Comptes Rendus Mathématique

Yuan Zhang

Sobolev regularity of the canonical solutions to $\bar{\partial }$ on product domains

Marco Schlichting

Comptes Rendus
Mathématique
Marco Schlichting
Corrigendum to “Symplectic and orthogonal K -groups of the integers”
Volume 362 (2024), p. 307-308
Online since: 2 May 2024
https://doi.org/10.5802/crmath.606
This article is licensed under the
Creative Commons Attribution 4.0 International License.
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C EN T R E
MER S ENN E
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Comptes Rendus. Mathématique
2024, Vol. 362, p. 307-308
https://doi.org/10.5802/crmath.606
Erratum / Erratum
Number Theory / Théorie des nombres
Corrigendum to “Symplectic and orthogonal
K -groups of the integers”
Corrigendum à « K -groupes symplectiques et
orthogonaux de l’anneau des entiers »
Marco Schlichting a
a Marco Schlichting, Mathematics Institute, Zeeman Building, University of Warwick,
Coventry CV4 7AL, UK
E-mail: m.schlichting@warwick.ac.uk
Abstract. The action of the duality functor on the odd torsion of Kn (Z) was stated incorrectly in [3], in half of
the cases, and lead to incorrect formulas for the odd primary torsion of πn BSp(Z)+ and πn BO∞,∞(Z)+.
Résumé. L’action du foncteur de dualité sur la torsion impaire de Kn (Z) était énoncé incorrectement dans [3],
dans la moitié des cas, et a conduit à des formules incorrectes pour la torsion primaire impaire deπn BSp(Z)+
et πn BO∞,∞(Z)+.
Manuscript received 6 November 2023, accepted 7 January 2024.
Let R be a ring of integers in a number field and ℓ an odd prime. The formula [3, (2.3)] for the
ℓ-primary torsion subgroups of Kn(R) is false. The correct formulas extracted from [4, Proof of
Theorem 70] are for i > 1
K2i (R){ℓ} = K2i (R ′){ℓ} = H 2et (R ′, (K2i+2)ℓ)
where the right hand term is the inverse limit limν H 2et (R
′, (K2i+2)/ℓν). In particular, the duality
acts by (−1)i+1 on this group. In odd degree for i > 1 we have
K2i−1(R){ℓ} = K2i−1(R ′){ℓ} = H 0et (R ′, (K2i )/ℓ∞)
where the right hand term is the direct limit colimν H 0et (R
′, (K2i )/ℓν). In particular, the duality
acts by (−1)i on this group. These formulas are not new; see [4, Theorem 70 and proof thereof];
see also [1, Lemma 3.4.4]. Together with [3, Lemma 2.1] this yields now a correction of [3,
Theorem 2.2] in the cases n ≡ 0,2 mod 4.
Theorem 1. Let R be a ring of integers in a number field, and ℓ ∈ Z an odd prime. Then for all
n ≥ 1 we have isomorphisms
GWn(R){ℓ} ∼= K Spn(R){ℓ} ∼= KQn(R){ℓ} ∼=
{
Kn(R){ℓ} n ≡ 2,3 mod 4
0 n ≡ 0,1 mod 4.
[3, Section 3] is unaffected by this error and thus, we obtain the following table of homotopy
groups correcting [3, Théorème 0.1 and Theorem 1.1] where the 2-primary part comes from [2,
4.7.2] and [3, Theorem 3.3] and the odd primary part from Theorem 1. See also [1, Theorem 3.2.1].
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
308 Marco Schlichting
Theorem 2. The homotopy groups of the spaces B Sp(Z)+ and BO∞,∞(Z)+ for n ≥ 1 are given in
the following table
n mod 8 0 1 2 3 4 5 6 7
πnB Sp(Z)+ 0 0 Z⊕Kn(Z)odd Kn(Z) Z/2 Z/2 Z⊕Kn(Z)odd Kn(Z)
πnBO∞,∞(Z)+ Z⊕Z/2 (Z/2)3
(Z/2)2
⊕
Kn(Z)odd
Z/8
⊕
Kn(Z)odd
Z 0 Kn(Z)odd Kn(Z)
Finally, this leads to a correction of [3, Remark 1.3] and the following table for n > 0. Denote by
Bk the k-th Bernoulli number [4, Example 24] and let dn denote the denominator of
1
n+1 B(n+1)/4
for n = 3 mod 4. By [4, Introduction, Lemma 27] we have Kn(Z) = Z/2dn for n = 3 mod 8 and
Kn(Z) =Z/dn for n = 7 mod 8. Similarly, denote by ck the numerator of Bk /4k. Then K4k−2(Z) is a
finite group of order ck when k is even and of order 2ck when k is odd. This group is conjectured
to be cyclic.
n mod 8 0 1 2 3 4 5 6 7
πnB Sp(Z)+ 0 0 Z⊕|Z/ck | Z/2dn Z/2 Z/2 Z⊕|Z/ck | Z/dn
πnBO∞,∞(Z)+ Z⊕Z/2 (Z/2)3 (Z/2)2 ⊕|Z/ck | Z/dn Z 0 |Z/ck | Z/dn
where |Z/m| denotes a finite group of order n conjectured to be cyclic and n = 4k −2.
Full proof of the claims in [3] are now available in [1].
References
[1] B. Calmès, E. Dotto, Y. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin, T. Nikolaus, W. Steimle, “Hermitian K -theory
for stable ∞-categories, Part III: Grothendieck-Witt groups of rings”, https://arxiv.org/abs/2009.07225, 2020.
[2] M. Karoubi, “Bott periodicity in topological, algebraic and Hermitian K -theory”, in Handbook of K -theory. Vol. 1, 2,
Springer, 2005, p. 111-137.
[3] M. Schlichting, “Symplectic and orthogonal K -groups of the integers”, C. R. Math. Acad. Sci. Paris 357 (2019), no. 8,
p. 686-690.
[4] C. Weibel, “Algebraic K -theory of rings of integers in local and global fields”, in Handbook of K -theory. Vol. 1, 2,
Springer, 2005, p. 139-190.


Corrigendum to “Symplectic and orthogonal $K$-groups of the integers”

Nguyen Manh Linh

Groupes de Brauer algébriques modulo les constantes d’espaces homogènes et leurs compactifications

Julian Tugaut

Comptes Rendus
Mathématique
Julian Tugaut
Granular media equation with double-well external landscape: limiting steady
state
Volume 362 (2024), p. 775-778
Online since: 17 September 2024
https://doi.org/10.5802/crmath.595
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 775-778
https://doi.org/10.5802/crmath.595
Research article / Article de recherche
Partial differential equations, Probability theory / Equations aux dérivées
partielles, Probabilités
Granular media equation with double-well
external landscape: limiting steady state
Équation des milieux granulaires avec potentiel externe
à double puits : état stationnaire limite
Julian Tugaut ,a
a Université Jean Monnet, CNRS UMR 5208, Institut Camille Jordan, Maison de
l’Université, 10 rue Tréfilerie, CS 82301, 42023 Saint-Étienne Cedex 2, France
E-mail: julian.tugaut@univ-st-etienne.fr (J. Tugaut)
Abstract. In this paper, we give a simple condition on the initial state of the granular media equation which
ensures that the limit as the time goes to infinity is the unique steady state with positive center of mass. To do
so, we use functional inequalities, Laplace method and McKean–Vlasov diffusion (which corresponds to the
probabilistic interpretation of the granular media equation).
Résumé. Dans ce papier, nous donnons une condition simple portant sur l’état initial de l’équation des
milieux granulaires qui assure que la limite en temps long est l’unique probabilité invariante avec un centre
de masse strictement positif. Pour ce faire, nous utilisons des inégalités fonctionnelles, la méthode de Laplace
et la diffusion de McKean–Vlasov (qui correspond à l’interprétation probabiliste de l’équation des milieux
granulaires).
2020 Mathematics Subject Classification. 35K55, 60J60, 60G10, 39B72.
Funding. This work is supported by the French ANR grant METANOLIN (ANR-19-CE40-0009).
Manuscript received 21 October 2023, accepted 29 November 2023.
In this work, we are interested in the long-time behavior of the following granular media
equation in the one-dimensional setting:
∂
∂t
µσt (x) =
σ2
2
∂2
∂x2
µσt (x)+
∂
∂x
{
µσt (x)
(∇V (x)+α(x −mσ1 (t )))} (1)
with α> 0 and mσ1 (t ) :=
∫
R xµ
σ
t (d x). Since m
σ
1 (t ) depends on µ
σ
t , (1) is nonlinear. We assume the
following on V :
(V-1) Polynomial function: V is a polynomial function with deg(V ) ≥ 4.
(V-2) Symmetry: V is an even function.
(V-3) Double-well potential: The equation V ′(x) = 0 admits exactly three solutions: a, −a and
0 with a > 0; V ′′(a) > 0 and V ′′(0) < 0. The bottoms of the wells are reached for x = a and
x =−a. Moreover, V (2k)(0) ≥ 0, for all k ≥ 2.
(V-4) limx→±∞V ′′(x) =+∞ and for any x ≥ a, V ′′(x) > 0.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
776 Julian Tugaut
The nonconvexity of V is measured by the following constant: θ := supR−V ′′. We also assume,
eventually, that synchronization occurs that means: α > θ. The initial measure µσ0 is assumed
to be absolutely continuous with respect to the Lebesgue measure with a density that we denote
for simplicity by µσ0 . Moreover,
∫
R x
2kµσ0 (x)d x <∞ for any k ∈N and
∫
Rµ
σ
0 (x) log(µ
σ
0 (x))d x <+∞
that is the initial entropy is finite and so the same is true for the initial free-energy.
In this setting, it is well-known that if σ is small enough, then there are several steady states,
see [4, 7]. We have proven that if µσ is a steady state with total mass equal to 1 of the granular
media equation (1) then there exists m ∈R such thatµσ =µm,σ where the measureµm,σ is defined
as
µm,σ(d x) :=
exp
[
− 2
σ2
(
V (x)+ α2 x2 −αmx
)]
∫
R exp
[
− 2
σ2
(
V (y)+ α2 y2 −αmy
)]
d y
d x .
Moreover, m is a zero of the following function:
χσ(m) :=
∫
R x exp
[
− 2
σ2
(
V (x)+ α2 x2 −αmx
)]
d x∫
R exp
[
− 2
σ2
(
V (x)+ α2 x2 −αmx
)]
d x
−m .
In [7], we have shown that ifσ<σc (α), then there exists m(σ) > 0 such thatχσ is positive on the
interval (0;m(σ)) and negative on (m(σ);+∞). Thus, due to the symmetry of V , there exist exactly
three steady states : νσ+ with positive expectation, νσ− with negative expectation and νσ0 with zero
expectation. We point out that we can extend our work to non symmetrical case provided that
χσ is positive on the interval (0;m(σ)). Despite this non-uniqueness, the convergence towards
one of the invariant probability measures has been proven in [6]. Nevertheless, very few is known
about the basins of attraction. Here, we exhibit a simple condition ensuring that µσt converges
towards νσ+.
Theorem 1. We assume there exists δ ∈ (0;m(σ)) such that ∫R xµσ0 (d x) > m(σ)−δ and
χσ(m(σ)−δ) >
{
W2
(
µσ0 ;µ
m(σ)−δ,σ) if α> θ
L2
(
µσ0 ;µ
m(σ)−δ,σ) if α≤ θ , (2)
whereW2 stands for the quadratic Wasserstein distance. Then, µσt weakly converges towards ν
σ+ as
t goes to infinity.
Proof. We introduce the self-stabilizing diffusion:
Xσt = X0 +σBt −
∫ t
0
∇V (Xσs )d s −α
∫ t
0
(
Xσs −E
[
Xσs
])
d s . (3)
Here, X0 is a random variable, independent from the Brownian motion B and we assume that
X0 follows the law µσ0 . This kind of processes has been introduced in the seminal work [5]. We
do not discuss the wellposedness of Equation (3). About this, we refer to [3]. We point out that
L (Xσt ) =µσt for any t ≥ 0. We can rewrite the equation in this way: Xσt = X0+σBt−
∫ t
0 ∇V (Xσs )d s−
α
∫ t
0 (X
σ
s −mσ1 (s))d s. We introduce the following diffusion
Y σt = X0 +σBt −
∫ t
0
∇V (Y σs )d s −α∫ t
0
(
Y σs − (m(σ)−δ)
)
d s . (4)
We consider the deterministic time Tσ0 := inf{t ≥ 0 : E(Xσt ) ≤ m(σ)−δ} where Xσ is defined
in Equation (3). Then, for any t ∈ [0;Tσ0 ), we have Xσt ≥ Y σt so E(Xσt ) ≥ E(Y σt ). We remark
E(Y σt ) = E(Y σ∞)+E(Y σt −Y σ∞) where Y σ∞ follows the law µm(σ)−δ,σ, which is the unique invariant
probability of Diffusion (4).
Julian Tugaut 777
On the one hand, we assume that α > θ. Then, x 7→ V (x)+ α2 (x − (m(σ)−δ))2 is uniformly
convex. Also, the unique invariant probability measure of Equation (4) is the measure µm(σ)−δ,σ.
Hence, by applying [2, Theorem 2.1.], we deduce:
W2
(
L
(
Y σt
)
;µm(σ)−δ,σ
)
≤ e−(α−θ)tW2
(
L (X0) ;µ
m(σ)−δ,σ
)
.
With Ỹ σ∞ and Ỹ σt realizing the optimal coupling inW2, we have:
E
(
Y σt −Y σ∞
)=E(Ỹ σt − Ỹ σ∞)≥−E(∣∣∣Ỹ σt − Ỹ σ∞∣∣∣)≥−W2 (L (Y σt ) ;µm(σ)−δ,σ)
≥−e−(α−θ)tW2
(
L (X0) ;µ
m(σ)−δ,σ
)
≥−W2
(
L (X0) ;µ
m(σ)−δ,σ
)
.
Hence, for any t ≤ Tσ0 , we have E(Y σt ) ≥
∫
R xµ
m(σ)−δ,σ(d x)−W2
(
L (X0) ;µm(σ)−δ,σ
)
. We deduce
E(Y σt ) ≥ (m(σ)−δ)+χσ (m(σ)−δ)−W2
(
L (X0) ;µm(σ)−δ,σ
)
. Consequently, for any t ≤ Tσ0 , we have
E(Xσt ) ≥ (m(σ)−δ)+χσ (m(σ)−δ)−W2
(
L (X0) ;µ
m(σ)−δ,σ
)
> m(σ)−δ .
Hence, Tσ0 =+∞ so that for any t ≥ 0: E(Xσt ) ≥ m(σ)−δ.
By applying main theorem in [6], we deduce that L (Xσt ) weakly converges towards one of the
steady state: νσ−, νσ0 or ν
σ+.
Since
∫
xνσ−(d x) < 0 =
∫
xνσ0 (d x) <
∫
xL (Xσt )(d x) for any t ≥ 0, we deduce that L (Xσt ) does
not converge towards either νσ− nor νσ0 . Consequently, L (X
σ
t ) weakly converges towards ν
σ+ as t
goes to infinity.
On the other hand, if α ≤ θ, we can use Poincaré inequality from [1, Corollary 1.6.] since
V ′′(x)+α> 0 for sufficiently large x. We deduce the existence of C (σ) > 0 such that
L2
(
L
(
Y σt
)
;µm(σ)−δ,σ
)
≤ e−C (σ)t L2
(
L (X0) ;µ
m(σ)−δ,σ
)
so that
W2
(
L
(
Y σt
)
;µm(σ)−δ,σ
)
≤ e−C (σ)t L2
(
L (X0) ;µ
m(σ)−δ,σ
)
.
We then complete the proof as previously with
L2
(
L (X0);µ
m(σ)−δ,σ
)
instead ofW2
(
L (X0);µ
m(σ)−δ,σ
)
. □
As mentioned above, χσ(m) ≤ 0 for any m ≥ m(σ) so we immediately deduce that the theorem
can not be applied if
∫
R xµ
σ
0 (d x) ≥ m(σ). We have an immediate corollary:
Corollary 2. We assume that there exists m ∈ (0;m(σ)) such that µσ0 = µm,σ. Then, µσt weakly
converges towards νσ+ as t goes to infinity.
Proof. Hypothesis (2) is satisfied since for any m ∈ (0;m(σ)), there exists δ ∈ (0;m(σ)) satisfying
m = m(σ)−δ. Then, we just remark:
χσ(m(σ)−δ) > 0 =W2
(
µm(σ)−δ,σ;µm(σ)−δ,σ
)
= L2
(
µm(σ)−δ,σ;µm(σ)−δ,σ
)
.
First condition is also satisfied due to
∫
R xµ
m(σ)−δ,σ(d x) = m(σ)−δ+χσ(m(σ)−δ) > m(σ)−δ.
Thus, we may apply Theorem 1. □
We stress that we only used the decrease of the Wasserstein distance in the proof of the
theorem. Hence, it is not a priori required that the decay is exponential.
We point out that we have a similar result with the other half-line: if µσ0 =µ−m(σ)+δ,σ. Then, µσt
converges weakly towards νσ− as t goes to infinity.
778 Julian Tugaut
References
[1] D. Bakry, F. Barthe, P. Cattiaux and A. Guillin, “A simple proof of the Poincaré inequality for
a large class of probability measures including the log-concave case”, Electron. Commun.
Probab. 13 (2008), pp. 60–66.
[2] F. Bolley, I. Gentil and A. Guillin, “Convergence to equilibrium in Wasserstein distance for
Fokker–Planck equations”, J. Funct. Anal. 263 (2012), no. 8, pp. 2430–2457.
[3] S. Herrmann, P. Imkeller and D. Peithmann, “Large deviations and a Kramers’ type law for
self-stabilizing diffusions”, Ann. Appl. Probab. 18 (2008), no. 4, pp. 1379–1423.
[4] S. Herrmann and J. Tugaut, “Non-uniqueness of stationary measures for self-stabilizing
processes”, Stochastic Processes Appl. 120 (2010), no. 7, pp. 1215–1246.
[5] H. P. J. McKean, “A class of Markov processes associated with nonlinear parabolic equations”,
Proc. Natl. Acad. Sci. USA 56 (1966), pp. 1907–1911.
[6] J. Tugaut, “Convergence to the equilibria for self-stabilizing processes in double-well land-
scape”, Ann. Probab. 41 (2013), no. 3A, pp. 1427–1460.
[7] J. Tugaut, “Phase transitions of McKean–Vlasov processes in double-wells landscape”,
Stochastics 86 (2014), no. 2, pp. 257–284.


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Sobolev regularity of the canonical solutions to $\bar{\partial }$ on product domains

Corrigendum to “Symplectic and orthogonal $K$ -groups of the integers”