Annales de la Faculté des Sciences de Toulouse

Jean-Marc Rinkel

Inverses et propriétés spectrales des matrices de Toeplitz à symbole singulier

T.J. Rabeherimanana

Résultats sur l'irréductibilité et la conservativité de la chaîne de différences absolues

Claude Mitschi

Michael F. Singer

Solvable-by-finite groups as differential Galois groups

Luigi D'Onofrio

Luigi Greco

ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE
LUIGI D’ONOFRIO
LUIGI GRECO
On the regularity of solutions to a nonvariational
elliptic equation
Annales de la faculté des sciences de Toulouse 6e série, tome 11,
no 1 (2002), p. 47-56
<http://www.numdam.org/item?id=AFST_2002_6_11_1_47_0>
© Université Paul Sabatier, 2002, tous droits réservés.
L’accès aux archives de la revue « Annales de la faculté des sciences de
Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l’accord avec les
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doit contenir la présente mention de copyright.
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- 47 -
On the regularity of solutions
to a nonvariational elliptic equation (*)
LUIGI D’ONOFRIO (1) AND LUIGI GRECO (1),(2)
e-mail : ldonofri@unina.it ; luigreco@unina.it
Annales de la Faculte des Sciences de Toulouse Vol. XI, n° 1, 2002
pp. 47-56
R,ESUME. - Nous réalisons une étude détaillée du degre de régularité de
toutes les solutions d’une equation elliptique aux derivees partielles non-
variationelle. Cette equation a souvent ete une source de contre-exemples
importants en théorie des equations aux derivees partielles.
ABSTRACT. - We make a detailed study of the degree of regularity of
all solutions of a nonvariational elliptic partial differential equation. Such
equation has often turned out to be a source of counter examples which
are critical to the theory of PDE’s.
1. Introduction
In this paper we study solutions to the elliptic partial differential equa-
tion of nondivergence form
with the coefficient matrix
~ * ~ Reçu le 23 octobre 2001, accepte le 11 decembre 2002
(1) Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Universita di Napoli
"Federico II", via Cintia, 80126 Napoli, Italy.
(2) Was supported by M.U.R.S.T., P.R.I.N. 2000 "Calcolo delle Variazioni".
Here I denotes the identity matrix of order n , x 0 x is the matrix whose
entries are Xi and K > 1. We recall that the coefficient matrix A at (1.2)
is often used to provide interesting examples in the theory of PDE’s, both
variational and nonvariational ones [6], [10], 1 l~ , see also [3]. The entries
aij are not continuous at the origin 0; they are weakly differentiable with
derivatives in Ln near 0, compare with [5], [9]. In [1] equations
with weakly differentiable coefficients whose derivatives are in are
considered. However, for K large, Equation (1.1) does not fall under the
scope of the results of [1]. We note also that for large K (and n 2: 3)
Equation (1.1) is not of Cordes type; for equations of Cordes type we refer
to [13].
We consider strong solutions, namely twice weakly differ entiable func-
tions u satisfying the Equation (1.1) a.e., and examine their behavior near
the origin 0 of in the spirit of the results of [8], aiming to show that all
solutions of Equation (1.1) have a definite degree of regularity at 0, which is
precisely among those exhibited by certain particular solutions. The work
[8] concerns a variational equation. As it is well known, the theory of non-
divergence equations with discontinuous coefficients in dimension n > 3 is
less complete than the one in the divergence case. The argument used in
[8] does not extend to our context, hence our approach here is completely
different.
We consider the Dirichlet problem for equation (1.1) in a ball centered
at 0 and investigate how solutions depend on the boundary data. It is not
restrictive to consider the unit ball B = {x E JRn : Ixl  1~. So we consider
the problem
where cp is a given function on 9~ = {x . 1 ~ . We shall show
that for each 03C6 E problem (1.3) has a unique solution of class
n CO(B). Of course, uniqueness is a consequence of the celebrated
maximum principle of Aleksandrov-Bakelman-Pucci, see e.g. [7]. Concern-
ing the existence of a solution, as may be expected, there are a number of
ways to get it. Here we use the familiar method of superposition of solu-
tions [4], which is both elementary and yields a representation formula for
the solution.
The preliminary step in doing so is to find a decomposition for the boun-
dary data, and for this we recall [12] that a complete orthogonal system in
L2(~B) can be formed by (the restriction to o~B of) harmonic homogeneous
polynomials. Thus we find the expansion
where Hi is a harmonic homogeneous polynomial of degree l = 0, 1,.... Note
that these polynomials in the decomposition (1.4) are uniquely determined.
Then, for each l we consider the Dirichlet problem
It can be shown (see Lemma 2.1 below) that problem (1.5) has a solution
vi of the form
where .~l is a nonpositive number depending only on n, K and l, not on Hi .
The final step will be then to prove that for p E the function
belongs to n C° {B) and solves (1 .3) .
Using formula (1.7), as mentioned we extend the results of [8] proved in
the variational case, to the nonvariational equation (1.1). Essentially, these
results read as follows. Defining the moment of u by setting
where H is an arbitrary harmonic homogeneous polynomial of degree l G N,
if tt;(l) ~ 0, then the solution u of Lu = 0 has at the origin 0 at most the
same regularity as the function v = Note that w ( 1 ) depends on the
boundary value p of u: we!) = JaB pH ds.
2. The Dirichlet problem
In this section we solve the Dirichlet problem (1.3) for cp E 
using the method of superposition of solution, hence we begin by finding a
solution to the problem
for H a harmonic homogeneous polynomial. Let l be the degree of H.
Clearly, for all the function |x|03BBH satisfies the boundary condition.
On the other hand, an elementary calculation shows that for .r ~ 0
Then we find a solution to problem (2.1) of the following type:
with Ao = Ai = 0, and for l 2: 2
Thus for l = 0 and l = 1, v = H E For all l, we have v E
~0~). To study the regularity of v at 0 for l > 2, we examine the
quantity
Note that the function l ~ l + ~l is increasing and diverging at as
L -~ -f-oo. The following is trivially seen
LEMMA 2.1. For all l > 2 we have l + 03BBl > 1 and
Hence v E b’l > 2. Moreover, the regularity of v increases with
l > 2.
We shall use the following property of harmonic homogeneous polyno-
mials, see e.g. [12] , Appendix C on pg. 274-276; if H has degree l  1, we
have:
Now we prove some estimates for functions of the form
H being a harmonic polynomial homogeneous of degree l > 2, and l + 03BB-1 >
LEMMA 2.2. - We have the following
and for all p E ~ 0,1 ~ [
Proof. - Integrating in spherical coordinates we find
which immediately gives (2.6).
To prove (2.7) we compute
and hence
Moreover
and taking into account homogeneity of H
We then conclude, recalling that 1 - l  ~  0
To prove (2.8) we use (2.7) and get
Inequality (2.8) now follows easily using (2.5) for k = 2.
Recalling the expansion at (1.4) it is natural to consider the series at
(1.7), that is
For each l = 0, 1,..., here vi is the solution to problem (2.1) and is defined
by formulas (2.2)-(2.3). By means of the estimates of Lemma 2.2, we can
easily see the following
LEMMA 2.3. - The series at (,~.10~ converges in L2 (B) and in .
Its sum u solves the equation Lu = 0.
For the proof, it is enough to note that by Bessel inequality we have
hence applying (2.6) and (2.8) to each term Vi proves the claim.
Remark 2.1. Actually, the series at (2.10) converges in for
We are now in a position to state the main result of this section
PROPOSITION 2.1. - For every c~ E C° (aB) the Dirichlet problem
has a unique solution of class W o~ (B) n CO(B). . Moreover the solution is
expressed by
What remains is to show the continuity of u = B. Clearly, by
Lemma 2.3 and the Sobolev embedding theorem, u is continuous in B. On
the other hand, the series at (1.4) need not converge uniformly even for cp E
C’° (aB) For this reason, to prove continuity up to the bundary we do not
rely directly on the expansion (1.4). Instead, we recall how (1.4) is usually
proved. We approximate (/? uniformly by polynomials m = 1, 2, .... Each
on BB coincides with a finite sum of harmonic homogeneous polynomials
see [12], pg. 70. Here L has degree l. Then it is easily seen that the
polynomials occurring in the expansion at (1.4) are given by Hl = limm i
in (We mean that = 0 if 1 > ~~.,-z.) This implies that u =
limm w~ in L2 ( B ) , where
Note that E CO(B). On the other hand wm solves the problem
As converges uniformly, by the maximum principle converges
uniformly on B. Clearly the limit is u, which therefore belongs to CO(B). .
Remark 2.2. Actually, the series at (2.10) is converging in a much
stronger way then merely in the sense of W o~ (B) For all there exist
lk E N such that l + Àz > k and hence Vz = Hl E if l > lk. Then
the series
converges in the sense of for any p E J o,1 ~. To see this, we note
that (2.7) can be generalized as follows
and then by (2.5), as in the proof of inequality (2.8) we find
where ak = inf ~ (l + : l > > 0. Applying (2.13) to each term
vi = Hl in the series (2.12) we obtain the desired result.
We close this Section mentioning that in [2] equations with radially ho-
mogeneous coefficients are studied. However, the results presented there in
this more general context are not suitable for the purpose of this paper, as
we rely heavily on the representation formula (2.11).
3. The degree of regularity of solutions
Here, we examine the regularity of solutions to Equation (1.1) with k > 1
in a neighborhood of the origin 0 E Of course, solutions are of class C°°
away from the origin, so we are concerned with their behavior at 0. The regu-
larity we want to study is best expressed in terms of the spaces of Holder
continuous functions. We recall the notation used in [8]. For a given open set
n C k = 1, 2,... and 0  c~  1, C o’~ (S2) denotes the space of func-
tions whose k-th order partial derivatives are locally Holder continuous with
exponent a. The completion of is denoted by C~+Q (SZ) . For a = 1,
it coincides with C’~+1 (SZ), while if 0  c~  1, v E means that
uniformly on compact subsets of 03A9 x n. Also, note the embedding
1.
Now we introduce the so-called momenta of a function u E 
where R > 0; for a given harmonic homogeneous polynomial H, we set
We can now state our first result, which provide an upper bound for the
degree of regularity of the solutions at 0.
THEOREM 3.1.2014 Let u E C2 (BR - ~0~) be a solution to Equation (1.1).
For a given harmonic polynomial H homogeneous of degree 1 > 2, if the
moment wH does not vanish, then u ~ the number ~l being
defined at (2.3). .
~roo f. By rescaling, we may assume R > 1. Also, we may assume u E
Then we show that u coincides with the solution we constructed in
Section 2 to the Dirichlet problem for L, with boundary value 03C6 = Let
us denote by U this solution. We observe that u, U E C2{B - ~0~) n GI(B)
both solve Lu = L U = 0 in B - {0} and u == U on 9B. If ~ ~ 
then the identity u = U follows by an extended maximum principle (in the
sense of [6]) proved by Pucci, see Theorem XI on pg. 157 and the subsequent
remark in [10]. In the case K > we recall that the continuous func-
tion Ixll-(n-l)/K2 solves the equation in B - ~0~. Applying the maximum
principle of Aleksandrov on B - ~0~, we see that
Moreover this function is C~ if and only if tt(0) = ~7(0), that is, u = U.
By the results of the previous Section, u has the representation
with Hk a harmonic polynomial homogenous of degree k. Hence by ortho-
gonality
We write l + Àl = m + cx, with m integer, 1 ( m  l - 1 and 0 x Q  1, and
denote by Q the NlacLaurin polynomial of order m of u. Recall that Q and
H are orthogonal to each other. Assuming u E = we would
have as :r 2014~ 0
and in turn
which is false as w(l) ~ 0.
By means of the momenta of u we can find also a "lower bound" for the
degree of regularity.
THEOREM 3.2. Every solution u E of Equation (1.1) be-
longs (BR) , If there exists an integer 1 > 2 such that
for every harmonic polynomial H homogenous of degree k with 1  k  l,
then
Here I + ÀI = m + a, as above.
Proof. For the first part, we note that in the series (3.1) the term
is that with the lowest degree of regularity. To complete the proof,
we see that, by the assumptions, the representation (3.1) reduces to
that is, the terms corresponding to k for all 1  k  l vanish. In the series
(3.2), the worst term is now (assuming that Hl ~ 0). The thesis
can be then proved by the argument used in the Remark 2.2.
- 56 -
As a complement to the above theorems we have
PROPOSITION E a solution to Equation (1.1).
If w~ - 0, for all harmonic homogenous polynomials H, then u - ~.
The claim follows directly by our representation (2.11) of solutions of
class Z~ o~ .
Bibliography
[1] ALVINO (A.) and TROMBETTI (G.). 2014 Second Order Elliptic Equations whose Co-
efficients Have Their First Derivates Weakly Ln, Ann. Mat. Pura Appl., (4) 138
(1985), 331-340.
[2] BASS (R.F.).2014 The Dirichlet problem for radially homogeneuos elliptic operator,
Trans. Amer. Mat. Soc., 320, no. 2 (1990), 593-614.
[3] BÉNILAN (P.) and BOUHSISS (F.). 2014 Une remarque sur l’unicité des solutions pour
l’opérateur de Serrin, C. R. Acad. Sci. Paris, t. 325, Série I, (1997), 611-616.
[4] COURANT (R.) and HILBERT (D.). - Methods of Mathematical Physics, Interscience
Publishers, Inc., New York, 1953.
[5] FRANCIOSI (M.) and FUSCO (N.). 2014 A W2,2 bound for a class of elliptic equations
with discontinuous coefficients, Ricerche Mat., 31 (1982), 207-218.
[6] GILBARG (D.) and SERRIN (J.). 2014 On isolated singularities of solutions of second
order elliptic differential equations, J. Analyse Math., 4 (1955-56), 309-340.
[7] GILBARG (D.) and TRUDINGER (N.S.).2014 Elliptic Partial Differential Equations of
Second Order, Second Edition, Springer-Verlag, Berlin, 1983.
[8] GRECO (L.) and SBORDONE (C.). - Sharp Upper Bounds for the Degree of Regular-
ity of the Solutions to an Elliptic Equation, Comm. Partial Differential Equations,
(5&#x26;6) 27 (2002), 945-952.
[9] MIRANDA (C.). 2014 Sulle equazioni ellittiche del secondo ordine di tipo non vari-
azionale a coefficienti discontinui, Ann. Mat. Pura Appl., (4) 63 (1963), 353-386.
[10] PUCCI (C.). 2014 Opemtori ellittici estremanti, Ann. Mat. Pura Appl., (4) 72 (1966),
141-170.
[11] SERRIN (J.). 2014 Pathological solutions of elliptic differential equations, Ann. Scuola
Norm. Sup. Pisa, (3) 18 (1964), 385-387.
[12] STEIN (E.M.). 2014 Singular Integrals and Differentiability Properties of Functions,
Princeton University Press, Princeton, 1970.
[13] TALENTI (G.). 2014 Sopra una classe di equazioni ellittiche a coefficienti misurabili,
Ann. Mat. Pura Appl., (4) 69 (1965), 285-304.


On the regularity of solutions to a nonvariational elliptic equation

Salomon Sambou

Résolution du $\bar{\partial }$ pour les courants prolongeables définis dans un anneau

Alexander Bendikov

Laurent Saloff-Coste

Invariant local Dirichlet forms on locally compact groups

Grégoire-Thomas Moniot

Existence de surfaces de Willmore qui ne sont pas minimales

Henri Caussinus

ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE
HENRI CAUSSINUS
Some concluding observations
Annales de la faculté des sciences de Toulouse 6e série, tome 11,
no 4 (2002), p. 587-591
<http://www.numdam.org/item?id=AFST_2002_6_11_4_587_0>
© Université Paul Sabatier, 2002, tous droits réservés.
L’accès aux archives de la revue « Annales de la faculté des sciences de
Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l’accord avec les
conditions générales d’utilisation (http://www.numdam.org/conditions).
Toute utilisation commerciale ou impression systématique est constitu-
tive d’une infraction pénale. Toute copie ou impression de ce fichier
doit contenir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
- 587 -
Some concluding observations
HENRI CAUSSINUS (1)
Annales de la Faculté des Sciences de Toulouse Vol. XI, n° 4, 2002
pp. 587-591
First of all I wish to thank all the authors for their friendly and autho-
ritative contributions to this special issue of the ’Annales de la Faculté des
Sciences de l’Université de Toulouse’, the editor of the ’Annales’ for wel-
coming them, Stephen Fienberg and Peter van der Heijden who served as
editors of this special issue. In addition to this special issue, Antoine de
Falguerolles originated the idea of a web site devoted to quasi-symmetry.
Alan Agresti, Murray Aitkin, Erling Andersen, Stephen Fienberg, Antonio
Forcina, Maria Kateri and Takis Papaioannou, Tamas Rudas and Wicher
Bergsma, Yoshio Takane, Sadao Tomizawa, Peter van der Heijden sent
interesting contributions to this site: I would like to warmly thank all of
them.
Since the topic and the development of my early work in the sixties may
seem surprising (see the introduction to this issue by Stephen Fienberg and
Peter van der Heijden), I shall start this reply by a few considerations on its
origin. Actually, at the end of the fifties, there were few statisticians in the
French universities. Fortunately one of them, Roger Huron, was professor
at the Faculté des Sciences de Toulouse where I was studying mathematics.
Since I was fascinated by the understanding of hazard (which, I presume,
came from my childhood: my father was an insurance agent and I heard
the word risk much more than my turn), I took his course ’probability and
statistics’ to learn probabilities - I was quite unaware that statistics had
something to do with mathematics! Fortunately again, Professor Huron was
a wonderful and stimulating teacher and, finally, he made an offer to me to
become his assistant and to undertake a research work in his laboratory.
Roger Huron’s basic course was a ’classical’ one in the spirit of statistical
modeling and inference. This was consistent with his own research in the
field of genetics which involved modeling, estimation, and testing, most
(1) Laboratoire de Statistique et Probabilités, Universite Paul Sabatier, Toulouse,
France. Email: caussinu@cict.fr
often for categorical data: see, e.g., Huron (1955), Huron (1958), Huron
and Ruffie (1959). For example, the second of these papers is devoted to
maximum likelihood estimation of some models while the first considers the
test of specific deviations in a multinomial distribution, that is the analysis’
of the chi-square test (in a sense, this was also to be one of my goals when
dealing with truncated contingency tables). I do not remember exactly why
I decided (with the help of Roger Huron) to work on contingency tables:
perhaps it was from the vague feeling that was a good time to do so (thanks
to Stephen Stigler’s paper I am now more aware of this fact). But the main
reasons for which I continued are clear: the discovery of log-linear modeling
and the reading of Leo Goodman’s papers. I should mention also some
papers by Maurice Frechet (1959,1960) and Pierre Thionet (1963,1964) on
the existence and construction of tables with given marginals.
Meanwhile, Jean-Paul Benzecri was developing his research in Rennes,
then in Paris, but I was not at all aware of his work which remained hardly
available up to, say 1965 or 1966: see, e.g., Benzecri (1977). The French
atmosphere of the seventies, however, was certainly important for the further
orientation of my work: prevalence of techniques related to the singular value
decomposition and the associated graphical displays, new points of view on
the role of models in statistics, etc. Actually, one of my main interests was
then to bring together the concepts of statistical modeling and data analysis
as an attempt to a better understanding of statistical methods: see, for
example, Caussinus and Falguerolles (1987) or Besse, Caussinus, Ferre, and
Fine ( 1986) . It is indeed a great pleasure that several of my friends continue
to go deeply into this approach as evidenced by the papers by Simplice
Dossou-Gbété and Axel Grorud and by Antoine de Falguerolles and Peter
van der Heijden in this issue.
Coming back to my 1966 paper, since it has been published in French,
it would have been obviously overlooked has it not been for Leo Goodman
who wrote a review in the Mathematical Reviews (vol. 39, number 3, March
1970) and sent me a very kind letter to initiate a contact, Stephen Fienberg
who gave a thorough account in his authoritative book with Yvonne Bishop
and Paul Holland, and Alan Agresti who, later on, kindly emphasized the
importance of this work in his books. Additionally, all of you provided fur-
ther interesting contributions to quasi-symmetry including your papers in
this issue. It is quite amazing to see the variety of situations in which quasi-
symmetry has been used as also exemplified in the web site.
Claude Thélot recalls some applications and some developments he did
in the seventies (I had then the pleasure to write a paper with him) and
explains why he gave up this model for his researches on social mobility.
As for me, I also gave up the topic progressively except for a few further
contributions including the paper with Claude Thélot and another one with
Antoine de Falguerolles (unfortunately all are still only available in French! ) .
I also supervised the thesis of Arnold Izsak (1972). Incidentally, Arnold
Izsak found a paper analyzing migration by means of an approach implicitly
equivalent to the model of quasi-symmetry (Mossin, 1971), a paper which
is worth mentioning as yet another equivalent model to the ones considered
in this issue. But the main purpose of Izsak’s thesis was to consider the
identification of the parameters, a difficult question which does not have
a general answer. I provided also some hints in Caussinus (1976). In fact,
apart from the French atmosphere evoked above, I think that this difficulty
(which I considered as a serious drawback of the model) was the main reason
for my abandoning work this topic. I continued, however, to consider the
question of identification from time to time. Let me add a few words about
it, mostly inspired by my above-mentioned paper.
Let us consider a square table whose entries have probabilities 
i, j = 1,..., r. Denote by P the r x r corresponding matrix and by T
the matrix whose elements are tij = where ~2, is the sum over j of
the Pij’S . In many problems (social mobility, migrations, etc.) it is natu-
ral and convenient to consider the r categories as the states of a discrete
Markov chain with transition matrix T. (Note incidentally that there is
equivalence between quasi-symmetry of P, or T , , and reversibility of the
associated Markov process: see Caussinus, 1976, McCullagh, 1982.) In prac-
tice, this chain is regular and there exists a column-vector, say f, of statio-
nary probabilities. This vector has clearly an intrinsic meaning and can be
used to identify the parameters of a quasi-symmetric model in the following
way. Assume that the matrix T is quasi-symmetric and write
with Sij which is a possible way to identify the parameters up to a
multiplicative scalar. To achieve identification let us assume that the fj’s
add to one. Let L be the diagonal matrix of the fi’s, S the matrix of the
Sij and ll the column vector with coordinates equal to 1. Then T = SL,
T’ = LS and T’L1 = LSL1 = LTH = LH, hence f = LII. Thus. in
equation (1), fj has a sensible interpretation in terms of attractivity since it
is the stationary probability of category j. Coming back to P, pjj = 
where pj, is the propensity to leave i at the time when the process is observed
and can be interpreted as the relative attractivity of j at this time.
Furthermore, let us consider the transition matrix T n for n steps. It is easy
to prove that T n is still quasi-symmetric and can be written as T’z = ,5’,~ L
where Sn is symmetric for all n and converges to 11 11’ when n tends t.o
infinity (see Caussinus, 1976, for a proof in the more general framework of
continuous time processes). If the elements of S (respectively Sn ) are consi-
dered as distances (or rather as similarities) between the categories for one
step (respectively n steps) this is consistent with the fact that the distance
effects should decrease with the time and vanish - in practice become equal -
for large n. Nevertheless, the previous approach to interpret the parameters
suffers from a serious drawback: it depends both on the stayers and the
movers, that is it depends on the diagonal elements of P. Of course, it can
be used with a matrix P where the diagonal elements are set equal to zero,
but there still remain a large degree of arbitrariness.
At this place, I would like to mention that I introduced the model of
quasi-symmetry while looking for log-linear models beyond the more ele-
mentary ones. Defining such log-linear models with good properties did not
seem a trivial question and Peter McCullagh’s contribution to this issue
brings interesting arguments that help us understand why it was not tri-
vial. Finally, even if the model of quasi-symmetry was the main part of my
early work, it is important to remember that I started from the description
of association by means of log-linear models for ordinary as well as mul-
tidimensional contingency tables. Since that time, much progress has been
realized, as evidenced, e. g., by the contribution of Wicher Bergsma and
Tamas Rudas to this issue.
More generally, it is a great pleasure to realize that the problems I consi-
dered several years ago are now much better understood. Thanks again to
you all for the best gift I could receive.
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- 591 -
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Some concluding observations

M. Ledoux

ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE
M. LEDOUX
Note of the editor
Annales de la faculté des sciences de Toulouse 6e série, tome 11,
no 1 (2002), p. 5
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NOTE OF THE EDITOR
M. Ledoux
Editor-in-Chief
J.-P. Ramis asked me to succeed to him as Editor-in-Chief of the Annales
de la Faculté des Sciences de Toulouse, starting January lst, 2003. I would
like to thank him for his work during the past years, as well as all the
members of the preceding Editorial Board.
A new Editorial Commitee has been formed, and I would like to welcome
the new members and to thank them for accepting to join the Board. Due
to recent delays in publication, the new Editorial Board already appears on
the front page of this first issue of 2002. We expect to resorb these delays
within the next few months.


Note of the editor

Pierre Del Moral

Laurent Miclo

On the stability of nonlinear Feynman-Kac semigroups

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