Comptes Rendus Mathématique

Nguyễn H. V. Hưng

Comptes Rendus
Mathématique
Nguyễn H. V. Hu,ng
The mod 2 Margolis homology of the Dickson algebra
Volume 358, issue 4 (2020), p. 505-510.
<https://doi.org/10.5802/crmath.68>
© Académie des sciences, Paris and the authors, 2020.
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Comptes Rendus
Mathématique
2020, 358, n 4, p. 505-510
https://doi.org/10.5802/crmath.68
Algebraic Topology / Topologie algébrique
The mod 2 Margolis homology of the Dickson
algebra
Nguyễn H. V. Hu,nga
a Department of Mathematics, HUS, Vietnam National University, Hanoi, 334 Nguyễn
Trãi Street, Hanoi, Vietnam.
E-mail: nhvhung@vnu.edu.vn.
In memory of Nguyễn Thi. Thanh Bình
Abstract. We completely compute the mod 2 Margolis homology of the Dickson algebra Dn , i.e. the homology
of Dn with the differential to be the Milnor operation Q j , for every n and j . The motivation for this problem
is that, the Margolis homology of the Dickson algebra plays a key role in study of the Morava K-theory
K ( j )∗(BSm ) of the symmetric group on m letters Sm .
We show that Pengelley–Sinha’s conjecture on H∗(Dn ;Q j ) for n ≤ j is true if and only if n = 1 or 2. For
3 ≤ n ≤ j , our result proves that this conjecture turns out to be false since the occurrence of some “critical
elements” hs1 ,...,sk ’s of degree (2
j+1 −2n )+∑ki=1(2n −2si ) in this homology for 0 < s1 < ·· · < sk < n and k > 1.
Résumé. Dans cette note on calcule entièrement l’homologie de Margolis modulo 2 de l’algèbre de Dickson
Dn , i.e. l’homologie de Dn en choisissant pour différentielles les opérations de Milnor Q j , pour tous n et j .
La motivation pour cette étude est le rôle clé joué par cette homologie dans l’étude de la K-théorie de Morava
K ( j )∗(BSm ) du groupe symétrique Sm en m lettres.
Nous montrons que la conjecture de Pengelley–Sinha sur H∗(Dn ;Q j ) pour n ≤ j est vraie si et seulement
si n = 1,2. Pour 3 ≤ n ≤ j notre résultat montre que la conjecture est fausse à cause de l’occurence d’éléments
« critiques » hs1 ,...,sk de degré (2
j+1 − 2n )+∑ki=1(2n − 2si ) dans cette homologie pour 0 < s1 < ·· · < sk < n
et k > 1.
Mathematical subject classification (2010). 55S05, 55S10, 55N99.
Funding. This research is funded by the Vietnam National Foundation for Science and Technology Develop-
ment (NAFOSTED) under grant number 101.04-2019.300.
Manuscript received 24th February 2020, revised 2nd May 2020, accepted 4th May 2020.
Let A be the mod 2 Steenrod algebra, genenated by the cohomology operations Sq j with j ≥ 0
and subject to the Adem relation with Sq0 = 1. Further A is a Hopf algebra, whose coproduct is
given by the formula ∆(Sq j ) =∑ ji=0 Sq i ⊗Sq j−i .
Let A∗ be the Hopf algebra, which is dual to A . Let ξ j = (Sq2 j · · ·Sq2Sq1)∗ be the Milnor
element of degree 2 j+1−1 in A∗, for j ≥ 0, where the duality is taken with respect to the admissible
basis of A . According to Milnor [4], as an algebra, A∗ ∼= F2[ξ0,ξ1, . . . ,ξ j , . . . ], the polynomial
algebra in infinitely many generators ξ0,ξ1, . . . ,ξ j , . . . .
Let Q j , for j ≥ 0, be the Milnor operation (see [4]) of degree (2 j+1 −1) in A , which is dual to
ξ j with respect to the basis of A∗ consisting of all monomials in the generators ξ0,ξ1, . . . ,ξ j , . . . .
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
506 Nguyễn H. V. Hu,ng
Remarkably, Q j is a differential, that is Q2j = 0 for every j . In fact, Q0 = Sq1, Q j = [Q j−1,Sq2
j
], the
commutator of Q j−1 and Sq2
j
in the Steenrod algebra A , for j > 0.
In the article, we compute the Margolis homology of the Dickson algebra Dn , i.e. the homology
of Dn with the differential to be the Milnor operation Q j .
The real goal that we persue is to compute the Morava K -theory K ( j )∗(BSm) of the symmetric
group Sm on m letters. It was well known that, the Milnor operation is the first non-zero
differential, Q j = d2 j+1−1, in the Atiyah–Hirzebruch spectral sequence for computing K ( j )∗(X ),
the Morava K -theory of a space X . So, the Q j -homology of H∗(X ) is the E2 j+1 -page in the Atiyah–
Hirzebruch spectral sequence for K ( j )∗(X ). (See e.g. Yagita [10, §2], although the fact was well
known before this article.)
A key step in the determination of the symmetric group’s cohomology is to apply the Quillen
restiction from this cohomology to the cohomologies of all elementary abelian subgroups of
the symmetric group. For m = 2n and the “generic” elementary abelian 2-subgroup (Z/2)n
of the symmetric group S2n , the image of the restriction H∗(BS2n ) → H∗(B(Z/2)n) is exactly
the Dickson algebra Dn (see Mùi [5, Thm. II.6.2]). So, the E2 j+1 -page in the Atiyah–Hirzebruch
spectral sequence for K ( j )∗(BS2n ) maps to the Margolis homology H∗(Dn ;Q j ). This is why the
Margolis homology of the Dickson algebra is taken into account.
Let us study the range n Dickson algebra of invariants
Dn = F2[x1, . . . , xn]GL(n,F2),
where each generator xi is of degree 1, and the general linear group GL(n,F2) acts canonically on
F2[x1, . . . , xn]. Following Dickson [1], let us consider the determinant
[e1, . . . ,en] = det

x2
e1
1 . . . x
2e1
n
...
. . .
...
x2
en
1 . . . x
2en
n

for non-negative integers e1, . . . ,en . Then ω[e1, . . . ,en] = det(ω)[e1, . . . ,en], for ω ∈ GL(n,F2)
(see [1]). Set
Ln,s = [0,1, . . . , ŝ, . . . ,n], (0 ≤ s ≤ n),
where ŝ means s being omitted, and Ln = Ln,n . The Dickson invariant cn,s of degree 2n − 2s is
originally defined as follows:
cn,s = Ln,s /Ln , (0 ≤ s < n).
Dickson proved in [1] that Dn is a polynomial algebra on the Dickson invariants
Dn = F2[cn,0, . . . ,cn,n−1].
To be explicit, the Dickson invariant can be expressed as in Hưng–Peterson [3, §2]:
cn,s =
∑
i1+···+in=2n−2s
xi11 · · ·xinn , (0 ≤ s < n).
where the sum is over all sequences i1, . . . , in with ik either 0 or a power of 2.
We are interested in the following element of the Dickson algebra Dn :
A j ,n,s = [0, . . . , ŝ, . . . ,n −1, j ]/Ln ,
for 0 ≤ s < n ≤ j . By convention, A j ,n,−1 = 0.
In this article, when j and n are fixed, the elements cn,s and A j ,n,s will respectively be denoted
by cs and As for abbreviation.
C. R. Mathématique, 2020, 358, n 4, 505-510
Nguyễn H. V. Hu,ng 507
Lemma 1. For 0 ≤ j , 0 ≤ s < n,
Q j (cs ) =

c0, 0 ≤ j < n −1, j = s −1,
0, 0 ≤ j < n −1, j 6= s −1,
c0cs , j = n −1,
c0
(
cs A2n−1 + A2s−1
)
, 0 ≤ s < n ≤ j .
The action of the Steenrod algebra on the Dickson one is basically computed in [2]. Related
and partial results concerning the lemma can be seen in [7–9].
The next two theorems are stated in Sinha [6]. Their proofs are straightforward from Lemma 1.
Theorem 2. For 0 ≤ j < n −1,
H∗(Dn ,Q j ) ∼= F2[c2j+1]⊗F2[c1, . . . , ĉ j+1, . . . ,cn−1],
where ĉ j+1 means c j+1 being omitted.
Let F2[c1, . . . ,cn−1]ev be the F2-submodule of F2[c1, . . . ,cn−1] generated by all the monomials
c i11 · · ·c in−1n−1 with i1 +·· ·+ in−1 even.
Theorem 3.
H∗(Dn ;Qn−1) ∼= F2[c1, . . . ,cn−1]ev.
Proposition 4. For 0 ≤ s1, . . . , sk < n ≤ j ,
Q j (cs1 · · ·csk ) = c0
(
kcs1 · · ·csk A2n−1 +
k∑
i=1
cs1 . . . ĉsi . . .csk A
2
si−1
)
,
where ĉsi means csi being omitted.
Conjecture 5 (D. Pengelley – D. Sinha, see [6]). For n ≤ j ,
H∗(Dn ;Q j ) ∼= D2n/
(
Q j (c0),Q j (c0c1), . . . ,Q j (c0cn−1)
)
.
Let Doddn be the F2-submodule of Dn spanned by all monomials c
i0
0 · · ·c in−1n−1 with at least
one of the exponents i0, . . . , in−1 odd. Note clearly that Doddn is not a Q j -submodule of Dn , but
ImQ j ∩Doddn is, since Q j vanishes on this module.
Pengelley–Sinha’s conjecture is equivalent to the equality: KerQ j =
(
ImQ j ∩Doddn
)⊕D2n . In
other words, there is no class in H∗(Dn ;Q j ) represented by an element in Doddn . The following
two theorems show that Pengelley–Sinha’s conjecture is true for n = 1 or 2 and every j .
Theorem 6. For n = 1, 0 ≤ j ,
H∗(D1;Q j ) ∼= F2[c20 ]/(c2
j+1
0 ).
In particular, H∗(D1;Q0) = F2 (this is also a special case of Theorem 3), H∗(D1;Q1) = Λ(c20 ),
whereΛ(c20 ) denotes the F2-exterior algebra on c
2
0 .
Theorem 7. DenoteΛ(c20 ) =Λ(c20 )/(F2 ·1). If n = 2, 0 ≤ j ,
H∗(D2;Q j ) ∼=

F2[c21 ], for j = 0,1,
Λ(c20 )⊕F2[c21 ], for j = 2,
F2[c20 ,c
2
1 ]/(c
2
0 A
2
0,c
2
0 A
2
1), for j > 2,
where A0 = (x21 x2
j
2 +x2
j
1 x
2
2)/(x1x
2
2 +x21 x2), A1 = (x1x2
j
2 +x2
j
1 x2)/(x1x
2
2 +x21 x2).
The cases j = 0,1 in the previous theorem are special cases of Theorems 2 and 3.
Proposition 8. Pengelley–Sinha’s Conjecture for n ≤ j is true if and only if 1 ≤ n ≤ 2.
C. R. Mathématique, 2020, 358, n 4, 505-510
508 Nguyễn H. V. Hu,ng
How can we adjust Pengelley–Sinha’s conjecture to make a correct one in the problem for
3 ≤ n ≤ j ? The critical elements hs1,...,sk ’s, defined below in the Margolis homology of the
Dickson algebra Dn , are the main ingredient in our correction of Pengelley–Sinha’s conjecture for
3 ≤ n ≤ j .
Note that, c20 divides Q j (c0cs1 · · ·csk ) in Dn , if s1, . . . , sk are pairwise distinct.
Definition 9. For n ≤ j , 0 ≤ s1, . . . , sk < n, and s1, . . . , sk pairwise distinct:
hs1,...,sk =
1
c20
Q j (c0cs1 · · ·csk ).
To be more explicit, under the hypotheses of the definition:
hs1,...,sk = (k +1)cs1 · · ·csk A2n−1 +
k∑
i=1
(cs1 . . . ĉsi . . .csk )A
2
si−1.
Note that, hs1,...,sk ∈ Doddn if k > 1, and that hs1,...,sk depends also on n and j .
Lemma 10.
(i) As =
{
0 mod (c0, . . . ,cr ), 0 ≤ s ≤ r < n,
6= 0 mod (c0, . . . ,cr ), 0 ≤ r < s < n.
(ii) Ar and As are coprime in Dn for 0 ≤ r 6= s < n.
(iii) If n ≤ j , then hs1,...,sk ∈ KerQ j and [hs1,...,sk ] 6= 0 in the Margolis homology H∗(Dn ;Q j ) for
0 < s1 < ·· · < sk < n, 1 < k.
The lemma is based on the following inductive formula, in which the complete notations A j ,n,s
and cn,s are used instead of the simplified ones As and cs :
A j ,n,s = A2j−1,n−1,s−1 + A2j−1,n,n−1cn−1,s
Ln
Ln−1
,
for 0 ≤ s < n ≤ j . Here, by convention, An−1,n,n−1 = 1, cn−1,n−1 = 1.
Note that Q j is a (total) derivation, that is Q j (ab) =Q j (a)b +aQ j (b). We study the s-th partial
derivation for 0 < s ≤ n, and its “inverse”, the so-called integral on a direction. These notions will
play key roles in the remaining part of the article.
Definition 11. Let s1, . . . , sk be pairwise distinct, with 0 ≤ s1, . . . , sk < n, and R ∈ Dn . The s-th
partial derivation is the morphism defined for 0 ≤ s ≤ n by
∂s (cs1 · · ·csk R2) =

c0cs1 · · ·csk A2n−1R2, k odd, s = n,
c0cs1 · · · ĉsi · · ·csk A2si−1R2, s = si ,
0, otherwise.
Since A−1 = 0, it yields ∂0 = 0. If ∂s (cs1 · · ·csk ) 6= 0, then s should be one of the indices s1, . . . , sk
or n. Obviously, Im∂s ⊂ c0 A2s−1Dn . Proposition 4 leads to:
Lemma 12. Let s1, . . . , sk be pairwise distinct, with 0 ≤ s1, . . . , sk < n ≤ j , and R ∈ Dn . Then
Q j (cs1 · · ·csk R2) =
n∑
s=1
∂s (cs1 · · ·csk )R2.
Definition 13. The integral on the r -th direction Ir : c0 A2r−1Dn → Doddn , for 0 < r ≤ n, is the
morphism given by:
Ir (c0cs1 · · ·csk A2r−1R2) =

cs1 · · ·csk R2, k odd, r = n,
cs1 · · ·csk cr R2, r 6= s1, . . . , sk ,n,
0, otherwise.
where s1, . . . , sk are pairwise distinct, 0 ≤ s1, . . . , sk < n, 0 ≤ k, and R ∈ Dn .
C. R. Mathématique, 2020, 358, n 4, 505-510
Nguyễn H. V. Hu,ng 509
Lemma 14. Let s1, . . . , sk be pairwise distinct, with 0 ≤ s1, . . . , sk < n, 0 < s ≤ n, and R ∈ Dn . Then
(i) Is∂s (cs1 · · ·csk R2) =
{
cs1 · · ·csk R2, either k odd, s = n, or s ∈ {s1, . . . , sk },
0, otherwise.
(ii) ∂s Is (c0cs1 · · ·csk A2s−1R2) =
{
c0cs1 · · ·csk A2s−1R2, either k odd, s = n, or s 6= s1, . . . , sk ,n,
0, otherwise.
Let hc20 D
2
n and hD
2
n be the submodules of Dn generated by the generators {hs1,...,sk | 0 < s1 <
·· · < sk < n,1 < k} over c20 D2n and D2n = F2[c21 , . . . ,c2n−1] respectively. Let h0D2n be the submodule of
Dn generated by {h0,s2,...,sk | 0 = s1 < s2 < ·· · < sk < n,1 < k} over D2n .
Theorem 15. For 3 ≤ n ≤ j ,
KerQ j ∩Doddn =
(
ImQ j ∩Doddn
)+hD2n ,
where ImQ j ∩Doddn = h0D2n ⊕hc20 D2n , and h0D2n ∩hD2n = {0}.
The exponent of c0 in each Dickson monomial of h0D2n is odd, whereas the exponent of
c0 in every Dickson monomial of hc20 D
2
n or of hD
2
n is even. It yields h0D
2
n ∩ hc20 D2n = {0} and
h0D2n ∩hD2n = {0}.
The smallest natural number n such that there exists a sequence 0 < s1 < ·· · < sk < n with k > 1
is n = 3.
Remark 16. The sum in Theorem 15 is not a direct sum. This is a consequence of the fact that
the critical elements are not linear independent over D2n .
Let S = (s1, . . . , sk ) be a sequence with 0 < s1 < ·· · < sk < n and k > 2. It is remarkable that
HS = khs1,...,sk A2n−1 +
k∑
i=1
hs1,...,ŝi ,...,sk A
2
si−1 = 0.
Let π : D2n → F2[c21 , . . . ,c2n−1] be the projection, whose kernel is c20 D2n . We denote π(Z 2) by Z 2
for abbreviation. So Z 2 +Z 2 ∈ c20 D2n for Z 2 ∈ D2n . The equality HS = 0 implies
HS +H S = khs1,...,sk (A2n−1 + A2n−1)+
k∑
i=1
hs1,...,ŝi ,...,sk (A
2
si−1 + A2si−1)
= khs1,...,sk A2n−1 +
k∑
i=1
hs1,...,ŝi ,...,sk A
2
si−1 = H S .
The left hand side belongs to hc20 D
2
n ⊂
(
ImQ j ∩ Doddn
)
(it is in Doddn as k − 1 > 1), while the
right hand side is in hD2n with at most one “coefficient” A
2
si−1 being zero. (The zero-coefficient
occurs when s1 = 1, since A2si−1 6= 0 for si > 1 by Lemma 10.) Therefore,
(
ImQ j ∩Doddn
)∩hD2n =
hc20 D
2
n ∩hD2n 6= {0}.
The following main result of the article is a consequence of the preceding one and the
equalities: Q j (c0) = c20 A2n−1, Q j (c0cs ) = c20 A2s−1 (0 < s < n).
Theorem 17. For 3 ≤ n ≤ j ,
H∗(Dn ;Q j ) =
D2n
(c20 A
2
0, . . . ,c
2
0 A
2
n−1)
⊕ hD
2
n
hc20 D
2
n ∩hD2n
.
Example 18. For j = n ≥ 3, we have As = cs for 0 ≤ s < n. So the critical element, which also
depends on n and j , is explicitly given by
hs1,...,sk = (k +1)cs1 · · ·csk c2n−1 +
k∑
i=1
(cs1 . . . ĉsi . . .csk )c
2
si−1,
C. R. Mathématique, 2020, 358, n 4, 505-510
510 Nguyễn H. V. Hu,ng
for 0 < s1 < ·· · < sk < n, 1 < k. Theorem 17 yields
H∗(Dn ;Qn) =
D2n
(c40 ,c
2
0 c
2
1 , . . . ,c
2
0 c
2
n−1)
⊕ hD
2
n
hc20 D
2
n ∩hD2n
=Λ(c20 )⊕F2[c21 , . . . ,c2n−1]⊕
hD2n
hc20 D
2
n ∩hD2n
.
For k > 2, by Remark 16,
hs2,...,sk c
2
0 = kh1,s2,...,sk c2n−1 +
k∑
i=2
h1,s2,...,ŝi ...,sk c
2
si−1
is a nonzero element in hc20 D
2
n ∩hD2n for 1 = s1 < ·· · < sk < n, while
khs1,s2,...,sk c
2
n−1 +
k∑
i=1
hs1,...,ŝi ...,sk c
2
si−1 = 0
is a linear relationship of the critical elements over D2n for 1 < s1 < ·· · < sk < n.
The contains of this note will be published in detail elsewhere.
Acknowledgment
This research was carried out when the author visited the Vietnam Institute for Advanced Study
in Mathematics (VIASM), Hanoi, in the academic year 2019-2020. He would like to express his
warmest thank to the VIASM for the hospitality and for the wonderful working condition.
The author hearty thanks Dev Sinha for an interesting discussion on the subject during the
Vietnam-US Mathematical joint Metting, Quynhon June 10-13, 2019. He is deeply indebted to
H. Miller, D. Ravenel, L. Schwartz, and S. Wilson for generous discussions on the goal of the
problem and the references of this article.
References
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problem”, Trans. Am. Math. Soc. 12 (1911), p. 75-98.
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113 (1991), no. 4, p. 1097-1104.
[3] N. H. V. Hưng, F. P. Peterson, “Spherical classes and the Dickson algebra”, Proc. Camb. Philos. Soc. 124 (1998), no. 2,
p. 253-264.
[4] J. W. Milnor, “The Steenrod algebra and its dual”, Ann. Math. 67 (1958), p. 150-171.
[5] H. Mùi, “Modular invariant theory and the cohomology algebras of symmetric group”, J. Fac. Sci., Univ. Tokyo, Sect.
I A 22 (1975), p. 319-369.
[6] D. P. Sinha, “Cohomology of symmetric groups”, 2019, Lecture on the Vietnam-US Mathematical joint Metting,
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[7] L. Smith, R. M. Switzer, “Realizability and nonrealizability of Dickson algebras as cohomology rings”, Proc. Am. Math.
Soc. 89 (1983), p. 303-313.
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C. R. Mathématique, 2020, 358, n 4, 505-510


The mod 2 Margolis homology of the Dickson algebra

Alexandros Margaris

James C. Robinson

Comptes Rendus
Mathématique
Alexandros Margaris and James C. Robinson
Some comments on Laakso graphs and sets of differences
Volume 358, issue 4 (2020), p. 515-521.
<https://doi.org/10.5802/crmath.70>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 4, p. 515-521
https://doi.org/10.5802/crmath.70
Mathematical Analysis, Topology / Analyse mathématique, Topologie
Some comments on Laakso graphs and sets of
differences
Alexandros Margarisa and James C. Robinson∗, a
a Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.
E-mail: j.c.robinson@warwick.ac.uk.
Abstract. We recall a variation of a construction due to Laakso [3], also used by Lang and Plaut [3] of a
doubling metric space X that cannot be embedded into any Hilbert space. We give a more concrete version
of this construction and motivated by the results of Olson & Robinson [6], we consider the Kuratowski
embeddingΦ(X ) of X into L∞(X ) and prove thatΦ(X )−Φ(X ) is not doubling.
Manuscript received 23rd July 2019, revised 17th April 2020, accepted 8th May 2020.
1. Introduction
We say that a metric space (X ,d) is doubling, with doubling constant K , if for every given x ∈ X
and r > 0, there exist y1, . . . , yK in X such that
B(x,r ) ⊂
K⋃
i=1
B(yi ,r /2).
We say that a metric space (X ,d) embeds into a normed space (Y ,‖ ·‖) in a bi-Lipschitz way if
there exists f : X → Y and some constant L > 0, such that for all x, y ∈ X
d(x, y)
1
L
≤ ‖ f (x)− f (y)‖ ≤ L d(x, y).
We know that when a metric space embeds into an Euclidean space in a bi-Lipschitz way then it
must be doubling, but there are examples due to Laakso [3], Lang & Plaut [4] and Semmes [9] that
show that this condition is not sufficient.
In 1983, Assouad [1] proved that any doubling metric space X can be embedded into an
Euclidean space in a bi–Hölder way, i.e. for any 0 <α< 1, there exists some k ∈N and aφ : X →Rk
such that
1
L
d(x, y)α ≤ |φ(x)−φ(y)| ≤ Ld(x, y)α,
for all x, y ∈ X .
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
516 Alexandros Margaris and James C. Robinson
We also want to recall the notion of a homogeneous metric space. A subset A of a metric space
(X ,d) is said to be (M , s)-homogeneous if for every x ∈ A and r > ρ > 0
N (A∩B(x,r ),ρ) ≤ M
(
r
ρ
)s
,
where N (A ∩ B(x,r ),ρ) denotes the minimum number of balls of radius ρ required to cover
A∩B(x,r ). It can be easily shown that a metric space is homogeneous if and only if it is doubling
(see Chapter 9 in the book of Robinson [7]). In 2010, Olson and Robinson [6] used Assouad’s
construction [1] and proved that if X is homogeneous then it can be also embedded in an almost
bi-Lipschitz way into a Hilbert space H , i.e. for any γ> 12 , there exists a map f : (X ,d) → H such
that for some positive constant L
1
L
d(x, y)
|logd(x, y)|γ ≤ ‖ f (x)− f (y)‖ ≤ L d(x, y).
Moreover, in 2014, Robinson [8] generalised the above result and and proved the following
embedding theorem for subsets of Banach spaces.
Theorem 1. Suppose X is a compact subset of a real Banach space B such that the set X − X =
{x − y : x, y ∈ X } is homogeneous. Then for any γ> 1, there exists a natural number N and a dense
set of linear maps L : B → RN , that are injective on X and γ-almost bi-Lipschitz, i.e. for some
constant cL
1
cL
‖x − y‖
|log‖x − y‖|γ ≤ |L(x)−L(y)| ≤ cL ‖x − y‖,
for all x, y ∈ X .
The above theorem can be used to provide embeddings of subsets of compact metric spaces,
using the isometric embedding Φ : (X ,d) → L∞(X ), given by x 7→ d(x, · ), due to Kuratowski (see
the notes from Heinonen [2] for a detailed proof). In particular, we can define “X − X ” in this
context to mean
X −X ≡Φ(X )−Φ(X ) = { f ∈ L∞(X ) : f = d(x, · )−d(y, · ), for x, y in X }.
It is an open problem whether we can prove there exist almost bi-Lipschitz embeddings into
Euclidean spaces under a weaker condition than the one in Theorem 1. It is also a question
whether we can prove “better” embeddings than almost bi-Lipschitz when X − X is homoge-
neous. There are known examples of homogeneous subsets of Banach spaces for which the set
of differences is not homogeneous (see for example Chapter 9 in the book of Robinson [7]) but
there is no information on the embedding properties of these sets.
In this paper, we consider a variation of the construction due to Laakso [3], which was used
by Lang & Plaut [4] to construct a doubling metric space X that cannot be embedded in a bi-
Lipschitz way into any Hilbert space. We prove that Φ(X ) −Φ(X ) is not doubling as a subset
of L∞(X ), thus giving motivation towards the direction of studying the set of differences more
closely. In this way, we also rule out the possibility of using Theorem 1 to prove the existence of
almost bi-Lipschitz maps into a Euclidean space for this set.
2. The Laakso graphs
We first recall the definition of the Gromov–Hausdorff distance of compact metric spaces.
Definition 2. Suppose X ,Y are compact metric spaces. Then
dG H (X ,Y ) = infmax{dist( f (X ), g (Y )),dist(g (Y ), f (X ))},
where the infimum is taken over all metric spaces M and all possible isometric embeddings
f : X → M and g : Y → M.
C. R. Mathématique, 2020, 358, n 4, 515-521
Alexandros Margaris and James C. Robinson 517
It is easy to check that dG H (X ,Y ) = 0 if and only if X is isometric to Y (see the lecture notes
from Heinonen [2] for a proof), proving that the set of all isometry classes of compact metric
spaces equipped with dG H forms a metric space, which is compact (see Heinonen [2] again).
We now recall the construction due to Lang and Plaut [4] of a metric space that is homoge-
neous but does not embed in a bi-Lipschitz way into any Hilbert space. Here we use a construc-
tion that is somewhat more concrete than that of Lang and Plaut [4]. We define the limiting metric
space explicitly and then prove that it coincides with the one defined by Lang and Plaut.
Let X0 be a single edge of length 1. To construct Xi+1 from Xi , we take six copies of Xi and
rescale them by the factor of 14 as in the following Figure.
Figure 1. The first stages of the construction. At each step i the dotted subset is isomorphic
to Xi−1.
We define a metric %i (x, y) on each of the Xi to be the geodesic distance, i.e. the shortest path
that we need to travel on the graph to get from x to y . We note that each Xi has diameter 1, has
two endpoints, and comprises of 6i edges of length 4−i each. Every Xi , for i > 0 also includes
6i−1 “squares”, which we call “edge cycles” for the rest of the paper. For any j > i , we construct an
isometric embedding of Xi into X j , by identifying vertices in Xi with vertices in X j and endpoints
with endpoints. The image of Xi into X j is represented with the dotted lines in the above figure.
It is also easy to see that dG H (Xi , X j ) <
( 1
4
)i
, and so {(Xi ,%i )}∞i=1 forms a Cauchy sequence in the
Gromov–Hausdorff metric and it follows that it converges to a limiting metric space X , which is
used by Lang and Plaut in their argument. We now construct this space X explicitly using the
following procedure
C. R. Mathématique, 2020, 358, n 4, 515-521
518 Alexandros Margaris and James C. Robinson
Let (Xi ,%i ) be as above for any j > i let hi→ j denote an isometric embedding of Xi into X j .
Then, we take X ∗ =∪∞i=1Xi and define the following function on X ∗×X ∗, by setting
%∗(x, y) =
{
%i (x, y) if x, y ∈ Xi .
% j (hi→ j (x), y) if x ∈ Xi , y ∈ X j and i < j .
We note that the above function is a pseudometric on X ∗ as long as we choose the embeddings
hi→ j such that
hi→ j ◦h j→k = hi→k , ∀ i < j < k. (1)
We now define a new metric space X , by identifying points in Xi with their respective images in
all X j for j > i . For all x, y ∈ X ∗, we define the following equivalence relation
x ∼ y ⇐⇒ %∗(x, y) = 0,
and we set X =∪∞i=1[Xi ]. Then, for any [x], [y] ∈ X , we define
%([x], [y]) = %∗(x, y).
This definition of X does not depend on the embedding hi→i+1 we choose at each step, as long
as (1) is satisfied. If we consider another embedding we end up with an isometric metric space.
Using the above construction, it is easy to check that
dG H (X , Xi ) → 0.
Indeed, let π : Xi → X be such that for any x ∈ Xi ,
π(x) = [x].
It is immediate that π is an isometry from Xi onto [Xi ]. Therefore,
dG H (X , Xi ) = dG H (X , [Xi ]) ≤ dX (X , [Xi ]).
Let x ∈ X \ [Xi ]. Then, there exists k > i such that x ∈ [Xk ]. Then,
dX (x, [Xi ]) = dXk (x,hi→k (Xi )) ≤
(
1
4
)i+1
i→∞−−−→ 0,
which proves that X coincides with the metric space defined by Lang and Plaut. For the rest of
the argument when we mention a point x ∈ Xi we refer to the class [x] with respect to the above
equivalence relation.
Lang & Plaut [4] showed that X is doubling with doubling constant 6. Using the above
construction, their proof becomes somewhat more transparent. For a proof see [5].
Now, we recall the Kuratowski embedding
Φ : X → L∞(X ) (2)
x 7→ %(x, · ).
and we define
X −X = {%(x, · )−%(y, · ) : x, y ∈ X }.
We now prove that X −X is not doubling.
Theorem 3. If X is the metric space defined above and Φ : X → L∞(X ) is the Kuratowski embed-
ding defined in (2) then,Φ(X )−Φ(X ) is not doubling.
Proof. We assume thatΦ(X )−Φ(X ) is doubling.
Let r = ( 14 )i , for some i ∈ N and take the ball BX−X (0,2r ). Suppose that there exist M and
g j ∈ X −X that satisfy
BX−X (0,2r ) ⊂∪Mj=1BX−X (g j ,r ).
C. R. Mathématique, 2020, 358, n 4, 515-521
Alexandros Margaris and James C. Robinson 519
Now, let f ∈ BX−X (0,2r ). Then, there exist x, y ∈ X such that
f (z) = %(x, z)−%(y, z), ∀ z ∈ X .
We can easily check that ‖ f ‖∞ = %(x, y) < 2r. Similarly, let t j , s j ∈ X such that
g j (z) = %(t j , z)−%(s j , z), ∀ z ∈ X .
Any time we choose x, y ∈ X such that %(x, y) < 2r , we obtain an element of BX−X (0,2r ). Let
[Xk ] ⊂ X such that
t j , s j ∈ [Xk ],
for all j ≤ M . We now have two cases:
If k ≤ i , we show that for any edge cycle in Xi , there exist copies of some t j or s j that belong to
this edge cycle. Suppose that there exist a cycle in Xi that does not contain any images of t j , s j .
Then, we choose x, y ∈ Xi as in the Figure 2, where we zoom in at that specific cycle. Then, x, y ∈ X
satisfy
%(t j , x) > r, ∀ j ≤ M
%(s j , x) > r, ∀ j ≤ M
%(s j , y) > 0, ∀ j ≤ M
r < %(x, y) < 2r.
Figure 2. The edge cycle in Xi , which does not contain any t j , s j .
Since f ∈ BX−X (0,2r ), there exist j ≤ M such that
‖ f − g j ‖∞ < r ⇐⇒‖%(x, z)−%(y, z)−%(t j , z)+%(s j , z)‖∞ < r,
for some j ≤ M . Choosing z as in the above figure, depending on the position of t j , s j we have
‖%(x, z)−%(y, z)−%(t j , z)+%(s j , z)‖∞ = %(t j , x)+%(s j , y) > r
or
‖%(x, z)−%(y, z)−%(t j , z)+%(s j , z)‖∞ = %(x, y)+%(s j , t j ) > r,
a contradiction.
C. R. Mathématique, 2020, 358, n 4, 515-521
520 Alexandros Margaris and James C. Robinson
We conclude that any edge cycle in Xi contains one of the ti , si and since there are 6i−1 edge
cycles contained in Xi , we deduce that
M ≥ 6i−1,
a contradiction.
If k > i , we consider the endpoints vi j ,ui j that enclose an edge cycle in Xi . We rescale the
cycle by the appropriate factor to create an edge cycle in Xk , with the same endpoints in Xk
(with respect to the equivalence relation we have). Since distances are preserved, we only need
to repeat the above argument for all these cycles in Xk (see also Figure 3). 
Figure 3. The case k > i .
3. Conclusion
The above result gives us an indication that we might expect better embedding properties, if we
impose some condition on the set of differences. Moreover, following the results due to Olson and
Robinson [6], which are mentioned in the introduction, we arrive to the following open problems
(1) If X is as in Theorem 3, is there an almost bi-Lipschitz embedding f : X →Rk ?
(2) Does every doubling metric space admit an almost bi-Lipschitz embedding into an
Euclidean space? (A positive answer would yield a positive answer to the first question.)
(3) If the set of differences X − X is homogeneous, can we obtain “better” embedding
properties than almost bi-Lipschitz?
References
[1] P. Assouad, “Plongements Lipschitziens dans Rn ”, Bull. Soc. Math. Fr. 111 (1983), p. 429-448.
[2] J. Heinonen, Geometric Embeddings of Metric Spaces, Report. University of Jyväskylä. Department of Mathematics and
Statistics, vol. 90, University of Jyväskylä, 2003.
[3] T. J. Laakso, “Plane with A∞-weighted metric not bilipschitz embeddable to Rn ”, Bull. Lond. Math. Soc. 34 (2002),
no. 6, p. 667-676.
[4] U. Lang, C. Plaut, “Bilipschitz embeddings of metric spaces into space forms”, Geom. Dedicata 87 (2001), no. 1-3,
p. 285-307.
C. R. Mathématique, 2020, 358, n 4, 515-521
Alexandros Margaris and James C. Robinson 521
[5] A. Margaris, “Dimensions, Embeddings and Iterated Function Systems”, PhD Thesis, University of Warwick (UK),
2019.
[6] E. J. Olson, J. C. Robinson, “Almost bi-Lipschitz embeddings and almost homogeneous sets”, Trans. Am. Math. Soc.
362 (2010), no. 1, p. 145-168.
[7] J. C. Robinson, Dimensions, Embeddings and Attractors, Cambridge Tracts in Mathematics, vol. 186, Cambridge
University Press, 2011.
[8] ——— , “Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing products of
balls”, Proc. Am. Math. Soc. 142 (2014), no. 4, p. 1275-1288.
[9] S. Semmes, “On the nonexistence of bilipschitz parameterizations and geometric problems about A∞-weights”, Rev.
Mat. Iberoam. 12 (1996), no. 2, p. 337-410.
C. R. Mathématique, 2020, 358, n 4, 515-521


Some comments on Laakso graphs and sets of differences

David Rottensteiner

Michael Ruzhansky

Comptes Rendus
Mathématique
David Rottensteiner and Michael Ruzhansky
The Harmonic Oscillator on the Heisenberg Group
Volume 358, issue 5 (2020), p. 609-614.
<https://doi.org/10.5802/crmath.78>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 5, p. 609-614
https://doi.org/10.5802/crmath.78
Analysis on non commutative Lie groups / Analyse sur les groupes de Lie non
commutatifs
The Harmonic Oscillator on the Heisenberg
Group
L’oscillateur harmonique sur le groupe de Heisenberg
David Rottensteiner∗, a andMichael Ruzhanskya, b
a Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent
University, Krijgslaan 281, S8, 9000 Gent, Belgium
b School of Mathematical Sciences, Queen Mary University of London, Mile End
Road, London E1 4NS, United Kingdom.
E-mails: david.rottensteiner@ugent.be, michael.ruzhansky@ugent.be.
Abstract. In this note we present a notion of harmonic oscillator on the Heisenberg group Hn which forms
the natural analogue of the harmonic oscillator on Rn under a few reasonable assumptions: the harmonic
oscillator on Hn should be a negative sum of squares of operators related to the sub-Laplacian on Hn ,
essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions
and form an orthonormal basis of L2(Hn ). This approach leads to a differential operator on Hn which is
determined by the (stratified) Dynin–Folland Lie algebra. We provide an explicit expression for the operator
as well as an asymptotic estimate for its eigenvalues.
Résumé. Dans cette note, nous présentons une notion d’oscillateur harmonique sur le groupe de Heisenberg
Hn qui forme l’analogue naturel de l’oscillateur harmonique sur Rn sous quelques hypothèses raisonnables
: l’oscillateur harmonique sur Hn devraît être une somme négative de carrés d’opérateurs liée au sous-
laplacien sur Hn , être essentiellement auto-adjoint avec un spectre purement discret, et les vecteurs propres
doivent former une base orthonormée de L2(Hn ). Cette approche conduit à un opérateur différentiel sur Hn
qui est déterminé par l’algèbre de Dynin–Folland de Lie (stratifiée). Nous fournissons une expression explicite
pour l’opérateur ainsi qu’une estimation asymptotique pour ses valeurs propres.
2020 Mathematics Subject Classification. 35R03, 35P20.
Funding. David Rottensteiner was supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial
Differential Equations and the Austrian Science Fund (FWF) project [I 3403].
Michael Ruzhansky was supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differen-
tial Equations, by the EPSRC grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151.
Manuscript received 21st February 2020, revised and accepted 25th May 2020.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
610 David Rottensteiner and Michael Ruzhansky
1. Introduction
The aim of this note is to introduce a canonical harmonic oscillator on the Heisenberg group Hn .
For n = 1 and exponential coordinates (t1, t2, t3) ∈ R3 ∼= H1 the harmonic oscillator we propose is
explicitly given by
QH1 =−
(
∂2t1 +∂2t2
)− 1
4
(
t1
2 + t22
)
∂2t3 −
(
t1∂t2 − t2∂t1
)
∂t3 +4π2t 23 .
Our approach is motivated by the following three realisations of the classical harmonic oscillator
QRn on Rn :
(R1) the negative sum of squares −∆+4π2 |t |2 of partial derivatives of order 1 and coordinate
multiplication operators;
(R2) the Weyl and Kohn–Nirenberg quantizations on Rn of the symbol σ(t ,ξ) := 4π2(|ξ|2+|t |2)
with t ,ξ ∈Rn ;
(R3) the image dρ1(−LHn ) of the negative sub-Laplacian −LHn on Hn under the infinitesimal
Schrödinger representation dρ1 (of Planck’s constant equal to 1) of the Heisenberg Lie
algebra hn 1, for n = 1.
The operator QRn is usually defined by the expression −∆+ 4π2 |t |2, or some scaled version of
it.2 However, the Schrödinger representation ρ1 of Hn acting on L2(Rn) and the associated Lie
algebra representation, naturally acting on S (Rn), clearly relate each of the realisations (R1)–(R3)
to the others. Moreover, each of these realisations features a sum of squares. It ought therefore to
be natural to assume that similar realisations should be available for the canonical harmonic
oscillator on Hn .
The special role of the Heisenberg Lie algebra hn in this context is not coincidental: it is
precisely the Lie algebra which is generated by the partial derivatives ∂t j and the multiplication
operators for the coordinate functions tk , j ,k = 1, . . . ,n. It is well known that hn is stratified,
therefore permits a (canonical) homogeneous structure, and that the sums of squares in the
identities above are essentially related to the first stratum of hn .
An operator on Hn satisfying criteria analogous to (R1)–(R3) should clearly involve left-
invariant (or alternatively right-invariant) vector fields on Hn , which are uniquely determined
by some vectors in hn , and a scalar potential expressed in terms of the coordinate functions on
Hn . It ought therefore to be natural to study the Lie algebra generated by the standard basis of
left-invariant vector fields, here denoted by X1, . . . , X2n+1, and the multiplication operators de-
fined by the coordinates t1, . . . , t2n+1 on hn ∼= R2n+1 which determine the coordinates in which
the vector fields are written. The resulting Lie algebra and its representation theory were first
studied in Dynin [1], and in more detail in Folland [5]. This Lie algebra, which we shall call the
Dynin–Folland Lie algebra, is in fact stratified and thus admits a sub-Laplacian. Endowed with
the canonical homogeneous structure arising from the stratification, the Dynin–Folland Lie al-
gebra together with its associated connected, simply connected Lie group, the group’s generic
irreducible unitary representations, and the associated negative sub-Laplacian (a positive Rock-
land operator) give rise to the harmonic oscillator on the Heisenberg group. We provide a concrete
formula for this operator and describe the asymptotic growth of its eigenvalues, using results by
Mohamed, Lévy-Bruhl and Nourrigat [9]
1extended to the universal enveloping algebra u(hn ).
2The factor 4π2 is due to our choice of realising the Schrödinger representation; our expression agrees with the
versions in Folland [4] or Stein [12] up to scaling.
C. R. Mathématique, 2020, 358, n 5, 609-614
David Rottensteiner and Michael Ruzhansky 611
2. The Dynin–Folland Group and its Representations
2.1. The Dynin–Folland Lie Algebra
In order to present our results, in this and the next sections we fix the notation and recall the
fundamental results about the Dynin–Folland Lie algebra and group and its generic unitary
irreducible representations due to Dynin [1] and Folland [5]. For more details we refer to [3,5,10].
We choose the usual exponential coordinates for the Heisenberg group Hn , thus express the
group law by
(t2n+1, t2n , . . . , t1)(t ′2n+1, t
′
2n , . . . , t
′
1) =
(
t2n+1 + t ′2n+1 +
1
2
n∑
j=1
(t j t
′
n+ j − t ′j tn+ j ), t2n + t ′2n , . . . , t1 + t ′1
)
.
We can also group the variables as t̃ 3 := t2n+1, t̃ 2 := (t2n , . . . , tn+1), t̃ 1 := (tn , . . . , t1) and rewrite the
group law as
(t̃ 3, t̃ 2, t̃ 1)(t̃
′
3, t̃
′
2, t̃
′
1) =
(
t̃ 3 + t̃ ′3 +
1
2
(〈
t̃ 1, t̃
′
2
〉−〈t̃ 2, t̃ ′1〉), t̃ 2 + t̃ ′2, t̃ 1 + t̃ ′1) . (1)
In these coordinates, one can realise the Schrödinger representation ρκ of formal dimension |κ|n ,
κ ∈R\ {0}, acting on f ∈ L2(Rn), as(
ρκ(t̃ 3, t̃ 2, t̃ 1) f
)
(t̃ ′1) = e2πiκ
(
t̃ 3+ 12 〈t̃ 1,t̃ 2〉+〈t̃ 2,t̃ ′1〉
)
f (t̃ ′1 + t̃ 1).
The real Lie algebra g̃ of operators generated by the left-invariant vector fields on Hn
X j = ∂t j −
1
2
tn+ j∂t2n+1 , Xn+ j = ∂tn+ j +
1
2
t j∂t2n+1 , X2n+1 = ∂t2n+1 , j = 1, . . . ,n,
and by the multiplication operators Yk = 2πi tk for k = 1, . . . ,2n +1 is 3-step nilpotent and, as a
vector space, isomorphic to R×R2n+1 ×hn ∼= R4n+3. If we denote by Z the multiplication by the
constant 2πi and identify the operators Z ,Y1, . . . , X2n+1 with the standard basis vectors in R4n+3,
the isomorphism is realised by equipping R4n+3 with the Lie bracket defined by
[X j , Xn+ j ] = X2n+1, [X j ,Y2n+1] =−1
2
Yn+ j , [Xn+ j ,Y2n+1] = 1
2
Y j , [Xk ,Yk ] = Z , (2)
for j = 1, . . . ,n and k = 1, . . . ,2n + 1, and vanishing brackets otherwise. We will denote this
Lie algebra by hn,2 and refer to it as the Dynin–Folland Lie algebra. The connected, simply
connected Lie group obtained by exponentiation will be referred to as the Dynin–Folland group
and denoted by Hn,2. The Lie bracket relations immediately reveal that the Lie sub-algebra
generated by Z ,Y1, . . . ,Y2n+1 is Abelian, and hence Hn,2 can be viewed as a semi-direct product of
the form R2n+2 oHn . Using exponential coordinates and identifying any element of Hn,2 with its
corresponding coordinate vector
(
z, y1, . . . , y2n+1, x2n+1, . . . , x1
) =: (z, y, x) ∈ R×R2n+1 ×R2n+1, the
Hn,2-group law can be expressed by
(z, y, x)(z ′, y ′, x ′) =
(
z + z ′+ 1
2
(〈
x, y ′
〉−〈y, x ′〉)− 1
8
〈y − y ′, [x, x ′]〉,
y + y ′+ 1
4
(
ad∗(x)y ′−ad∗(x ′)y), x · x ′) (3)
provided we denote by
〈
x, y ′
〉
the inner product
∑2n+1
j=1 x j y
′
j , by [x, x
′] the hn-Lie bracket of the
vectors
∑2n+1
j=1 x j X j and
∑2n+1
j=1 x
′
j X j in hn , by x · x ′ the x ′′-coordinates of (0,0, x ′′) = (0,0, x) ·
(0,0, x ′) ∈ H ≤ Hn,2, and by ad∗ the coadjoint representation of hn ∼=R2n+1 on h∗n ∼=R2n+1 given by
ad∗(t̃ 3, t̃ 2, t̃ 1)(t̃ ′3, t̃
′
2, t̃
′
1) = (0,−t̃ 3 t̃ ′1, t̃ 3 t̃ ′2).
C. R. Mathématique, 2020, 358, n 5, 609-614
612 David Rottensteiner and Michael Ruzhansky
2.2. Stratification and Unitary Irreducible Representations
The Lie bracket (2) admits the stratification
g3 :=RZ , g2 :=R-span{Y1, . . . ,Y2n , X2n+1}, g1 :=R-span{Y2n+1, X2n , . . . , X1},
which possesses a canonical family of homogeneous dilations {Dr }r>0 on hn,2 given by
Dr (Z ) = r 3Z ,
Dr (Yk ) = r 2Yk , Dr (X2n+1) = r 2X2n+1,
Dr (Xk ) = r Xk , Dr (Y2n+1) = r Y2n+1,
(4)
for k = 1, . . . ,2n. The sub-Laplacian LHn,2 induced by the above stratification is the left-invariant
differential operator on Hn,2 corresponding to the sum of squares
X 21 +·· ·+X 22n +Y 22n+1 ∈ u(hn,2).
Moreover, for l := κZ∗ ∈ z(g)∗ with κ ∈ R \ {0} the matrix representation of the corresponding
symplectic form Bl = l ([ · , · ]) immediately reveals that Bl is non-degenerate on hn,2/z(hn,2) ×
hn,2/z(hn,2), i.e., up to Plancherel measure zero all unitary irreducible representations are square-
integrable modulo the centre Z (Hn,2), and its Pfaffian (characterizing the Plancherel measure) is
given by Pf(l ) = |κ|2n+1. These generic representations of Hn,2, denoted by πκ,κ ∈ R \ {0}, can be
induced by the characters χκ := e2πi 〈κZ∗, · 〉 of the normal Abelian subgroup R2n+2 ≤ Hn,2: for a
fixed κ ∈R\ {0} the action of πκ on the representation space Hκ ∼= L2(Hn) is explicitly given by(
πκ(z, y, x) f
)
(t ) = e2πiκz e2πiκ
〈
t · 12 x,y
〉
f (t · x) (5)
for f ∈ L2(Hn), where t · 12 x and t · x again denote the Hn-group products of the corresponding
coordinate vectors.
3. The Harmonic Oscillator on Hn
The representation π :=π1 for κ= 1 defined in (5) was the object of interest in Dynin’s account [1]
since it served the purpose of introducing a Weyl quantization on Hn . For our definition of the
harmonic oscillator QHn on Hn this representation plays the same crucial role as the Schrödinger
representation does for QRn . For this reason the analogue of (R3) yields a canonical definition of
QHn . The analogues of (R1) and (R2) will be an immediate consequence of our choice.
Definition 1. For the basis {Z ,Y1, . . . , X2n+1} of the Dynin–Folland Lie alebra hn,2 and the represen-
tation π=π1 ∈ Ĥn,2 realised on the representation space L2(Hn), we define the harmonic oscillator
on Hn to be the positive essentially self-adjoint operator
QHn :=dπ
(−LHn,2)=−dπ(X1)2 −·· ·−dπ(X2n)2 −dπ(Y2n+1)2,
whose natural domain includes the space of smooth vectors H ∞π ∼=S (Hn).
The essentially self-adjoint differential operator QHn on Hn admits the following three reali-
sations:
(R1′) the differential operator −LHn +4π2t 22n+1;
(R2′) the Dynin–Weyl quantization on Hn of the symbolσ(t ,ξ) := 4π2(ξ21+·· ·+ξ22n +t 22n+1) with
t ,ξ ∈R2n+1;
(R3′) the image dπ(−LHn,2 ) of the negative sub-Laplacian −LHn,2 under the infinitesimal
representation dπ of the Dynin–Folland Lie algebra hn,2.
C. R. Mathématique, 2020, 358, n 5, 609-614
David Rottensteiner and Michael Ruzhansky 613
Since the Lie algebra isomorphism hn,2 → g̃ defined by (2) is precisly dπ, we immediately have
QHn =−LHn +4π2t 22n+1
=−
(
∂t1 −
1
2
tn+1∂t2n+1
)2
−·· ·−
(
∂t2n +
1
2
tn∂t2n+1
)2
+4π2t 22n+1,
(6)
thus (R1′). As for Dynin’s Weyl quanization ofσ(t ,ξ) := 4π2(ξ21+·· ·+ξ22n+t 22n+1), it suffices to recall
that every monomial (2πi ξ j )m , j = 1, . . . ,2n+1, m ∈N, is mapped to the left-invariant differential
operator X mj , and that multiplication by the monomial (2πi tk )
m , k = 1, . . . ,2n +1, is mapped to
the multiplication operator Y mk = (2πi tk )m . For more details we refer to [1] and [10, § 5].
Note that equivalently we could realiseπ as a direct summand of the left regular representation
of Hn,2, thereby replacing the left-invariant sub-Laplacian LHn in (6) by the right-invariant one.
The spectral asymptotics, however, would not change (cf. [9]).
4. Spectral Properties
The harmonic oscillator QHn has purely discrete spectrum in (0,∞) and we obtain the asymptotic
growth rate of its eigenvalues by employing a powerful method, which in the setting of stratified
groups was developed in Mohamed, Lévy-Bruhl and Nourrigat [9] (see also [8] for a short version
in English), and which was extended to the setting of graded groups in ter Elst and Robinson [2]. If
G is nilpotent but non-Abelian, then for almost every unitary irreducible representationπ ∈ Ĝ the
representation space Hπ is infinite-dimensional. Moreover, if G is graded, then a homogeneous
left-invariant differential operator R on G is said to be a Rockland operator if for every non-trivial
π ∈ Ĝ the operator dπ(R) is injective on the space of smooth vectors H ∞π ⊂Hπ. This is the case
if and only if the operator R is hypoelliptic. The equivalence was conjectured by Rockland and
settled in Helffer-Nourrigat [6]. Hulanicki, Jenkins and Ludwig [7] showed that if R is positive,
then for every π ∈ Ĝ the operator dπ(R) has purely discrete spectrum in (0,∞). In particular,
every negative sub-Laplacian −L on a stratified group is a positive Rockland operator. A concrete
description of the spectrum of −dπ(L ) is due to [9], and for general dπ(R) due to [2]: in both
cases the authors showed that the number of eigenvalues of a given −dπ(L ) or dπ(R), counted
with multiplicities, asymptotically grows like the volumes of certain subsets of the corresponding
coadjoint orbit Oπ. The subsets in question are determined (up to a multiplicative constant) by
a (any) homogeneous quasi-norm on g∗. These estimates also give an asymptotic value for the
magnitude of a given eigenvalue.
In the case of G = Hn,2 and R =−LHn,2 , the realisation of π= π1 ∈ Ĥn,2 in Hπ = L2(Hn) given
by (5) makes these results readily available for dπ(R) = QHn . The choice of a convenient quasi-
norm on h∗n,2 and the fact that the coadjoint orbit Oπ is flat facilitate the computation of the
volumes in question substantially. One can use this to show:
Theorem. The harmonic oscillator QHn on the Heisenberg group Hn has a purely discrete spec-
trum spec(QHn ) ⊂ (0,∞). The number of its eigenvalues, counted with multiplicities, which are
less or equal to λ> 0 is asymptotically given by
N (λ) ∼λ 6n+32 ,
and the magnitude of the eigenvalues is asymptotically equal to
λk ∼ k
2
6n+3 for k = 1,2, . . . .
Moreover, the eigenvectors of QHn are in S (Hn) and form an orthonormal basis of L
2(Hn).
The power 6n+32 bears a specific relation to the canonical homogeneous structure of hn : the
nominator 6n+3 is the homogeneous dimension of the first two strata of hn,2, i.e., of the subspace
C. R. Mathématique, 2020, 358, n 5, 609-614
614 David Rottensteiner and Michael Ruzhansky
g1 ⊕g2 ⊆ hn,2, while the denominator 2 is the homogeneous degree of −LHn,2 . For a proof of the
spectral asymptotics we refer to the preprint [11], especially the proof of Proposition 6.3. The
eigenvectors are clearly elements of
⋂∞
l=1 dom(QHn
l
) ⊆ L2(Hn); since ⋂∞l=1 dom(dπ(R)l ) = H ∞π
(see [2, Prop. 2.1]), this set coincides with S (Hn). The eigenvectors form an orthonormal basis of
L2(Hn) because QHn possesses a compact resolvent as was shown for general dπ(R) in [6].
References
[1] A. S. Dynin, “Pseudodifferential operators on the Heisenberg group”, Dokl. Akad. Nauk SSSR 225 (1975), p. 1245-
1248.
[2] A. F. M. ter Elst, D. W. Robinson, “Spectral estimates for positive Rockland operators”, in Algebraic groups and Lie
groups, Australian Mathematical Society Lecture Series, vol. 9, Cambridge University Press, 1997, p. 195-213.
[3] V. Fischer, D. Rottensteiner, M. Ruzhansky, “Heisenberg-Modulation Space on the Crossroads of Coorbit Theory and
Decomposition Space Theory”, https://arxiv.org/abs/1812.07876, 2018.
[4] G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989.
[5] ——— , “Meta-Heisenberg groups”, in Fourier analysis (Orono, ME, 1992), Lecture Notes in Pure and Applied
Mathematics, vol. 157, Marcel Dekker, 1994, p. 121-147.
[6] B. Helffer, J. Nourrigat, “Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un
groupe de Lie nilpotent gradué”, Commun. Partial Differ. Equations 4 (1979), no. 8, p. 899-958.
[7] A. Hulanicki, J. W. Jenkins, J. Ludwig, “Minimum eigenvalues for positive, Rockland operators”, Proc. Am. Math. Soc.
94 (1985), no. 4, p. 718-720.
[8] A. Morame, P. Lévy-Bruhl, J. Nourrigat, “Spectral theory and representations of nilpotent groups”, Bull. Am. Math.
Soc. 26 (1992), no. 2, p. 299-303.
[9] ——— , “Étude spectrale d’opérateurs liés à des représentations de groupes nilpotents”, J. Funct. Anal. 113 (1993),
no. 1, p. 65-93.
[10] D. Rottensteiner, “Time-Frequency Analysis on the Heisenberg Group”, PhD Thesis, Imperial College London, 2014,
https://spiral.imperial.ac.uk/handle/10044/1/24568, 154 pages.
[11] D. Rottensteiner, M. Ruzhansky, “Harmonic and anharmonic oscillators on the Heisenberg group”, https://arxiv.org/
abs/1812.09620, 2018.
[12] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathemat-
ical Series, vol. 43, Princeton University Press, 1993, with the assistance of Timothy S. Murphy, xiv+695 pages.
C. R. Mathématique, 2020, 358, n 5, 609-614


The Harmonic Oscillator on the Heisenberg Group

Pierre Del Moral

Sumeetpal S. Singh

Comptes Rendus
Mathématique
Pierre Del Moral and Sumeetpal S. Singh
A backward Itô–Ventzell formula with an application to stochastic
interpolation
Volume 358, issue 7 (2020), p. 881-886.
<https://doi.org/10.5802/crmath.110>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
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Comptes Rendus
Mathématique
2020, 358, n 7, p. 881-886
https://doi.org/10.5802/crmath.110
Probabilities / Probabilités
A backward Itô–Ventzell formula with an
application to stochastic interpolation
Pierre Del Moral∗, a and Sumeetpal S. Singhb
a INRIA Bordeaux Research Center, University of Bordeaux, Talance, France
b Department of Engineering, University of Cambridge, United Kingdom
E-mails: pierre.del-moral@inria.fr (P. Del Moral), sss40@cam.ac.uk (S.S. Singh)
Abstract. This Note and its extended version [7] present a novel backward Itô–Ventzell formula and an
extension of the Aleeksev–Gröbner interpolating formula to stochastic flows. We also present some natural
spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between
the two stochastic flows when their drift and diffusion functions are not the same.
2020 Mathematics Subject Classification. 47D07, 93E15, 60H07.
Funding. The first author is supported by the Chair Stress Test, RISK Management and Financial Steering, led
by the French Ecole polytechnique and its Foundation and sponsored by BNP Paribas.
Manuscript received 23rd June 2020, accepted 3rd September 2020.
Version française abrégée
Cette Note et sa version étendue [7] présentent une nouvelle formule à rebours de type Itô–
Ventzell (Theorème 1). Cette formule permet notamment de déduire une extension de la formule
d’interpolation d’Aleeksev–Gröbner aux flots stochastiques (Corollaire 2). Ces formules d’inter-
polation peuvent aussi s’obtenir par extension naturelle du calcul intégral stochastique bilatéral
développé par Pardoux and Protter dans l’article [10] aux interpolations de flots stochastiques.
Ces extensions sont développées en détails dans la version étendue [7]. Les approximations as-
sociées à cette approche offrent une variante de la construction originale de l’intégrale de Itô. Ces
dernières sont décrites sur la base d’un échantillonnage temporel dans la formule (9).
Supposons donnés deux flots stochastiques Xs, t (x) et X s, t (x) associés à des équations (3)
pour des champs de vitesse et diffusion (bt ,σt ) et (bt ,σt ) distincts. On notera par la suite
at (x) :=σt (x) σt (x)′ et at (x) :=σt (x)σ′t (x).
Un flot interpolant naturel dans l’analyse de la différence entre ces deux flots stochastiques est
donné le flot composé u ∈ [s, t ] 7→ Xu, t ◦X s,u . Comme il est précisé dans le Corollaire 3, la formule
à rebours de type Itô–Ventzell décrite dans le Theorème 1, ainsi que les formules d’interpolations
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
882 Pierre Del Moral and Sumeetpal S. Singh
associées développées dans le Corollaire 2, nous permettent d’appliquer la règle différentielle de
type Itô décrite dans la formule (12).
La formule d’interpolation (6) ainsi obtenue s’exprime en fonction du gradient et de la matrice
Hessienne du flot de référence Xs, t . Cette formulation variationnelle permet d’analyser finement
la distance entre les deux flots Xs, t et X s, t en fonctions des propriétés spectrales de ces flots.
Le Théorème 4 présente des estimations uniformes par rapport au temps (16) entre ces deux
flots stochastiques. La version étendue de cette Note [7] illustre ces formules d’interpolation en
théorie des pertubations, ainsi que dans le cadre des diffusions en interaction et dans l’étude des
approximations temporelles de processus stochastiques.
1. A generalized backward Itô–Ventzell formula
We represent the gradient ∇ f of a real valued function f of several variables as a column vector
while the gradient ∇F and the Hessian ∇2F of a (column) vector valued function F as tensors
of type (1,1) and (2,1). The transpose A′ of a (p, q)-tensor A is the (q, p)-tensor with entries
A′i , j = A j , i for any multiple indices i = (ik )1≤k ≤p and j = ( jl )1≤ l ≤q .
We denote byλmax (S) the maximal eigenvalue of a symmetric matrix S, and byρ(A) =λmax ((A+
A′)/2) the logarithmic norm of some matrix A.
We denote by Ω := C (R+,Rr ) the set of continuous paths from R+ into Rr , for some integer
parameter r ≥ 1.
Let Wt be an r -dimensional Brownian motion and denote by Ws,t the σ-field overΩ generated
by the increments (Wu −Wv ), with u, v ∈ [s, t ]. Without further mention, all the anticipating
stochastic integrals discussed in the present article are understood as Skorohod integrals.
We denote by D t the Malliavin derivative from some dense domainD2,1 ⊂ L2(Ω) into the space
L2(Ω×R+;Rr ). For multivariate d-column vector random variables F with entries F j , we use the
same rules as for the gradient and D t F is the (r, p)-matrix with entries (D t F )i , j := D it F j . For
(p ×q)-matrices F with entries F jk we let D t F be the tensor with entries (D t F )i , j ,k = D it F
j
k .
Let F be some function from Rp into Rq , and let y ∈Rp be some given state, for some p, q ≥ 1.
Suppose we are given a forward p-dimensional continuous semi-martingale Ys,t and a backward
random field Fs, t from Rp into Rq with a column-vector type canonical representation of the
following form : 
Ys, t = y +
∫ t
s
Bs,u du +
∫ t
s
Σs,u dWu
Fs, t (x) = F (x)+
∫ t
s
Gu, t (x)du +
∫ t
s
Hu, t (x)dWu
(1)
for some Ws, t -adapted functions Bs, t ,Gs, t , Hs,t ,Σs, t with appropriate dimensions and satisfying
the following conditions :
(H )1 : The functions Gu, t , ∇Hu, t , ∇2Fu, t and the derivatives Dv∇Fu, t (x) and Dv Hu, t (x) are
continuous w.r.t. x for any given u, v ∈ [s, t ] and ω ∈Ω.
(H )2 The function Gu, t ,∇Hu, t ,∇2Fu, t , and the derivatives Dv Hu, t ,Dv∇Fu, t have at most po-
lynomial growth w.r.t. the state variable, uniformly with respect to ω ∈Ω.
(H )2 The processes Bs,u ,Σs,u as well as the derivatives DvΣs,u have moments of any order.
Next Theorem 1 is the first main result of this Note.
C. R. Mathématique, 2020, 358, n 7, 881-886
Pierre Del Moral and Sumeetpal S. Singh 883
Theorem 1. Assume conditions (H i )i=1,2,3 are satisfied. In this situation, for any s ≤ u ≤ v ≤ t we
have the generalized backward Itô–Ventzell formula
Fv, t (Ys, v )−Fu, t (Ys,u) =
∫ v
u
(∇Fr, t (Ys,r )′ Bs,r + 1
2
∇2Fr, t (Ys,r )′Σs,rΣ′s,r −Gr, t (Ys,r ))dr
+
∫ v
u
(∇Fr,t (Ys,r )′Σs,r −Hr, t (Ys,r )) dWr (2)
The above Theorem 1 can be seen as the backward version of the generalized Itô–Ventzell
formula presented in [9] (see also [8, Theorem 3.2.11]).
2. Stochastic flows interpolation
Let bt (x) be a vector-valued function from Rd into Rd and σt (x) = [σt ,1(x), . . . , σt ,r (x)] be a
matrix-valued function from Rd into Rd ×r , for some parameters d ,r ≥ 1.
For any time horizon s ≥ 0 we denote by Xs, t (x) be the stochastic flow defined for any t ∈ [s,∞[
and any starting point Xs, s (x) = x ∈Rd by the stochastic integral equation
Xs, t (x) = x +
∫ t
s
bu
(
Xs,u(x)
)
du +
∫ t
s
σu
(
Xs,u(x)
)
dWu (3)
We assume that the drift bt (x) and the diffusion matrix σt (x) have continuous and uniformly
bounded derivatives up to the third order. This condition is met for linear Gaussian models as
well as for the geometric Brownian motion. It ensures that the stochastic flow x 7→ Xs, t (x) is a
twice differentiable function of the initialisation x. In addition, all absolute moments of the flow
and the ones of its first and second order derivatives exists for any time horizon.
For any p ≥ 1 and any twice differentiable function f from Rd into Rp with at most polynomial
growth the function
Fs, t (x) =Ps,t ( f )(x) := f (Xs, t (x)) (4)
satisfies the backward formula (1) with F (x) = f (x) and the random fields
Gu, t (x) :=∇Fu, t (x)′ bu(x)+ 1
2
∇2Fu, t (x)′ au(x) and Hu, t (x) :=∇Fu, t (x)′σu(x)
In addition, the regularity conditions on the drift and the diffusion function ensure that condi-
tions (H )i with i = 1,2,3 are satisfied.
Let X s, t (x) be the stochastic flow associated with a stochastic integral equation defined as (3)
by replacing (bt ,σt ) by some drift and diffusion functions (bt ,σt ) with the same regularity
properties. Also let Ps, t be the operator defined as in (4) by replacing Xs, t by X s, t .
Theorem 1 allows to describe the differences of operators Ps, t −Ps, t in terms of the difference
of their corresponding drifts and diffusion functions,
∆at := at −at ∆bt := bt −bt and ∆σt :=σt −σt (5)
where at (x) := σt (x)σt (x)′ and at (x) := σt (x)σ′t (x). More precisely, rewritten in terms of the
stochastic semigroups Ps, t and Ps, t the generalized backward Itô–Ventzell formula (2) yields the
following corollary.
Corollary 2. For any twice differentiable function f from Rd into Rp with at most polynomial
growth we have the forward-backward multivariate interpolation formula
Ps, t ( f )(x)−Ps, t ( f )(x) =Ts, t ( f ,∆a,∆b)(x)+Ss, t ( f ,∆σ)(x) (6)
C. R. Mathématique, 2020, 358, n 7, 881-886
884 Pierre Del Moral and Sumeetpal S. Singh
with the stochastic integro-differential operator
Ts, t ( f ,∆a,∆b)(x)
:=
∫ t
s
[
∇Pu, t ( f )
(
X s,u(x)
)′
∆bu
(
X s,u(x)
)
+ 1
2
∇2Pu,t ( f )
(
X s,u(x)
)′
∆au
(
X s,u(x)
)]
du (7)
and the Skorohod stochastic integral term given by
Ss, t ( f ,∆σ)(x) :=
∫ t
s
∇Pu, t ( f )
(
X s,u(x)
)′
∆σu
(
X s,u(x)
)
dWu (8)
The interpolation formula (6) with a fluctuation term given by the Skorohod stochastic inte-
gral (8) can be seen as a Aleeksev–Gröbner formula of Skorohod type [1]. For a more thorough
discussion on these stochastic interpolation formulae and their applications we refer to [7] and
the references therein.
Consider the discrete time interval [s, t ]h := {u0, . . . , un−1} associated with some refining time
mesh ui+1 = ui + h from u0 = s to un = t , for some time step h > 0. In this notation, the
Skorohod stochastic integral (8) can alternatively be defined by the L2-approximation of two-
sided stochastic integrals
Ss, t ( f ,∆σu)(x) = lim
h→0
∑
u∈ [s, t ]h
∇Pu+h, t ( f )
(
X s,u(x)
)′
∆σu
(
X s,u(x)
)
(Wu+h −Wu) (9)
This alternative approach can be seen as a variation of Itô original construction of the stochastic
integral and it relies on an extended version [7] of the two-sided stochastic integration calculus
introduced by Pardoux and Protter in [10].
Using elementary differential calculus, for twice differentiable (column vector-valued) func-
tion f from Rd into Rp we readily check the gradient and the Hessian formulae
∇Ps, t ( f )(x) =∇Xs, t (x) Ps, t (∇ f )(x)
∇2Ps, t ( f )(x) =
[∇Xs, t (x)⊗∇Xs, t (x)] Ps, t (∇2 f )(x)+∇2Xs, t (x) Ps, t (∇ f )(x) (10)
Choosing p = d and the identity function f (x) = e(x) := x the above formula reduces to
Ps,t (e)(x) =∇Xs, t (x) and ∇2Ps, t (e)(x) =∇2Xs, t (x)
In this context, we have backward stochastic flow differential equation
ds Xs, t (x) =−
[(
∇Xs, t (x)′ bs (x)+ 1
2
∇2Xs, t (x)′ as (x)
)
d s +∇Xs, t (x)′σs (x) dWs
]
(11)
In the above display, ds X is, t (x) represents the change in X
i
s,t (x) w.r.t. the variable s. A proof of the
above formula based on Taylor expansions is presented in [6], see also the appendix of [3].
In differential form, the forward-backward multivariate interpolation formula (6) applied to
the identity function yields the following Corollary 3.
Corollary 3. For any u ∈ [s, t ] and any x ∈ Rd we have the forward-backward stochastic interpo-
lation differential equation
du
(
Xu, t ◦X s,u
)
(x) = (du Xu, t )(X s,u(x))+ (∇Xu, t )(X s,u(x))′ du X s,u(x)
+ 1
2
(∇2Xu, t )(X s,u(x))′ au (X s,u(x)) du (12)
Forward-backward interpolation formulae of the same form as (12) without the Skorohod
fluctuation term for stochastic matrix Riccati diffusion flows are also discussed in [5]. The
articles [2, 4] also discuss similar interpolation formulae for mean field particle systems and
deterministic nonlinear measure valued semigroups.
C. R. Mathématique, 2020, 358, n 7, 881-886
Pierre Del Moral and Sumeetpal S. Singh 885
3. Uniform perturbation estimates
The second order perturbation methodology developed in the present article takes advantage
of the stability properties of the tangent and the Hessian flow in the estimation of Skorohod
fluctuation term and this sharpen analysis of the difference between stochastic flows.
For some multivariate function ft (x), for (t , x) ∈ [0,∞)×Rd , let ‖ f (x)‖ := supt ‖ ft (x)‖ and the
uniform norm be ‖ f ‖ := supt , x ‖ ft (x)‖. For any n ≥ 1 we also set∣∣∣∣∣∣ f (x)∣∣∣∣∣∣n := sup
s≥0
sup
t ≥ s
E
(∥∥∥ ft (X s, t (x))∥∥∥n)1/n and ρ(∇σ) := sup
1≤k ≤r
sup
t , x
ρ(∇σt ,k (x)) (13)
We denote by κn and κδ,n some constants that depends on some parameters n and (δ,n) but
do not depend on the time horizon, nor on the space variable. We also consider the following
collection of regularity conditions indexed by α≥ 1 :
(M )α For any x ∈Rd we have the uniform moment inequality w.r.t. the time horizon
sup
s≤ t
E
(∥∥Xs, t (x)∥∥α)1/α∨ sup
s≤ t
E
(∥∥∥X s, t (x)∥∥∥α)1/α ≤ κα (1∨‖x‖)
(T )α There exists some parameters λ> 0 such that
∇bt + (∇bt )′+
∑
1≤k ≤r
∇σt ,k (∇σt ,k )′ ≤−2λ I and λ>
d(α−2)
2
ρ(∇σ)2
In addition, the drift and diffusion matrix (bt ,σt ) satisfy the same condition for some λ> 0.
For constant diffusion functions
σt (x) =Σt and σt (x) =Σt for some matrices Σt and Σt . (14)
the condition (T )n is met for any n ≥ 2 as soon as the following log-norm inequalities are met
∇bt + (∇bt )′ ≤−2λ I and ∇bt +
(∇bt )′ ≤−2λ I for some λ∧λ> 0, (15)
We end this Note with a uniform interpolation theorem w.r.t. the time parameter.
Theorem 4. Assume conditions (M )2n/δ and (T )2n/(1−δ) are satisfied for some parameters n ≥ 2
and δ ∈]0,1[. In this situation, for any x ∈Rd and s ≤ t we have the time-uniform estimates
E
[∥∥∥Xs, t (x)−X s, t (x)∥∥∥n]1/n
≤ κδ,n
(|||∆a(x)|||2n/(1+δ) +|||∆b(x)|||2n/(1+δ) +|||∆σ(x)|||2n/δ (1∨‖x‖)) (16)
Whenever (14) and (15) are satisfied, for any n ≥ 2 we have the time-uniform estimates
E
[∥∥∥Xs, t (x)−X s, t (x)∥∥∥n]1/n ≤ κn (|||∆b(x)|||n +∥∥∥Σ−Σ∥∥∥) (17)
Illustrations of the forward-backward interpolation formulae presented in this Note in the
context of diffusion perturbation theory, interacting diffusions and discrete time approximations
are discussed in the extended version [7].
Références
[1] V. Alekseev, “An estimate for the perturbations of the solution of ordinary differential equations”, Vestn. Mosk. Univ.
(1961), no. 2, p. 28-36.
[2] M. Arnaudon, P. Del Moral, “A duality formula and a particle Gibbs sampler for continuous time Feynman–Kac
measures on path spaces”, https://arxiv.org/abs/1805.05044, 2018.
[3] ——— , “A variational approach to nonlinear and interacting diffusions”, Stochastic Anal. Appl. 37 (2019), no. 5,
p. 717-748.
[4] ——— , “A second order analysis of McKean–Vlasov semigroups”, 2020, To appear in The Annals of Applied Probabi-
lity.
C. R. Mathématique, 2020, 358, n 7, 881-886
886 Pierre Del Moral and Sumeetpal S. Singh
[5] A. N. Bishop, P. Del Moral, A. Niclas, “A perturbation analysis of stochastic matrix Riccati diffusions”, Ann. Inst. Henri
Poincaré, Probab. Stat. 56 (2020), no. 2, p. 884-916.
[6] G. Da Prato, L. Tubaro, “Some remarks about backward Itô formula and applications”, Stochastic Anal. Appl. 16
(1998), no. 6, p. 993-1003.
[7] P. Del Moral, S. Sidhu Singh, “Backward Itô–Ventzell and stochastic interpolation formulae”, https://hal.
archives-ouvertes.fr/hal-02161914v4, 2019.
[8] D. Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and Its Applications, vol. 1995, Springer,
2006.
[9] D. Ocone, E. Pardoux, “A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential
equations”, Ann. Inst. Henri Poincaré, Probab. Stat. 25 (1989), no. 1, p. 39-71.
[10] E. Pardoux, P. E. Protter, “A two-sided stochastic integral and its calculus”, Probab. Theory Relat. Fields 76 (1987),
no. 1, p. 15-49.
C. R. Mathématique, 2020, 358, n 7, 881-886


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