anny ce @L . fiAn
This version may be slightly different from the published version 27 pages
A. Lejay / Homogenization of divergence-form operators in random media
1 Introduction
Some averaging — or homogenization — properties for some elliptic or parabolic
partial differential equations (PDE) in a stationary, ergodic random media
are studied with probabilistic techniques.
This consists in finding constant coefficients which approximate in some
suitable sense highly oscillating coefficients that represent the random media.
In other words, we study the limit of the solutions of some PDEs when a
coefficient — which represents the scale of the heterogeneities — decreases
to 0.
Using the probabilistic representation of the solutions of parabolic and
elliptic PDE, this leads to establishing a Central Limit Theorem for the
stochastic process generated by a second-order partial differential operator.
More precisely, we are interested in PDEs with second-order partial dif-
ferential operators of the form
A
ε,ω
= L
ε,ω
+ b
i
(x/ε, ω)
∂
∂x
i
+ c(x/ε, ω) +
d(x/ε, ω)
ε
, (1)
where L
ε,ω
=
e
2V (x/ε,ω)
2
∂
∂x
i
Ã
a
i,j
(x/ε, ω)e
−2V (x/ε,ω)
∂
∂x
j
!
(2)
under the assumption that the coefficients are bounded stationary random
fields and that the matrix a is symmetric. The operator A
ε,ω
contains in
fact a lower-differential term of the form ∂
x
i
(e
i
(x/ε, ω)·) = e
i
(x/ε, ω)∂
x
i
+
ε
−1
(∂
x
i
e
i
)(x/ε, ω), assuming that e is differentiable.
The solution of the parabolic equation
∂u
ε
(t, x)
∂t
= A
ε,ω
u
ε
(t, x) (3)
with the initial condition u
ε
(0, ·) = f is given by the Feynman-Kac formula
u
ε
(t, x) =
e
E
ε
x,ω
·
exp
µ
1
ε
Z
t
0
d(X
s
/ε, ω) ds +
Z
t
0
c(X
s
/ε, ω) ds
¶
f(X
t
)
¸
,
where
e
E
ε
x,ω
is the distribution of the stochastic process generated by the
operator
e
L
ε,ω
= L
ε,ω
+ b
i
(x/ε, ω)∂
x
i
.
Studying of the convergence of the process
e
X
ε,ω
associated to
e
L
ε,ω
as ε
decreases to 0 is equivalent to studying the convergence of (ε ·
ε
e
X
ω
t/ε
2
)
t>0
,
where
ε
X
ω
is the process whose infinitesimal generator is
e
−2V (·,ω)
2
∂
∂x
i
Ã
a
i,j
(·, ω)e
2V (·,ω)
∂
∂x
j
!
+ εb
i
(x, ω)
∂
∂x
i
.
2
A. Lejay / Homogenization of divergence-form operators in random media
As it will be shown in Section 4, the Girsanov theorem allows to reduce this
problem, where the first-order coefficient of the operator is of order ε, to the
study of the Central Limit Theorem for the process X
1,ω
whose generator
is the self-adjoint divergence-form operator L
1,ω
. A rather similar use of
the Girsanov transform in a different context to prove some homogenization
results may be found in [23].
A Central Limit Theorem for the process
R
·
0
d(X
ω
s
) ds has to be proved to
deal with our initial problem ( i.e., with a highly-oscillating zero-order term).
The homogenization property for the divergence-form operator ∂
x
i
(a
i,j
(·/ε, ω)∂
x
j
)
with random coefficients has been studied with analytical tools first by S. Ko-
zlov [17, 18] and G. Papanicolaou and S.R.S. Varadhan [34] (see also [35]).
The probabilistic method consists in finding functions which are solutions
of auxiliary problems, so that our process is transformed as the sum of a
local martingale and a process that converges to 0. Then, the Central Limit
Theorem for the local martingale is applied with the help of the Ergodic
Theorem. See e.g., [10, 3, 16] for various applications of this procedure.
The difficulty for homogenization in random media lies in the resolution
of the auxiliary problem, that has to be done on a suitable space.
For a general random media — in contradistinction to what happens in
the case of periodic media where the Poincar´e inequality holds (see e.g., [24]
for results under weaker hypotheses in periodic media) —, the resolution
of the auxiliary problem cannot be considered in a direct way, because the
needed function is not a stationary random fields. Three strategies may be
used: 1) The solution is approximated by a sequence of stationary random
fields, and one studies the convergence of their gradients. This method is
especially well-suited to the case of an initial environment whose law is the
invariant distribution. (see e.g., [6, 29, 13, 31]). 2) The solution is directly
constructed using the spectral theory for the shift operators of the random
media (see e.g., [34, 9]). 3) The gradient of the solution is given directly in
an appropriate space with the help of the Lax-Milgram Theorem (see [15] for
an analytical use of this method).
The two last approaches may be used to prove that the family of processes
converges for almost every realization. We have chosen here to use the third
method. Our approach is close to that used by S. Kozlov in [19] for random
walks.
Furthermore, this approach does not really relies on the idea of mean
forward velocity as in [30, 6, 13, 31]. Our proof of the Central Limit Theorem
for the process associated to L
1,ω
is then rather different to that of [31], which
itself adapts to Dirichlet forms the ideas developed in [16, 6, 29, 13].
Solving the auxiliary problem or finding the invariant measure shows the
3
A. Lejay / Homogenization of divergence-form operators in random media
difficulty to study the limit behaviour of the processes associated to
1
2
∂
∂x
i
Ã
a
i,j
(x/ε, ω)
∂
∂x
j
!
+
1
ε
b
i
(x/ε, ω)
∂
∂x
i
for a general b, which so far remains an open problem. In fact, there exists
some counterexamples to the homogenization property for some stationary,
divergence-free random fields [1, 14]. We have also assumed that d is the
derivative of a bounded function.
Although some results may be given for general non-symmetric Dirichlet
forms [32] provided the mean forward velocity exists, two classes of problems
are generally considered: The first concerns the case where b is the derivative
of the skew-symmetric matrix which is a stationary random field [28, 9, 22,
34]. The second concerns the case where b is a gradient of a stationary
random field. In this case, the second-order differential operator is reduced
to a self-adjoint operator.
The term V is a potential. If a and V are regular enough, then the
operator L
ε,ω
can be written
L
ε,ω
=
1
2
a
i,j
(·/ε, ω)
∂
2
∂x
i
∂x
j
+
1
2ε
Ã
∂a
i,j
∂x
j
− 2a
i,j
∂V
∂x
j
!
(·/ε, ω)
∂
∂x
i
and a stochastic process may be associated to L
ε,ω
via the theory of stochastic
differential equations.
However, any regularity assumption on a and V may be dropped if one
use the theory of Dirichlet forms as developed e.g., in [11] instead of Itˆo
stochastic calculus. Hence, our results generalize those of [6, Section 6] and
[29, Chapter 2]. In fact, our proofs use some considerations on the semi-group
associated to a divergence-form operator, but hardly require the theory of
Dirichlet forms.
Afterwards, we prove that the solution to the parabolic PDE (3) converges
to the solution to the parabolic PDE
∂u(t, x)
∂t
= Au(t, x),
where the coefficients of the PDE operator A are constant and are averages
of the coefficients of A
1,ω
with respect to the law of the media. We use the
method introduced by
´
E. Pardoux in [33] to deal with the highly-oscillating
zero-order term, which also uses the Girsanov theorem.
In Section 5.2, we consider the case of the elliptic equations of the form
(α − A
ε,ω
)u
ε
= f. We prove that α − A
ε,ω
is invertible for α greater to some
4