Annales de la Faculté des Sciences de Toulouse

Mouna Kraïem

On some nonlinear partial differential equations involving the 1-Laplacian

Christer Borell

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
CHRISTER BORELL
Minkowski sums and Brownian exit times
Tome XVI, no 1 (2007), p. 37-47.
<http://afst.cedram.org/item?id=AFST_2007_6_16_1_37_0>
© Université Paul Sabatier, Toulouse, 2007, tous droits réservés.
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Annales de la Faculté des Sciences de Toulouse Vol. XVI, n◦ 1, 2007
pp. 37–47
Minkowski sums and Brownian exit times(∗)
Christer Borell (1)
ABSTRACT. — If C is a domain in Rn, the Brownian exit time of C
is denoted by TC . Given domains C and D in R
n this paper gives an
upper bound of the distribution function of TC+D when the distribution
functions of TC and TD are known. The bound is sharp if C and D are
parallel affine half-spaces. The paper also exhibits an extension of the
Ehrhard inequality.
RÉSUMÉ. — Si C est un domaine de Rn, le temps de sortie brownien
de C est noté TC . Donnant domaines C et D de R
n cet article montre
une borne supérieure pour la fonction de répartition de TC+D quand les
fonctions de répartition de TC et TD sont connues. En plus l’article exhibe
une généralisation de l’inégalité d’Ehrhard.
1. Introduction
Throughout W = (W (t))t0 denotes Brownian motion in Rn and if C
is a domain in Rn,
TC = TWC = inf {t > 0; W (t) /∈ C}
is called the exit time from C. Below the notation Px [·] or Ex [·] indicates
that Brownian motion starts at the point x at time zero.
The main aim of this paper is to prove an inequality of the Brunn-
Minkowski type for distribution functions of Brownian exit times from do-
mains in Rn, such that equality occurs for parallel affine half-spaces. Here
perhaps the most interesting point is the fact that the set of all Brownian
paths {W (ω); TC > t} is not an affine half-space if C is an affine half-space
in Rn. Recall that affine half-spaces often turn out to be extremals for Gaus-
sian measures (see Ehrhard [5] and Carlen, Kerce [3]). In connection with
(∗) Reçu le 28 avril 2005, accepté le 20 octobre 2005
(1) School of Mathematical Sciences, Chalmers University of Technology and Göteborg
University, S-412 96 Göteborg, (Sweden).
borell@math.chalmers.se
– 37 –
Christer Borell
our main result Theorem 1.1, the Bachelier formula for the distribution of
the maximum of real-valued Brownian motion (see e.g. Karatzas and Shreve
[6] , p. 96) plays an important part. Finally, to make comparisons with the
Ehrhard inequality (Ehrhard [4] , Borell [2]) we find it natural to extend
this inequality to more general linear combinations of sets.
Let us continue by giving some more definitions. First the so called vector
sum or Minkowski sum of two subsets A and B of Rn equals
A + B = {x + y; x ∈ A and y ∈ B} .
Moreover, if α > 0, the dilation αA = {αx; x ∈ A}.
In [1] I use a method based on the maximum principle for elliptic dif-
ferential equations to prove the following inequality for expected Brownian
exit times. Suppose C and D are bounded domains in Rn and x ∈ C, y ∈ D.
Then √
Ex+y [TC+D] 
√
Ex [TC ] +
√
Ey [TD]. (1.1)
Here equality occurs in many interesting cases. First recall that
Ex [TC ] =
∫ ∞
0
Px [TC > t] dt.
Therefore by the scaling property of Brownian motion√
Eαx [TαC ] = α
√
Ex [TC ], α > 0
and it follows that equality occurs in (1.1) if C is convex and D × {y} =
λ(C ×{x}) + (a, a) for appropriate λ > 0 and a ∈ Rn. In this paper we will
use a method similar to those in my papers [1] and [2] to prove inequalities
of the Brunn-Minkowski type for distribution functions of Brownian exit
times.
If H is an open affine half-space in Rn, the Bachelier formula for the
distribution of the maximum of real-valued Brownian motion yields
Px [TH > t] = Ψ(
d(x,Hc)√
t
), t > 0, x ∈ H
where d(x,Hc) = miny/∈H | x− y | and
Ψ(r) = 2
∫ r
0
exp(−λ
2
2
)
dλ√
2π
, 0  r ∞.
– 38 –
Minkowski sums and Brownian exit times
The main aim of this paper is to prove the following
Theorem 1.1. — Suppose C and D are domains in Rn and
f :C → [0, 1] , g:D → [0, 1] , and h:C + D → [0, 1] continuous functions
such that
Ψ−1(h(x + y))  Ψ−1(f(x)) + Ψ−1(g(y)), x ∈ C, y ∈ D. (1.2)
Then, if x ∈ C, y ∈ D, and t > 0,
Ψ−1(Ex+y [h(W (t));TC+D > t])
 Ψ−1(Ex [f(W (t));TC > t]) + Ψ−1(Ey [g(W (t));TD > t]). (1.3)
In particular,
Ψ−1(Px+y [TC+D > t])  Ψ−1(Px [TC > t]) + Ψ−1(Py([TD > t]). (1.4)
Equality occurs in (1.4) if C and D are parallel affine half-spaces.
It is not obvious to the author that Theorem 1.1 implies (1.1).
Next we introduce some additional definitions. Below F denotes a real,
separable Fréchet space and γ a centered Gaussian measure on F , that is γ is
a Borel probability measure on F such that each bounded linear functional
on F has a centered Gaussian distribution. The Borel field in F is denoted
by B(F ). The definitions of Minkowski sums and dilations of subsets of F
are as in the special case F = Rn.
If
Φ(r) =
∫ r
−∞
exp(−λ
2
2
)
dλ√
2π
, −∞  r ∞,
and 0 < θ < 1, my paper [2] proves the so called Ehrhard inequality
Φ−1(γ(θA + (1− θ)B))  θΦ−1(γ(A)) + (1− θ)Φ−1(γ(B))
for all A,B ∈ B(F ). As in the LataEla paper [8] we here follow the convention
that ∞−∞ = −∞+∞ = −∞.
The following result is slightly more informative than the Ehrhard in-
equality.
Theorem 1.2. — Suppose α, β > 0 are given. Then the inequality
Φ−1(γ(αA + βB)  αΦ−1(γ(A)) + βΦ−1(γ(B)) (1.5)
– 39 –
Christer Borell
is valid for all A,B ∈ B(F ) if
α + β  1 and | α− β | 1. (1.6)
Moreover, if γ is not a Dirac measure at origin and (1.5) is valid for all
A,B ∈ B(F ) then (1.6) holds.
Equality occurs in (1.5) if A and B are parallel affine half-spaces. If, in
addition, α + β = 1 equality occurs in (1.5) if A is convex and B = A.
To comment on a certain relation between Theorems 1.1 and 1.2 we
denote by C([0,∞[ ;Rn) the space of all continuous maps of [0,∞[ into Rn
equipped with its standard locally convex topology of uniform convergence
on compact subintervals of [0,∞[ . Furthermore, we will have the picture
that the identity map on C([0,∞[ ;Rn) gives a representation of Brownian
motion in Rn relative to Wiener measure on C([0,∞[ ;Rn). Now using the
shorthand notation Wx(t) = x + W (t) it follows that
[Wαx+βy(s) ∈ αC + βD, 0 < s  t]
⊇ α [Wx(s) ∈ C, 0 < s  t] + β [Wy(s) ∈ D, 0 < s  t] , α, β > 0
and, hence, by (1.5),
Φ−1(Pαx+βy [TαC+βD > t])  αΦ−1(Px [TC > t]) + βΦ−1(Py [TD > t])
(1.7)
for all reals α and β satisfying (1.6). The inequality (1.4) is not weaker than
the inequality (1.7) with α = β = 1 since
Ψ(Ψ−1(a) + Ψ−1(b))  Φ(Φ−1(a) + Φ−1(b)) (1.8)
for all 0  a, b  1, which follows from the fact that there is equality in
(1.4) when C and D are parallel affine half-spaces. In fact, strict inequality
holds in (1.8) for all 0  a < 1, 0  b < 1, such that (a, b) = (0, 0). To
see this, suppose 0  a  1, 0  b < 1 and let f(a) be the difference of
the members on the left-hand and right-hand side of (1.8). If b = 0, then
f(a) = a > 0 for all 0 < a < 1. Next suppose 0 < b < 1. The function f is
continuous and f(0) = b and f(1) = 0. Furthermore, if 0 < a < 1,
f ′(a) = exp(−Ψ−1(a)Ψ−1(b)− 1
2
(Ψ−1(b))2)
− exp(−Φ−1(a)Φ−1(b)− 1
2
(Φ−1(b))2).
If 12  b < 1, then Φ−1(b)  0 and since Ψ−1(y) > max(0,Φ−1(y)) if
0 < y < 1 we have that f ′(a) < 0 for all 0 < a < 1. Thus f is strictly
– 40 –
Minkowski sums and Brownian exit times
decreasing and we get f(a) > 0 if 0  a < 1. Next suppose 0 < b < 12 .
Then Φ−1(b) < 0 and it follows that f ′ is decreasing on ]0, 1[ . Thus f is
concave and, accordingly from this, f(a) > 0 if 0  a < 1.
It does not seem to exist any natural counterpart of the inequality (1.4)
for linear combinations of sets as in (1.7). For example, the inequality
Ψ−1(P 1
2x+
1
2y
[
T 1
2C+
1
2D
> t
]
)  1
2
Ψ−1(Px [TC > t]) +
1
2
Ψ−1(Py([TD > t])
is not true in general. In fact, if that was the case we use the concavity of
Ψ to get
P 1
2x+
1
2y
[
T 1
2C+
1
2D
> t
]
 1
2
Px [TC > t] +
1
2
Py([TD > t] .
Now if C is convex and D = C we integrate over 0  t < ∞ and have
that the expected exit time Ex [TC ] is a concave function of x ∈ C, which is
wrong for the plane domain
{
z ∈ C; 0 < arg z < π4 and | z |< 1
}
(see my
paper [1], Example 3.1).
By passing note that if α = β = 1 in (1.5) and the function Φ−1 is
everywhere replaced by Ψ−1 the resulting inequality is false since, otherwise,
(1.8) would be an equality for all 0 < a, b < 1.
Next suppose F = Rn and γ = γn in Theorem 1.2. Then if C ∈ B(Rn)
is convex and A = B = C and α = β = r2 
1
2 in (1.5) we get the following
result by Sudakov and Tsirelson [9] of independent interest.
Corollary 1.1. — Suppose C ∈ B(Rn) is convex and H an open
affine half-space in Rn such that
γn(C) = γn(H).
Then
γn(rC)  γn(rH) if r  1.
Stated otherwise,
Φ−1(γn(rC))  rΦ−1(γn(C)) if r  1.
See also Yurinsky´s book [10] and the early paper by Landau and Shepp
[7], which shows Corollary 1.1 in the special case γn(C)  12 .
The present paper is organized as follows. In Section 2 we prove Theorem
1.1. Finally, Section 3 is devoted to a (partly sketchy) proof of Theorem 1.2.
– 41 –
Christer Borell
2. Proof of Theorems 1.1
By monotone convergence there is no loss of generality to assume that
C and D are finite unions of open cubes with edges parallel to the coor-
dinate axes. In addition, given δ ∈ ]0, 1[, we may assume f :C → [0, δ] and
g:D → [0, δ] are continuous functions with compact supports in C and D,
respectively. Finally, there is no loss of generality to assume that h possesses
compact support in C + D and
h : C + D →
[
0,Ψ(2Ψ−1(δ))
]
.
Now for each q = f, g, h, set
uq(t, x) = Ex
[
q(W (t)); TDom q > t
]
for every t  0 and every x belonging to the closure of Dom q, where Dom
q denotes the domain of definition of q. Moreover, set
Uq = Ψ−1(uq)
and introduce the continuous function
V (t, x, y) = Uh(t, x + y)− Uf (t, x)− Ug(t, y)
defined for all t  0 and x ∈ C̄, y ∈ D̄. We will prove that V (t, x, y)  0,
which implies (1.3).
The construction shows that V (0, x, y)  0 for all x ∈ C̄, y ∈ D̄.
Furthermore, if x ∈ C and y ∈ ∂D, then V (t, x, y)  0 if and only if
uh(t, x + y)  uf (t, x). The latter inequality is obvious since
uf (t, x) = E [f(x + W (t));x + W (s) ∈ C, 0 < s  t]
 E [h(x + W (t) + y); x + W (s) + y ∈ C + y, 0 < s  t]
 E [h(x + W (t) + y); x + W (s) + y ∈ C + D, 0 < s  t] = uh(t, x + y).
In a similar way, it follows that V (t, x, y)  0 if x ∈ ∂C and y ∈ D. In the
next step we will show that V (t, x, y) is a solution of a certain parabolic
differential equation and the non-negativity of V (t, x, y) then follows from
the maximum principle.
Recall that Ψ(a) = 2Φ(a) − 1, 0  a  ∞, so that Ψ′(a) = 2ϕ(a),
0  a < ∞, where ϕ(a) = Φ′(a) if a ∈R. Moreover, if q = f, g, h we have
in the interior of Dom uq that
∂uq
∂t
=
1
2
∆uq
– 42 –
Minkowski sums and Brownian exit times
and, as uq = Ψ(Uq),
∂uq
∂t
= 2ϕ(Uq)
∂Uq
∂t
,
∇uq = 2ϕ(Uq)∇Uq
and
∆uq = 2ϕ(Uq)(∆Uq − Uq | ∇Uq |2)
Thus
∂Uq
∂t
=
1
2
∆Uq −
1
2
Uq | ∇Uq |2 .
To simplify notation, from now on let
ξ = (t, x), η = (t, y), and ς = (t, x + y)
so that, if t > 0, x ∈ C and y ∈ D,
∇xV = (∇Uh)(ς)− (∇Uf )(ξ),
∇yV = (∇Uh)(ς)− (∇Ug)(η),
∆xV = (∆Uh)(ς)− (∆Uf )(ξ),
∆yV = (∆Uh)(ς)− (∆Ug)(η)
and ∑
1in
∂2V
∂xi∂yi
= (∆Uh)(ς).
Thus introducing the differential operator
E =1
2
{
∆x −
∑
1in
∂2
∂xi∂yi
+ ∆y
}
we get
EV = 1
2
{(∆Uh)(ς)− (∆Uf )(ξ)− (∆Ug)(η)} .
Note here that the quadratic form
Q(r1, ..., rn, s1, ..., sn) =
∑
1in
r2i −
∑
1in
risi +
∑
1in
s2i
is positive semi-definite. From the above
EV = ∂Uh
∂t
(ς) +
1
2
Uh(ς) | (∇Uh)(ς) |2
– 43 –
Christer Borell
−∂Uf
∂t
(ξ)− 1
2
Uf (ξ) | (∇Uf )(ξ) |2
−∂Ug
∂t
(η)− 1
2
Ug(η) | (∇Ug)(η) |2
or
EV = ∂V
∂t
+ Ω(t, x, y)
with
Ω(t, x, y) =
1
2
Uh(ς) | (∇Uh)(ς) |2 −
1
2
Uf (ξ) | (∇Uf )(ξ) |2
−1
2
Ug(η) | (∇Ug)(η) |2 .
Here
| (∇Uf )(ξ) |2=| (∇Uh)(ς) |2
+
∑
1in
{
∂Uf
∂xi
(ξ) +
∂Uh
∂xi
(ς)
} {
∂Uf
∂xi
(ξ)− ∂Uh
∂xi
(ς)
}
and
| (∇Ug)(η) |2=| (∇Uh)(ς) |2
+
∑
1in
{
∂Ug
∂xi
(η) +
∂Uh
∂xi
(ς)
} {
∂Ug
∂xi
(η)− ∂Uh
∂xi
(ς)
}
.
From the above equations it follows that
Ω(t, x, y) =
1
2
| (∇Uh)(ς) |2 V − b(t, x, y) · ∇(x,y)V
for an appropriate function b(t, x, y). Moreover,
EV + b(t, x, y) · ∇(x,y)V =
∂V
∂t
+
1
2
| (∇Uh)(ς) |2 V.
The non-negativity of V (t, x, y) now follows from the maximum princi-
ple. For completeness we give a direct proof here. Let T ∈ ]0,∞[ be fixed. We
know that the function V (t, x, y) is non-negative on ({0}×(C×D))∪([0, T ]×
∂(C×D)). Therefore, if V (t, x, y) < 0 at some point (t, x, y) ∈ [0, T ]×(C̄×D̄)
there exists a strictly positive number ε such that the function εt+V (t, x, y)
possesses a strictly negative minimum in [0, T ]× (C̄ × D̄) at a certain point
ς0 = (t0, x0, y0) ∈ ]0, T ]× (C ×D). But then
V (ς0) < 0,
∂V
∂t
(ς0)  −ε, ∇(x,y)V (ς0) = 0, and EV (ς0)  0.
and we have got a contradiction. This proves the inequality (1.3).
– 44 –
Minkowski sums and Brownian exit times
The inequality (1.4) follows by choosing f = g = h = 1 in (1.3).
Furthermore, the distribution function of Brownian exit time from an affine
half-space shows that equality occurs in (1.4) if C and D are parallel affine
half-spaces. This completes the proof of Theorem 1.1.
3. A sketchy proof of Theorem 1.2
Let
dγn(x) = exp(−
1
2
| x |2) dx√
2π
n
be the canonical Gauss measure in Rn.
First suppose α, β > 0 and
Φ−1(γn(αA + βB))  αΦ−1(γn(A)) + βΦ−1(γn(B))
for all A,B ∈ B(Rn). We claim that (1.6) holds.
To see this suppose C ∈ B(Rn) is convex, symmetric, and
0 < γn(C) < 12 . Then
Φ−1(γn((α + β)C))  αΦ−1(γn(C)) + βΦ−1(γn(C))
Now, if α + β < 1 it follows that (α + β)C ⊆ C and we get
Φ−1(γn(C))  αΦ−1(γn(C)) + βΦ−1(γn(C))
or
0  (α + β − 1)Φ−1(γn(C))
which is a contradiction. On the other hand if | α− β |> 1 we get a contra-
diction as follows. Depending on symmetry there is no loss of generality to
assume that β − α > 1. Then
Rn \ C ⊇ αC + β(Rn \ C)
and we get
Φ−1(γn(Rn \ C))  αΦ−1(γn(C)) + βΦ−1(γn(Rn \ C))
or
−Φ−1(γn(C))  αΦ−1(γn(C))− βΦ−1(γn(C))
– 45 –
Christer Borell
since Φ−1(1− y) = −Φ−1(y) for all 0 < y < 1. Thus
0 > (α + 1− β)Φ−1(γn(C))
which is a contradiction.
To prove that (1.6) implies that (1.5) is valid for all A,B ∈ B(F ) there
is no loss of generality to assume F = Rn and γ = γn. Most parts of the
proof may be arranged in a similar way as the proof of Theorem 1.1 above
and, moreover, we may proceed almost in the same manner as in my proof
of Ehrhard’s inequality [2] (replace the pair (θ, 1− θ) by (α, β) and replace
the differential operator
E =1
2
{
∆x + 2
∑
1in
∂2
∂xi∂yi
+ ∆y
}
by the differential operator
Enew=
1
2
{
∆x +
1− α2 − β2
αβ
∑
1in
∂2
∂xi∂yi
+ ∆y
}
.
Finally, note that if, α, β > 0, the differential operator Enew is semi-elliptic
if and only if α + β  1 and| α− β | 1). The details are omitted here.
Acknowledgement. The author is very grateful to the referee for point-
ing out an error in an earlier version of this paper.
Bibliography
[1] Borell (Ch.). — Greenian potentials and concavity, Math. Ann. 272, p. 155-160
(1985).
[2] Borell (Ch.). — The Ehrhard inequality, C. R. Acad. Sci. Paris, Ser. I 337, p.
663-666 (2003).
[3] Carlen (E. A.), Kerce (C.). — On the cases of equality in Bobkov’s inequality
and Gaussian rearrangement, Calc. Var. 13, p. 1-18 (2001).
[4] Ehrhard (A.). — Symétrisation dans l’espace de Gauss, Math. Scand. 53, p. 281-
301 (1983).
[5] Ehrhard (A.). — Eléments extrémaux pour les inégalités de Brunn-Minkowski
gaussiennes. Annales de l’Institut Henri Poincaré 22, p. 149-168 (1986).
[6] Karatzas (I.), Shreve (S. E.). — Brownian Motion and Stochastic Calculus. Sec-
ond Edition, Springer-Verlag 1991.
– 46 –
Minkowski sums and Brownian exit times
[7] Landau (H. J.), Shepp (L. A.). — On the supremum of a Gaussian process, Sankhyā
32, ser A, p. 369-378 (1970).
[8] Latala (R. A.). — On some inequalities for Gaussian measures. Proceedings of the
ICM, 2, p. 813-822 (2002).
[9] Sudakov (V. N.), Tsirelson (B. S.). — Extremal properties of half-spaces for
spherically-invariant measures, Zapiski Nauchn, Seminarov LOMI 41, p. 14-24
(1974) (translated in J. Soviet Math. 9, p. 9-18 (1978)).
[10] Yurinsky (V.). — Sums and Gaussian Vectors. Lecture Notes in Mathematics 1617.
Springer-Verlag 1995.
– 47 –


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Pierre Milman

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
CAROLINE GRANT MELLES, PIERRE MILMAN
Erratum: article Melles-Milman, Lemma 4.5, p. 719-720, fascicule 4, 2006
Tome XVI, no 2 (2007), p. 425-426.
<http://afst.cedram.org/item?id=AFST_2007_6_16_2_425_0>
© Université Paul Sabatier, Toulouse, 2007, tous droits réservés.
L’accès aux articles de la revue « Annales de la faculté des sci-
ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique
l’accord avec les conditions générales d’utilisation (http://afst.cedram.
org/legal/). Toute reproduction en tout ou partie cet article sous quelque
forme que ce soit pour tout usage autre que l’utilisation à fin strictement
personnelle du copiste est constitutive d’une infraction pénale. Toute copie
ou impression de ce fichier doit contenir la présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
ERRATUM
(article Melles-Milman, Lemma 4.5, p. 719-720, fascicule 4, 2006)
”Due to an error of the authors, a few lines were missing in the fourth
paragraph of the original proof of Lemma IV.5 of our recently published
paper, leaving a gap in the proof.”
Lemma 4.5. — The same polynomials that generate J(0,0) also gener-
ate J over a neighborhood of (0, 0) in U × Cn+1 and generate I over a
neighborhood of {0} × Pn in U × Pn.
Proof. — Suppose that J is generated in a neighborhood of (0, 0) by
F1(x, y), ..., Fs(x, y), where Fi(x, y) is a homogeneous polynomial of degree
di in y with analytic coefficients in x. We will show that I is generated on
a neighborhood of {0} × Pn in U × Pn by the corresponding polynomials
Fi(x, ξ), where [ξ] = [ξ0 : ... : ξn] are homogeneous coordinates for Pn. More
precisely, we will show that I is generated on a neighborhood of any point
q ∈ {0} × Pn by dehomogenizations of F1, ..., Fs near q.
Choose homogeneous coordinates ξ on Pn such that q = (0, [1 : 0 :
... : 0]). Nonhomogeneous coordinates on the set W = {ξ0 = 0} ⊂ Pn are
wi = ξiξ0 for 1  i  n. We will check that I is generated in a neighborhood
of q by the polynomials
Fi(x, ξ)
ξdi0
= Fi
(
x,
ξ
ξ0
)
= Fi(x, 1, w1, ..., wn).
First we look at the maps σ1 and σ2 in local coordinates. We may use
(x, y0, w) as local coordinates in σ−12 (U×W ) ∼= U×C×W . Local coordinates
for U ×Cn+1 are (x, y0, y1, ..., yn), where yi = y0wi for 1  i  n. The maps
σ1 and σ2 are given by
σ1(x, y0, w) = (x, y0, y0w) and σ2(x, y0, w) = (x, w).
Suppose that G is a holomorphic section of I on a neighborhood of
q in U × Pn. Then σ∗2G is a holomorphic section of σ−12 I in a neighbor-
hood of σ−12 (q) = {(0, y0, 0) : y0 ∈ C}. The homogeneous polynomials
F1, ..., Fs that generate J(0,0) also generate J = σ1∗(σ−12 I) on a neighbor-
hood of (0, 0) ∈ U × Cn+1 (since J is coherent, by the Direct Image The-
orem), so their pullbacks σ∗1F1, ..., σ
∗
1Fs generate J̃ := σ−11 J on a neigh-
borhood of σ−11 (0, 0) ∈ U × C̃n+1 and therefore generate Ĩ := σ−12 I off
– 425 –
Erratum
H := σ−11 (U × {0}). Hence Supp(Ĩ/J̃ ) ⊂ H and therefore (by the complex
analytic nullstellensatz) there exist an integer d > 0 and holomorphic func-
tions A1, ..., As on a neighborhood of the point (x = 0, y0 = 0, w = 0) in
U × C̃n+1 such that
yd0σ
∗
2G(x, y0, w) =
s∑
i=1
Ai(x, y0, w)σ∗1Fi(x, y0, w)
on that neighborhood. But σ∗2G(x, y0, w) = G(x, w) is independent of the
value of y0 and σ∗1Fi(x, y0, w) = Fi(x, y0, y0w) = y
di
0 Fi(x, 1, w) since Fi is
homogeneous of degree di in y. Therefore, by comparing terms in yd0 of both
sides of the equation above, it follows that there are holomorphic functions
a1, ..., as (on a neighborhood of (x = 0, y0 = 0, w = 0)) depending only on
x and w such that
G(x, w) =
s∑
i=1
ai(x, w)Fi(x, 1, w).
Since the functions Fi(x, 1, w) are the local dehomogenizations of the ho-
mogeneous polynomials F (x, ξ), we are done. 
– 426 –


Erratum: article Melles-Milman, Lemma 4.5, p. 719-720, fascicule 4, 2006

Gabriela Putinar

Mihai Putinar

Reconstruction of algebraic sets from dynamic moments

Marius van der Put

Marc Reversat

Galois theory of $q$-difference equations

Christine Noot-Huyghe

Fabien Trihan

Sur l’holonomie de $\mathcal{D}$-modules arithmétiques associés à des $F$-isocristaux surconvergents sur des courbes lisses

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On some nonlinear partial differential equations involving the 1-Laplacian

Minkowski sums and Brownian exit times

Special Lagrangian submanifolds in the complex sphere

Weak analytic hyperbolicity of generic hypersurfaces of high degree in $\mathbb{P}^{4}$

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On functional linear partial differential equations in Gevrey spaces of holomorphic functions.

Erratum: article Melles-Milman, Lemma 4.5, p. 719-720, fascicule 4, 2006

Reconstruction of algebraic sets from dynamic moments

Galois theory of $q$ -difference equations

Sur l’holonomie de $\mathcal{D}$ -modules arithmétiques associés à des $F$ -isocristaux surconvergents sur des courbes lisses

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