Comptes Rendus Mathématique

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Yvan Martel

A new proof of a Liouville theorem for the one dimensional Gross–Pitaevskii equation

Igor Pak

Colleen Robichaux

Positivity of Schubert coefficients

Alonso Beaumont

Comptes Rendus
Mathématique
Alonso Beaumont
On the nilpotency of locally pro-p contraction groups
Volume 363 (2025), p. 267-270
Online since: 8 April 2025
https://doi.org/10.5802/crmath.728
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Comptes Rendus. Mathématique
2025, Vol. 363, p. 267-270
https://doi.org/10.5802/crmath.728
Research article / Article de recherche
Group theory / Théorie des groupes
On the nilpotency of locally pro-p contraction
groups
Sur la nilpotence des groupes de contraction localement
pro-p
Alonso Beaumont a
a IRMAR, Université de Rennes, 263 Avenue du Général Leclerc, 35042 Rennes Cedex, France
E-mail: alonso.beaumont@univ-rennes.fr
Abstract. H. Glöckner and G. A. Willis have recently shown [2] that locally pro-p contraction groups are
nilpotent. The proof hinges on a fixed point result [2, Theorem B]: if the local field Fp ((t )) acts on its d-th
power Fp ((t ))d additively, continuously, and in an appropriately equivariant manner, then the action has a
non-zero fixed point. We provide a short proof of this theorem.
Résumé. H. Glöckner et G. A. Willis ont récemment démontré [2] que les groupes de contraction localement
pro-p sont nilpotents. Leur démonstration repose sur un résultat de point fixe [2, Theorem B]: si le corps
local Fp ((t )) agit sur sa d-ième puissance Fp ((t ))d additivement, continûment et de manière convenablement
équivariante, alors l’action a un point fixe non nul. Nous présentons une démonstration courte de ce
théorème.
2020 Mathematics Subject Classification. 22D05, 20E36, 20F18.
Manuscript received 8 January 2025, revised 30 January 2025, accepted 1 February 2025.
1. Introduction
A locally compact topological group G is said to be a contraction group if it has a continuous
automorphism α whose forward orbits converge to the identity element e:
∀ g ∈G , αn(g ) −→
n→∞ e.
This dynamical property has interesting connections with the purely group-theoretic notion of
nilpotency. This relationship is best illustrated in the case where G is a Lie group: a contracting
automorphism on G provides a positive gradation for its Lie algebra g, proving in particular that g
is nilpotent (see [5]). A thorough study of totally disconnected locally compact (tdlc) contraction
groups is carried out in [1]. One of its main structural results asserts that any tdlc contraction
group is a direct product of a torsion-free contraction group, which is always nilpotent, and a
torsion contraction group [1, Theorem B]. The remaining questions concerning the torsion part
were treated in [2]. In order to accommodate for torsion, an additional hypothesis is needed: a
tdlc group is called locally pro-p if it contains an open pro-p subgroup. For example, analytic
groups over the field Qp of p-adic numbers, or the field Fp ((t )) of Laurent series with coefficients
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
268 Alonso Beaumont
in Fp , are locally pro-p. In [2] it is shown that locally pro-p contraction groups are nilpotent. The
major part of the proof consists in establishing the following fixed point result [2, Theorem B]:
Theorem 1. Let d ≥ 1 and consider a continuous action ψ : (Fp ((t )),+) × (Fp ((t ))d ,+) →(
Fp ((t ))d ,+
)
by group automorphisms. If
∀ x ∈ Fp ((t )), ∀ u ∈ Fp ((t ))d , ψ(t x, tu) = tψ(x,u)
then the action has a non-zero fixed point.
Glöckner and Willis’s proof of this result involves representing endomorphisms of the abelian
group Fp ((t ))d as infinite block matrices satisfying certain conditions. We refer the reader to
Section 3 of [2] for details. The purpose of this note is to provide an alternative, shorter proof
of this result. It has two main ingredients: a study of Fp -representations of elementary abelian
p-groups, and an application of Ascoli’s theorem to the automorphism group Aut
(
Fp ((t ))d
)
.
2. Proof of the theorem
We begin with a lemma. A linear representation V of a group G is said to be locally finite if its
finitely generated subrepresentations are finite-dimensional. We denote by V G the set of G-fixed
points in V .
Lemma 2. Let G be an elementary abelian p-group, and V a locally finite representation of G over
Fp . If V is infinite-dimensional, then either
V G or
(
V /V G
)G
is infinite-dimensional.
Proof. First, suppose that G is finite. We will show that V G is infinite-dimensional. In order to
give a self-contained presentation, we will provide a direct proof of the lemma in this case1. Write
G = 〈g1, . . . , gr 〉. By the local finiteness of V , there is a strictly increasing family (Vn)n≥0 of finite
subrepresentations. For all n ≥ 0, write dn = dimVn and consider the sequence
∀ i ∈ {0, . . . , p}, dn(i ) = dim
(
ker(g1 − id)i ∩Vn
)
.
By a standard application of the rank-nullity theorem, (dn(i + 1) − dn(i ))0≤i<p is a decreasing
sequence. Moreover, dn(p) = dn since (g1− id)p = g p1 −id = 0. We deduce that dn(1) = dimV
〈g1〉
n ≥
dn/p. The elements g1 and g2 commute, so we can consider the action of 〈g2〉 on V 〈g1〉. By
repeating the argument above we obtain dimV 〈g1,g2〉n ≥ dn/p2 and by a finite induction,
dimV G ≥ dimV Gn ≥ dn/pr −→n→+∞+∞.
Now suppose that G is infinite, and that both V G and (V /V G )G are finite-dimensional. Then
the preimage V ′ of (V /V G )G in V is finite-dimensional. Therefore, the G-action on V ′ factors
through a finite quotient of G . Since G is elementary abelian, this quotient is a direct summand.
We can therefore write G =G0 ×G1, where G0 is finite and G1 acts trivially on V ′. In particular, G1
acts trivially on V ′∩V G0 so V ′∩V G0 ⊂ (V G0 )G1 =V G . Hence,(
V ′∩V G0)/V G = (V G0 /V G )G = 0.
Suppose V G0 /V G is non-trivial. Since V G0 /V G is locally finite, it would have a non-trivial finite-
dimensional subrepresentation, and hence a non-zero G-fixed point, because G acts as an
abelian unipotent group. We deduce that V G0 /V G = 0. This is absurd since G0 is finite and thus
V G0 is infinite-dimensional. □
1An alternative line of argument is as follows. If G is finite then Fp [G] is a finite-dimensional Fp -algebra. If the Fp [G]-
module V G were finite then its essential extension V would also be finite (see for instance [4]), so V G must be infinite-
dimensional.
Alonso Beaumont 269
Let d ≥ 1. We consider Fp ((t ))d as an additive tdlc group, equipped with a continuous
automorphism given by scalar multiplication by t : u = (u1, . . . ,ud ) 7→ tu = (tu1, . . . , tud ). The
group Aut
(
Fp ((t ))d
)
of continuous automorphisms is endowed with the compact-open topology.
In particular, a continuous action G × Fp ((t ))d → Fp ((t ))d by group automorphisms induces a
continuous morphism ϕ : G → Aut(Fp ((t ))d ). We restate Theorem 1 in the following manner:
Theorem 3. A continuous morphism ϕ : Fp ((t )) → Aut
(
Fp ((t ))d
)
satisfying
∀ x ∈ Fp ((t )), ∀ u ∈ Fp ((t ))d , ϕ(t x)(u) = tϕ(x)(t−1u) (1)
has a non-zero fixed point: there exists u ∈ Fp ((t ))d \{0} such that ϕ(x)(u) = u for all x ∈ Fp ((t )).
Proof. The proof will be done in two steps. We will first construct a nested family of invariant
subgroups, and then we will use it to locate increasingly smaller orbits that will converge to a
fixed point. We will use the following notation: G = ϕ(Fp ((t ))), Gℓ = ϕ(t−ℓFpt) for all ℓ ≥ 0,
B = Fptd , and S = B\tB .
Step 1. Let M be the largest G-invariant subgroup contained in B . It can be defined as the group
generated by all G-invariant subgroups in B . We will prove the following: M is non-trivial and
closed, and A :=⋃n≥0 t−n M is a G-invariant Fp ((t ))-subspace of Fp ((t ))d in which M is open.
Since ϕ is continuous, Gℓ is compact, and so by Ascoli’s theorem [3, Theorem 47.1] it is an
equicontinuous family of automorphisms of Fp ((t ))d . In particular, there exists a neighbourhood
of 0 whose Gℓ-orbit is contained in B . The subgroup generated by this orbit is therefore open, Gℓ-
invariant, and contained in B . Let Mℓ be the largest Gℓ-invariant subgroup in B . By the previous
argument Mℓ contains an open subgroup and thus it is itself open. By property (1), we have
∀ x ∈ t−ℓFpt, ϕ(x)(t−1Mℓ) = t−1ϕ(t x)(Mℓ) = t−1Mℓ
and so t−1Mℓ is also Gℓ-invariant. This means that Mℓ and S have a non-empty intersection, for
otherwise the subgroup Mℓ+ t−1Mℓ, which is Gℓ-invariant and strictly larger that Mℓ, would be
contained in B . Therefore, the decreasing sequence (Mℓ∩S)ℓ≥0 consists of non-empty compact
sets, and hence its intersection is non-empty. We have proven that
M := ⋂
ℓ≥0
Mℓ
is a non-trivial subgroup of Fp ((t ))d . Every G-invariant subgroup of B is Gℓ-invariant, hence
contained in Mℓ for each ℓ. Therefore M is the largest G-invariant subgroup of B . Again using (1),
we see that t M is G-invariant and so t M ⊂ M . As an intersection of closed subgroups, M is closed.
We deduce that M is an Fpt-module and thus
A := ⋃
n≥0
t−n M
is a G-invariant Fp ((t ))-subspace of Fp ((t ))d . In particular, A is closed in Fp ((t ))d and therefore it is
a Baire space. As such, some t−n M , or equivalently all of them, are open in A.
Step 2. For all n ≥ 0, let Vn := t−n M/M . This quotient is an Fp -vector space of finite dimension
endowed with a linear G-action. This means that V := A/M =⋃n≥0 Vn satisfies the hypotheses of
Lemma 2. Let us show that V has infinitely many G-fixed points. Suppose otherwise: V G would
then be contained in Vn for some n ≥ 0. On the other hand, by Lemma 2 (V /V G )G would be
infinite: since V /V G surjects G-equivariantly onto V /Vn with finite kernel, (V /Vn)G would also
be infinite. We claim this is absurd because V G and (V /Vn)G are in bijection with one another.
Indeed, we may choose a section V /Vn = A/t−n M → A and compose it with t n · id: A → A
followed by A → V . This defines an isomorphism V /Vn → V which by property (1) preserves
G-fixed points.
270 Alonso Beaumont
We have shown that V G is infinite. In particular, for all n ≥ 0, V G is not contained in Vn : we
can find some vn ∈ (A/M)G that lifts to vn ∈ A\t−n M . Let kn > n be the unique integer such that
wn := t kn vn ∈ M\t M . Applying (1), we have
∀ x ∈ Fp ((t )), ϕ(x)(wn)−wn = t kn
(
ϕ(t−kn x)(vn)− vn
) ∈ t kn M .
Since M\t M is compact, the family (wn)n≥0 has an accumulation point w . For all x ∈ Fp ((t )) and
infinitely many n, we have ϕ(x)(w)−w ∈ t kn M , so ϕ(x)(w) = w : we have constructed a non-zero
G-fixed point in A. □
Remark 4. Glöckner and Willis state Theorem 3 slightly differently: they endow Aut
(
Fp ((t ))d
)
with the finer Braconnier topology, and assume that ϕ is continuous with respect to it. This
assumption is in fact no stronger than ours (see [6, Lemma 9.13]). In light of this, Step 1 of the
proof above may be started as follows. The subgroup
{
g ∈ Gℓ
∣∣g (B) = B} is open in Gℓ for the
Braconnier topology, and therefore it is of finite index. Thus, the intersection
⋂
g∈Gℓ g (B) is finite,
yielding an open Gℓ-invariant subgroup in B .
Remark 5. Let (G1,α1) and (G2,α2) be two tdlc contraction groups, equipped with their respec-
tive contracting automorphisms. A continuous actionψ : G1×G2 →G2 may be called equivariant
if ψ
(
α1(x),α2(y)
) = α2(ψ(x, y)) for all x ∈ G1, y ∈ G2. Then the argument of Step 1 can be ap-
plied in this more general setting: there exists a compact, non-trivial subgroup H of G2 such that
α2(H) ⊂ H and ψ(G1 ×H) = H .
Acknowledgments
I’d like to thank David J. Benson for his contribution to Lemma 2, as well as Adrien Le Boudec and
Helge Glöckner for their helpful comments.
Declaration of interests
The author does not work for, consult, own shares in or receive funding from any company
or organization that would benefit from this article, and has disclosed no relevant affiliations
beyond his research organization.
References
[1] H. Glöckner and G. A. Willis, “Classification of the simple factors appearing in composi-
tion series of totally disconnected contraction groups”, J. Reine Angew. Math. 2010 (2010),
no. 643, pp. 141–169.
[2] H. Glöckner and G. A. Willis, “Locally pro-p contraction groups are nilpotent”, J. Reine
Angew. Math. 781 (2021), pp. 85–103.
[3] J. Munkres, Topology, Second edition, Pearson Education, 2014, pp. i+503.
[4] A. Rosenberg and D. Zelinsky, “Finiteness of the injective hull”, Math. Z. 70 (1958), pp. 372–
380.
[5] E. Siebert, “Contractive automorphisms on locally compact groups”, Math. Z. 191 (1986),
pp. 73–90.
[6] M. Stroppel, Locally compact groups, EMS Textbooks in Mathematics, European Mathemat-
ical Society, 2006, pp. x+302.


On the nilpotency of locally pro-$p$ contraction groups

Karthik Elamvazhuthi

A linear test for global nonlinear controllability

Haesong Seo

A non-vanishing conjecture for cotangent bundles on elliptic surfaces

Thiago Fassarella

Nivaldo Medeiros

Rodrigo Salomão

Toric polar maps, normal crossings and mixed volumes

Claudemir Alcantara

Edgard Pimentel

José Miguel Urbano

Comptes Rendus
Mathématique
Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano
Hessian regularity in Hölder spaces for a semi-linear bi-Laplacian equation
Volume 363 (2025), p. 533-539
Online since: 26 May 2025
https://doi.org/10.5802/crmath.737
This article is licensed under the
Creative Commons Attribution 4.0 International License.
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MER S ENN E
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www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 533-539
https://doi.org/10.5802/crmath.737
Research article / Article de recherche
Partial differential equations / Équations aux dérivées partielles
Hessian regularity in Hölder spaces for a
semi-linear bi-Laplacian equation
Régularité du hessien dans les espaces de Hölder pour
une équation bi-laplacienne semi-linéaire
Claudemir Alcantara a, Edgard Pimentel b and José Miguel Urbano b,c
a Department of Mathematics, Pontifical Catholic University of Rio de Janeiro (PUC-Rio),
22451-900 Rio de Janeiro, Brazil
b CMUC, Department of Mathematics, University of Coimbra, 3000-143 Coimbra, Portugal
c Applied Mathematics and Computational Sciences Program (AMCS), Computer, Electrical
and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of
Science and Technology (KAUST), Thuwal, 23955 -6900, Kingdom of Saudi Arabia
E-mails: alcantara@mat.puc-rio.br, edgard.pimentel@mat.uc.pt, miguel.urbano@kaust.edu.sa
Abstract. We examine a semi-linear variant of the bi-Laplacian equation in the superlinear, subquadratic
setting and obtain C 2,σ-regularity estimates, depending on the growth regime of the nonlinearity. Our
strategy is to render this fourth-order problem as a system of two Poisson equations and explore the interplay
between the integrability and smoothness available for each equation taken isolated.
Résumé. Nous examinons une variante semi-linéaire de l’équation bi-laplacienne dans un cadre superli-
néaire et subquadratique, et obtenons des estimations de régularité C 2,σ dépendant du régime de croissance
de la non-linéarité. Notre stratégie consiste à transformer le problème du quatrième ordre en un système de
deux équations de Poisson, puis à explorer l’interaction entre l’intégrabilité et la régularité disponibles pour
chaque équation prise isolément.
Keywords. Bi-Laplacian operator, semi-linear equations, Hessian regularity, Hölder spaces.
Mots-clés. Opérateur bi-laplacien, équations semi-linéaires, régularité hessienne, espaces de Hölder.
2020 Mathematics Subject Classification. 35B65, 35J91, 35G20.
Funding. EP and JMU are supported by the Centre for Mathematics of the University of Coimbra (CMUC,
funded by the Portuguese Government through FCT/MCTES, UIDB/00324/2020). JMU is partially supported
by the King Abdullah University of Science and Technology (KAUST). CA was partially supported by the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) – Brasil and by Stone Instituição de
Pagamento S.A., under the Project StoneLab. This study was financed in part by the CAPES-Brazil – Finance
Code 001.
Manuscript received 29 January 2025, accepted 27 February 2025.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
534 Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano
1. Introduction
We examine the regularity of weak solutions to the semi-linear bi-Laplacian equation
∆2u = f (x,u,Du) inΩ, (1)
where Ω ⊂ Rd is a bounded smooth domain, ∆2u := ∆(∆u) denotes the bi-Laplacian operator,
and the nonlinearity f : Ω×R×Rd → R satisfies a polynomial growth condition. Our findings
report on C 2,σ-regularity estimates for the solutions to (1), depending on the growth regime of
the nonlinearity.
Elliptic equations driven by operators of higher order play an important role across disciplines
in pure mathematics and find relevant applications in various realms of life and social sciences.
We mention differential geometry, calculus of variations, free boundary problems, the mechanics
of deformable media (mainly in the mathematical theory of elasticity), and the dynamics of slow
viscous fluids; see the monograph [10] and the references therein.
From the perspective of partial differential equations (PDE), the study of bi-Laplacian equa-
tions has covered a plethora of topics. These include the existence of solutions, fundamental
properties (such as the validity of the maximum principle and the positivity of the Green’s func-
tion), and regularity estimates. We notice the existence of relevant literature examining those
properties in connection with the geometry of the domain. Of particular interest is the fact that
merely Lipschitz-regular domains entail further difficulties for the analysis. In this connection,
we refer the reader to [9] and the extensive list of references therein.
Regarding regularity estimates, the study of the bi-Laplacian operator has been pursued in
several contexts. In the realm of obstacle problems, it appears as the operator governing a
variational inequality. More precisely, given a function ϕ : Ω→R, let u ∈W 2,2(Ω) be such that
∆2u ≥ 0 inΩ, (2)
with u ≥ϕ, and
∆2u · (u −ϕ) = 0 inΩ. (3)
In [7], the author proves that weak solutions to this problem are in W 2,∞(Ω). The analysis of the
free boundary associated with (2)–(3) is the subject of [1]. In that paper, the authors establish the
local boundedness of the Hessian and verify that∆2u is a non-negative measure with finite mass.
In addition, they examine the planar case, showing that solutions are C 2-regular and that the free
boundary is contained in a continuously differentiable curve.
The bi-Laplacian operator has also been studied in [2] in the context of the two-phase free
boundary obstacle problem
∆2u = 0 in B+1 ,
u = g on (∂B1)+,
∂xd+1 u = 0 on B ′1,
∂xd+1∆u =λ−(u−)p−1 −λ+(u+)p−1 on B ′1,
(4)
where p > 1, λ− and λ+ are positive constants, g ∈ W 2,q (B+1 ) for q := max(2, p), and the sets B+1
and B ′1 are defined as
B+1 :=
{
(x, xd+1) ∈ B1 ⊂Rd ×R
∣∣ xd+1 > 0}
and
B ′1 := B1 ∩ {xd+1 = 0}.
The authors prove that both u and ∆u are locally bounded. They also establish regularity
estimates in Hölder spaces of the type C p+1,γ, for every γ ∈ (0,1), and show their findings are
optimal in the case of p ∈ N. They also consider an Almgren’s type frequency formula and a
Monneau monotonicity formula and perform a thorough analysis of the singular set associated
Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano 535
with (4). If B1 is replaced with Rd , it is worth noticing that the formulation in (4) can be regarded
as a Dirichlet-to-Neumann extension, in the spirit of Caffarelli and Silvestre, for the operator
(−∆)3/2u =λ−(u−)p−1 −λ+(u+)p−1 in Rd ,
with u → 0 as |x|→∞.
Interior regularity estimates for the pure equation∆2u = f prescribed in a domainΩ have also
been pursued in the literature. Of particular interest is the analysis of polyharmonic equations of
the form
(−∆)mu = f inΩ,
where 2 ≤ d ≤ 2m + 1 and f ∈ C∞c (Ω). In [9], the authors prove that solutions to this problem
satisfy
Dm−
d
2 + 12 u ∈ L∞(Ω) if d = 2k +1,
and
Dm−
d
2 u ∈ L∞(Ω) if d = 2k,
with k ∈N. In the concrete case of (1), were f = f (x) is a smooth function, the analysis would lead
to u ∈ L∞(Ω) for dimensions d = 4,5, and Du ∈ L∞(Ω) for dimensions d = 2,3.
Concerning the semi-linear formulation of the bi-Laplacian equation, we mention the devel-
opments reported in [3]. In that paper, the authors produce regularity estimates for the weak
solutions to
∆2u +a(x)u = g (x,u) in Rd . (5)
They work under natural assumptions on the functions a = a(x) and g = g (x,u), including
polynomial growth conditions on g , and prove that solutions to (5) are in W 4,2(Rd )∩W 2,s (Rd )
for every 1 ≤ s ≤∞. Their arguments rely on asymptotic properties of the fundamental solution
associated with the operator∆2+k2, for k ∈N. Apparently, these methods fall short in addressing
the dependence on the gradient Du, not covering the case of (1). To the best of our knowledge,
the analysis of semi-linear bi-Laplacian equations with explicit dependence on Du has hitherto
not been addressed in the literature.
Our reasoning in this note relies on a reduction argument. Namely, we write the fourth-order
PDE as a system of two second-order equations. This strategy is inspired by ideas introduced in
the works [5,6] of Evans. The gist of those papers is to design new PDE methods to address the
Mather minimization principle in dynamics, as well as the weak KAM theory.
Our approach stems from this class of ideas as we write the single equation in (1) as a coupling.
Indeed, the integrability available for D2u in the class of weak solutions allows us to define a
function m ∈ Lq (Ω) as m :=∆u. The definition of weak solution to (1) then implies that m is a very
weak solution to the Poisson equation ∆m = f . As a consequence, we render the bi-Laplacian
equation as the system {
∆u = m inΩ,
∆m = f (x,u,Du) inΩ. (6)
The unknown for (6) is a pair (u,m) in a suitable functional space, where u is a strong solution
for the first equation in the system, while m satisfies the second one in the very weak sense (we
make these matters precise further on in the note).
In this context, the regularity of the solutions to (1) benefits from the interplay between the
regularity estimates available at the level of the equations taken on their own. An Lq -regularity
theory for very weak solutions builds upon standard Sobolev embeddings to produce improved
integrability for m in Lebesgue spaces. In turn, the integrability of m affects the regularity of u.
Indeed, we prove that very weak solutions to
∆m = f (x,u,Du) inΩ
536 Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano
are in W 2,sloc (Ω), for some s ∈ (d/2,d ]. Hence, u solves a Poisson equation with right-hand side in
C 0,2−d/sloc (Ω). This fact unlocks a Schauder regularity theory for the solutions to (1).
The remainder of this note is split into two sections, one detailing preliminary definitions and
useful facts and the other encompassing our main result.
2. Notions of solution and auxiliary results
Because our strategy is to render the bi-Laplacian equation as a system, we rely on three different
notions of solutions. Namely, weak distributional solutions, very weak solutions, and Ld -strong
solutions. Requiring u to be in W 2,q (Ω), for values of q ∈ (d/2,d ] depending on the growth regime
satisfied by f , allows us, in particular, to switch between these different notions of solution
seamlessly. We start by defining a weak solution for the bi-Laplacian equation in a distributional
sense.
Definition 1 (Local weak solution of the bi-Laplacian equation). A function u ∈ W 2,2loc (Ω) is a
local weak solution to (1) if ∫
Ω
∆u∆ϕdx =
∫
Ω
f (x,u,Du)ϕdx,
for every ϕ ∈C∞c (Ω).
Concerning Poisson’s equation
∆w = g inΩ, (7)
we will explore two notions of solution, those of very weak and Lq -strong solution. For complete-
ness, we recall these notions in what follows.
Definition 2 (Very weak solution of the Poisson equation). Let g ∈ L1loc(Ω). A function w ∈
L1loc(Ω) is a very weak solution to (7) if∫
Ω
w∆ϕdx =
∫
Ω
g ϕdx,
for every ϕ ∈C∞c (Ω).
Definition 3 (Lq -strong solution of the Poisson equation). Let g ∈ Lq (Ω) for q > 1. We say that
w ∈W 2,q (Ω) is an Lq -strong solution to (7) if
∆w(x) = g (x), a.e. x ∈Ω.
We proceed by defining a notion of solution to the system in (6), relating (1) with the latter.
Definition 4. The pair (u,m) ∈W 2,q (Ω)×Lq (Ω) is a solution to (6) if u is an Lq -strong solution to
the first equation in (6) whereas m is a very weak solution to the second equation in (6).
Next, we recall a result on the regularity of very weak solutions to the Poisson equation.
Proposition 5 (Sobolev regularity for very weak solutions). Fix 1 < s <∞. Let w ∈ L1loc(Ω) be a
very weak solution to (7), with g ∈ Lsloc(Ω). Then Dw ∈ W 1,sloc (Ω). If w ∈ Lsloc(Ω), then w ∈ W
2,s
loc (Ω).
Moreover, forΩ′′ ⋐Ω′ ⋐Ω, there exists C > 0 such that
∥w∥W 2,s (Ω′′) ≤C
(∥w∥Ls (Ω) +∥g∥Ls (Ω)). (8)
Proof. To verify that w ∈ W 2,sloc (Ω), we resort to [4, Theorem 3]. Once we have w ∈ W
2,s
loc (Ω), w
becomes a strong solution to ∆w = g in Ω′. Standard results in elliptic regularity theory (see, for
instance, [8, Theorem 9.11]) yield (8). □
We conclude this section by verifying that a weak solution to (1) yields a pair (u,m) in a suitable
functional space, solving (6).
Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano 537
Lemma 6. Let u ∈W 2,q (Ω) be a local weak solution to (1), with q ≥ 2. Then, there exists m ∈ Lq (Ω)
such that (u,m) is a solution to (6) according to Definition 4.
Proof. It is clear that if u ∈W 2,q (Ω),∆u ∈ Lq (Ω). Set m :=∆u and notice that m ∈ Lq (Ω) is defined
almost everywhere inΩ. In addition, the weak formulation of (1) implies∫
Ω
m∆ϕdx =
∫
Ω
∆u∆ϕdx =
∫
Ω
f (x,u,Du)ϕdx,
for every ϕ ∈C∞c (Ω). Hence, m is a very weak solution to
∆m = f (x,u,Du) inΩ,
and the proof is complete. □
3. Improved regularity in Hölder spaces
In this section, we state and prove the main result in this note. For ease of presentation, we
set Ω ≡ B1, where B1 stands for the unit ball in Rd ; standard covering arguments ensure this
reduction entails no further restrictions on the problem.
Theorem 7. Let 2 ≤ q ≤ d and u ∈W 2,q (B1) be a local weak solution to
∆2u = f (x,u,Du) in B1.
Assume the nonlinearity satisfies the growth condition∣∣ f (x,r, p)∣∣≤ h(x)+C(|r |α+|p|β), (9)
for h ∈ Ld (B1) and fixed constants C > 0 and
α,β ∈ [1,2). (10)
Suppose further that
max(α,β)
d
2
< q ≤ d . (11)
Then u ∈C 2,σloc (B1) for
σ := 2− d max(α,β)
q
∈ (0,1).
Moreover, there exists C > 0 such that
∥u∥C 2,σ(B7/8) ≤C
(
∥h∥Ld (B1) +∥u∥
max(α,β)
W 2,q (B1)
)
.
Proof. We start by choosing
s := q
max(α,β)
,
noting that, due to (10) and (11), we have s ∈ (d/2, q]. Moreover, for B9/10 ⋐ B1 it follows from (9)
that ∥∥ f ( · ,u,Du)∥∥Ls (B9/10) ≤C(∥h∥Ls (B9/10) +∥u∥αLαs (B9/10) +∥Du∥βLβs (B9/10))
≤C
(
∥h∥Ld (B1) +∥u∥
max(α,β)
W 2,q (B1)
)
,
which is finite since u ∈ W 2,q (B1). The constant C here depends on q and the ingredients in (9).
By Lemma 6, there exists m ∈ Lq (B1) such that m is a very weak solution to
∆m = f (x,u,Du) in B1.
538 Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano
Due to Proposition 5, we conclude Dm ∈ W 1,sloc (B99/100). But, since d/2 < s ≤ q , we also have
m ∈ Ls (B1) and therefore m ∈W 2,sloc (B99/100). In addition, there exists C > 0 such that
∥m∥W 2,s (B9/10) ≤C
(
∥h∥Ld (B1) +∥m∥Ls (B1) +∥u∥
max(α,β)
W 2,q (B1)
)
≤C
(
∥h∥Ld (B1) +∥u∥W 2,q (B1) +∥u∥
max(α,β)
W 2,q (B1)
)
,
where the second inequality follows from the fact that
∥m∥Ls (B9/10) = ∥∆u∥Ls (B9/10) ≤ ∥u∥W 2,q (B1).
Because of Gagliardo–Nirenberg–Sobolev’s embedding theorem, we obtain m ∈C 0,σ(B8/9), with
σ := 2− d max(α,β)
q
.
Since u is an Lq -strong solution to ∆u = m, we have u ∈C 2,σ(B8/9) (see [8, Theorem 9.19]). Also,
by Schauder’s theory, there exists a positive constant C =C (d ,α,β, q), such that
∥u∥C 2,σ(B7/8) ≤C
(∥u∥L∞(B8/9) +∥m∥C 0,σ(B8/9)).
To complete the proof, we notice that
∥m∥C 0,σ(B8/9) ≤C∥m∥W 2,s (B8/9)
≤C∥∥ f ( · ,u,Du)∥∥Ls (B9/10)
≤C
(
∥h∥Ld (B1) +∥u∥
max(α,β)
W 2,q (B1)
)
,
and the estimate in the theorem follows. □
Remark 8. A fundamental question arises in the context of assumption (11): one must ensure
that
max(α,β)
d
2
< d ,
so the range for q is nonempty. The above inequality is indeed satisfied due precisely to (10).
The explicit description of the modulus of continuity is appealing, as it provides asymptotic
information. As q → d and the growth conditions for f approach the linear regime, the exponent
σ→ 1, yielding asymptotic estimates for (1) in C 2,1. We also notice the explicit gains of regularity
stemming from (1). Indeed, we start with a function u ∈ W 2,q (Ω) and the equation yields
u ∈C 2,σloc (Ω), with estimates.
We conclude this note with a corollary to Theorem 7 yielding smoothness of the solutions to (1)
in the case α=β= 1, under the assumption that h ∈C∞(Ω).
Corollary 9 (C∞-regularity estimates). Let u ∈ W 2,q (B1) be a weak solution to (1), with q ≥ 2
satisfying (11). Suppose (9) is in force, with
f (x,r, p) := h(x)+a(x)r + c(x) ·p,
where h, a ∈C∞(B1) and c ∈C∞(B1,Rd ). Suppose further there exists C > 0 such that
∥h∥C∞(B1) +∥a∥C∞(B1) +∥c∥C∞(B1,Rd ) ≤C .
Then u ∈C∞loc(B1). Moreover, for every k ∈N and every multi-index α with |α| = k, we have
sup
B7/6
|Dαu| ≤C(1+∥u∥W 2,q (B1)).
Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano 539
Proof. Fix a direction i ∈ {1, . . . ,d} and define
v := ∂u
∂xi
.
Clearly, v solves
∆2v = g (x)+a(x)v(x)+ c(x) ·Dv,
with g given by
g (x) := ∂
∂xi
h(x)+ ∂
∂xi
a(x)u(x)+ ∂
∂xi
c(x) ·Du(x).
One easily notices that∣∣g (x)+a(x)v(x)+ c(x) ·Dv∣∣≤C(1+∥u∥W 2,q (B1) + ∣∣v(x)∣∣+ ∣∣Dv(x)∣∣).
Hence, Theorem 7 implies v ∈ C 2,σloc (B1). Because the direction i is arbitrary, we conclude
u ∈C 3,σloc (B1). An induction argument on the order of differentiation completes the proof. □
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
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Sc. Norm. Super. Pisa, Cl. Sci. (4) 6 (1979), no. 1, pp. 151–184.
[2] D. Danielli and A. Haj Ali, “A two phase boundary obstacle-type problem for the bi-
Laplacian”, Nonlinear Anal., Theory Methods Appl. 214 (2022), article no. 112583 (26 pages).
[3] Y. Deng and Y. Li, “Regularity of the solutions for nonlinear biharmonic equations in RN ”,
Acta Math. Sci., Ser. B, Engl. Ed. 29 (2009), no. 5, pp. 1469–1480.
[4] G. Di Fratta and A. Fiorenza, “A short proof of local regularity of distributional solutions of
Poisson’s equation”, Proc. Am. Math. Soc. 148 (2020), no. 5, pp. 2143–2148.
[5] L. C. Evans, “Some new PDE methods for weak KAM theory”, Calc. Var. Partial Differ. Equ.
17 (2003), no. 2, pp. 159–177.
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(2009), no. 4, pp. 435–462.
[7] J. Frehse, “On the regularity of the solution of the biharmonic variational inequality”,
Manuscr. Math. 9 (1973), pp. 91–103.
[8] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprint
of the 1998 edition, Classics in Mathematics, Springer, 2001, pp. xiii+517.
[9] S. Mayboroda and V. Maz’ya, “Regularity of solutions to the polyharmonic equation in
general domains”, Invent. Math. 196 (2014), no. 1, pp. 1–68.
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Hessian regularity in Hölder spaces for a semi-linear bi-Laplacian equation

Perrine Jouteur

Burau representation of $B_4$ and quantization of the rational projective plane

Asaf Nachmias

Yuval Peres

Comptes Rendus
Mathématique
Asaf Nachmias and Yuval Peres
Every recurrent network has a potential tending to infinity
Volume 363 (2025), p. 793-798
Online since: 4 July 2025
https://doi.org/10.5802/crmath.758
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
2025, Vol. 363, p. 793-798
https://doi.org/10.5802/crmath.758
Research article / Article de recherche
Probability theory / Probabilités
Every recurrent network has a potential
tending to infinity
Tout réseau récurrent a un potentiel tendant vers l’infini
Asaf Nachmias a and Yuval Peres b
a Department of Mathematical Sciences, Tel Aviv University, Israel
b Beijing Institute of Mathematical Sciences and Applications, Beijing, China
E-mails: asafnach@tauex.tau.ac.il, yperes@gmail.com
Abstract. A rooted network consists of a connected, locally finite graph G , equipped with edge conductances
and a distinguished vertex o. A nonnegative function on the vertices of G which vanishes at o, has Laplacian 1
at o, and is harmonic at all other vertices is called a potential. We prove that every infinite recurrent rooted
network admits a potential tending to infinity. This is an analogue of classical theorems due to Evans and
Nakai in the settings of Euclidean domains and Riemannian surfaces.
Résumé. Un réseau enraciné est un graphe G localement fini et connexe, doté de conductances sur les arêtes
et d’un sommet distingué o. Un potentiel est une fonction sur les sommets de G qui est positive, s’annule
en o, a un laplacien 1 en o et est harmonique en tous les autres sommets. Nous prouvons que tout réseau
enraciné, récurrent et infini admet un potentiel tendant vers l’infini. Il s’agit d’un analogue de théorèmes
classiques de Evans et Nakai dans le cadre des domaines euclidiens et des surfaces riemanniennes.
Funding. This work was supported by ERC consolidator grant 101001124 (UniversalMap) as well as ISF
grants 1294/19 and 898/23.
Manuscript received 27 February 2025, revised and accepted 6 May 2025.
1. Introduction
A rooted network (G ,c,o) consists of a connected, locally finite graph G = (V ,E), endowed with
positive edge weights c : E → (0,∞) and a distinguished vertex o ∈V . The corresponding network
random walk with transition probabilities p(x, y) = cx y /cx is reversible, with reversing measure
cx =∑z∼x cxz . Define the network Laplacian of a function f : V →R by
∆ f (v) = ∑
x∼v
cv x
(
f (x)− f (v)).
A potential on (G ,c,o) is a function h : V → [0,∞) that is harmonic off o (i.e., ∆h(v) = 0 for
every vertex v ̸= o), normalized so that h(o) = 0 and ∆h(o) = 1. In every infinite rooted network
there is at least one potential, see Claim 5. Our focus is on recurrent networks.
We say that a function h : V → [0,∞) tends to infinity and write h → ∞, if for all M > 0 the
set
{
v ∈ V : h(v) ≤ M} is finite. Observe that on Z, the potentials h1(n) = max(n,0) for n ∈ Z and
h2(n) = max(−n,0) do not tend to infinity, but any non-trivial convex combination of them does.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
794 Asaf Nachmias and Yuval Peres
On Z2, on the other hand, it is well known that there is a unique potential h with h(v) =Θ(log|v |)
as v → ∞, see [6, Sections 12 and 31]. Recently there has been renewed interest in recurrent
potential theory, motivated by close connections to unimodular random networks and uniform
spanning trees in work of Berestycki, van Engelenburg and Hutchcroft, see [1,2].
Theorem 1. On any infinite recurrent rooted network there exists a potential which tends to
infinity.
The earliest theorem of this type is due to G. C. Evans [3] who showed that every compact
polar set K in Rd supports a probability measure such that its Newtonian potential is infinite
on K . Using Evans’ approach, Nakai [4] proved that every parabolic noncompact Riemannian
surface admits a function which is harmonic off a compact set and tends to infinity (i.e., its level
sets are compact). In Nakai’s proof, the set K is replaced by the Stone–Čech boundary of the
surface. Nakai’s argument can be adapted to prove Theorem 1, but the use of the Stone–Čech
compactification makes this argument non-constructive. Here we present a more constructive
probabilistic approach, using the minimax theorem.
Remark 2. Theorem 1 implies that if a rooted network admits a unique potential, then it must
tend to infinity. This answers a question raised following [1, Proposition 5.1].
Remark 3. It is well known that the existence of a potential tending to infinity implies recurrence.
Indeed, let h be a potential, let {X t } be the network random walk and consider the hitting time
τo = min
{
n ≥ 0 : Xn = o
}
. Then h(Xn∧τo ) is a nonnegative martingale, whence it almost surely
converges; this contradicts transience if h →∞.
2. Preliminaries
2.1. Dipoles and potentials
The Green kernel Go(x, y) of the random walk {Xn}n≥0 killed at o is defined by
Go(x, y) :=
∞∑
n=0
Px
(
Xn = y, τo > n
)
.
The corresponding Green density,
go(x, y) = Go(x, y)
cy
,
is symmetric by reversibility. A dipole from o to another vertex y in a network (G ,c) is a function
f : V → [0,∞), for which f (o) = 0 and
∀ v ∈V ∆ f (v) = 1o(v)−1y (v).
Claim 4. In a recurrent network, the Green density go( · , y) for the killed random walk is a dipole
from o to y. Moreover, supy go(x, y) <∞ for each x ∈V .
Proof. First observe that go( · , y) is harmonic off {o, y} and satisfies go(o, y) = 0. Second,
∆go( · , y)(y) =−1, since∑
x∼y
cx y
(
go(y, y)−go(x, y)
)=Go(y, y)− ∑
x∼y
cx y
cy
Go(x, y) = 1. (1)
Also,
∆go( · , y)(o) =
∑
x∼o
cox go(x, y) =
∑
x∼o
Go(y, x)
cox
cx
= 1, (2)
where the last equality is obtained by summing, over n ≥ 0, the identity∑
x∼o
Py
(
Xn = x, τo > n
)cox
cx
=Py
(
Xn+1 = o, τo > n
)=Py (τo = n +1)
and using recurrence. Thus, go( · , y) is a dipole.
Asaf Nachmias and Yuval Peres 795
Lastly, the identity (2) implies that go(x, y) ≤ c−1ox for every x ∼ o. More generally, since go( · , y)
is superharmonic off o, for any x ̸= o we have go(x, y) ≤ ∏ℓi=1 c−1xi−1xi , where x0, . . . , xℓ is a simple
path from o to x. □
Claim 5. The space of potentials on a recurrent infinite rooted (G ,c,o) is a nonempty, compact,
convex subset of RV endowed with the product topology.
Proof. By Claim 4, for any sequence of vertices yn →∞, Cantor diagonalization yields a subse-
quential pointwise limit of go( · , yn), and any such limit is a potential. Next, for any potential h
we have h(x) ≤ ∏ℓi=1 c−1xi−1xi , where x0, . . . , xℓ is a simple path from o to x, by the argument in the
proof of Claim 4, so the set of potentials is a closed subset of a compact product space. Finally,
convexity is obvious. □
2.2. The Doob transform determined by a potential
Any potential h defines a Markov chain on the state space Sh =
{
v : h(v) > 0} via the Doob h-
transform that has transition probabilities
ph(x, y) = p(x, y)h(y)
h(x)
.
The corresponding Markov chain {Yn}n≥0 is called the h-process. Write Phµ for the law of the h-
process with initial distribution µ and Ehµ for the corresponding expectation.
For any finite sequenceγ= (γ1, . . . ,γk ) in V , we writePγ1 (γ) :=
∏k
i=2 p(γi−1,γi ). Ifγ is contained
in Sh , we define P
h
γ1
(γ) similarly using ph(γi−1,γi ). We have the telescoping product
Phγ1 (γ) =Pγ1 (γ)
h(γk )
h(γ1)
. (3)
We will use the initial distribution µh defined by
µh(v) = cov h(v) for all v ∼ o. (4)
Observe that
Phµh (γ) = coγ1 h(γ1)Phγ1 (γ) = coγ1Pγ1 (γ)h(γk ) = coPo(oγ)h(γk ). (5)
There is a standard relation between the Green kernels of the h-process and the network
random walk killed at o.
Claim 6. Let h be a potential on a recurrent rooted network (G ,c,o). Then the h-process {Yn}n≥0
is transient. Moreover, its Green kernel Gh(x, y) =∑k≥0Phx (Yk = y), defined for x, y ∈ Sh , satisfies
Gh(x, y) =Go(x, y) h(y)
h(x)
, (6)
and for all v ∈ Sh
Gh(µh , v) = h(v)cv , (7)
where Gh(µ, v) :=∑x µ(x)Gh(x, v).
Proof. Transience will follow once we show Gh(x, y) <∞. To prove (6), we sum (3) over all paths
from x to y that avoid o. Note that by harmonicity of h off o, any such path is contained in Sh .
Next, (7) follows from (6) by
Gh(µh , y) =
∑
v
cov h(v)Gh(v, y) = h(y)
∑
v
cov Go(v, y) = h(y)cy ,
using (2) and the symmetry of go( · , · ) in the last step. □
796 Asaf Nachmias and Yuval Peres
Claim 7. Let h be a potential for a recurrent rooted network (G ,c,o) and consider the h-pro-
cess {Yn} started at v ∈ Sh . Then for every M > 0,
Phv
(
h(Yℓ) > M
)−→ 1 as ℓ→∞.
This was also observed in [2, Lemma 5].
Proof. Recall that {Xn} is the network random walk. By (3), we have
Phv
(
h(Yℓ) ≤ M
)= Ev [h(Xℓ)
h(v)
1{h(Xℓ)≤M and τo≥ℓ}
]
≤ M
h(v)
Pv (τo ≥ ℓ),
which tends to 0 as ℓ→∞ by recurrence. □
3. Proof of main theorem
Let d( · , · ) denote graph distance. For any integer r > 0, we set
B(o,r ) = {w : d(o, w) < r } and ∂B(o,r ) = {w : d(o, w) = r },
Next we state a formally weaker version of Theorem 1, where we can choose a different potential
for each radius, and show that it implies the theorem.
Lemma 8. LetΨ be the set of potentials on a recurrent infinite rooted network (G ,c,o). Then
M(r ) := max
ψ∈Ψ
min
v∈∂B(o,r )
ψ(v) −→∞ as r →∞. (8)
Remark 9. For every integer r > 0 and ψ ∈ Ψ, the minimum of ψ on V \ B(o,r ) is attained on
∂B(o,r ). Indeed, let {X t } be the network random walk started at w ̸∈ B(o,r ) and let τr be the
hitting time of ∂B(o,r ). Then ψ(X t∧τr ) is a positive martingale, so by recurrence and optional
stopping,
ψ(w) ≥ Ew
[
ψ(Xτr )
]≥ min
v∈∂B(o,r )
ψ(v).
Consequently, M(r ) = maxψ∈Ψminw∈V \B(o,r )ψ(w) is non-decreasing in r .
Proof of Theorem 1 given Lemma 8. For each integer r , let ψr be a potential where the maxi-
mum in the definition (8) of M(r ) is attained. For every n ≥ 1, choose rn such that M(rn) ≥ n2n .
By Claim 5,
h :=
∞∑
n=1
2−nψrn
is a potential. Next, if w ∉ B(o,rn), then h(w) ≥ 2−nψrn (w) ≥ n. Thus, for each n, the set{
v ∈V : h(v) < n} is finite. □
3.1. Preliminary lemmas
The following lemma shows that if the dipoles go( · , vn) converge to a potential h, then the
network random walks conditioned on τvn < τo converge weakly to the h-process. We actually
need this for dipole mixtures. Given a probability measure η on V , write
go( · ,η) :=
∑
v
η(v)go( · , v).
Lemma 10. Let (G ,c,o) be a recurrent rooted network. Let rn →∞ be a sequence of integers and for
each n let ηrn be a probability measure supported on ∂B(o,rn). Assume that the sequence go( · ,ηrn )
converges pointwise to a potential h as n → ∞. Then for any finite path γ in Sh starting at a
neighbor of o, we have ∑
v
ηrn (v)Po
(
oγ
∣∣τv < τ+o )−→Phµh (γ) as n →∞. (9)
Asaf Nachmias and Yuval Peres 797
Proof. First note that for v, w different from o, we have go(w, v) = Pw (τv < τo)go(v, v) and by
path reversal,
coPo(τv < τ+o ) = cvPv (τo < τ+v ) = cv Go(v, v)−1 = go(v, v)−1.
Therefore, if γ= (γ1, . . . ,γℓ) does not contain the vertex v , then
Po
(
oγ
∣∣τv < τ+o )=Po(oγ)Pγℓ (τv < τo)
Po(τv < τ+o )
= coPo(oγ)go(γℓ, v). (10)
Integrating this with respect to ηrn gives∑
v
ηrn (v)Po
(
oγ
∣∣τv < τ+o )= coPo(oγ)go(γℓ,ηrn ) −→ coPo(oγ)h(γℓ),
as n →∞ by our hypothesis. This yields the desired result by (5). □
Let h be a potential and set TM = inf
{
t > 0 : h(X t ) ≥ M
}
where {X t } is the network random
walk. Optional stopping for the nonnegative martingale h(X t∧τo ) gives Pv (TM < τo) ≤ h(v)/M .
Careful path reversal and conditioning yields the following.
Lemma 11. Let h be a potential for a recurrent rooted network (G ,c,o) and let v ̸= o be a vertex.
Then the network random walk {X t } satisfies
Po
(
h(Xℓ) ≥ M
∣∣τv < τ+o )≤ h(v)M ,
for all M > 0 and integer 0 < ℓ< d(o, v), where τ+o = min
{
t ≥ 1 : X t = o
}
.
Proof. We may assume that h(v) < M . Since ℓ< d(o, v) we have
Po
(
h(Xℓ) ≥ M
∣∣τv < τ+o )≤Po(TM < τv ∣∣τv < τ+o ). (11)
Recall that coPo(τv < τ+o ) = cvPv (τo < τ+v ) by path reversal. Similarly,
coPo(TM < τv < τ+o ) = cvPv (TM < τo < τ+v ),
so Po
(
TM < τv
∣∣τv < τ+o )=Pv (TM < τo ∣∣τo < τ+v ). (12)
Consider the random walk started at v and write L = max{t < τo : X t = v}. Then XL , . . . , Xτo is
distributed as a random walk started at v , conditioned on τo < τ+v , and stopped at o. Hence
Pv
(
TM < τo
∣∣τo < τ+v )=Pv (∃ s ∈ (L,τo) : h(Xs ) ≥ M)≤Pv (TM < τo). (13)
To bound the last probability we note that St = h(X t∧τo ) is a non-negative martingale, so by
optional stopping
h(v) = Ev S0 ≥ Ev STM ≥ MPv (TM < τo).
Combining this with (11), (12) and (13) concludes the proof. □
3.2. Proof of Lemma 8
Recall that Ψ denotes the set of potentials on (G ,c,o). Given an integer r > 0, let Υr be the
compact space of probability measures on ∂B(o,r ). Observe that for every ψ : V →R
min
v∈∂B(o,r )
ψ(v) = min
η∈Υr
∑
v
η(v)ψ(v). (14)
Next, we will apply Sion’s minimax theorem [5] to the function fr : Ψ×Υr →R, defined by
fr (ψ,η) =
∑
v∈∂B(o,r )
η(v)ψ(v).
As noted in Claim 5, the setΨ is a convex subset of the linear space RV endowed with the product
topology. Additionally, it is clear that fr is continuous and bilinear and that Υr is a compact
798 Asaf Nachmias and Yuval Peres
convex subset of the linear topological space R∂B(o,r ). Hence the assumptions of Sion’s theorem
hold, and we obtain, using (14), that M(r ) defined in (8) satisfies
M(r ) = max
ψ∈Ψ
min
η∈Υr
fr (ψ,η) = min
η∈Υr
max
ψ∈Ψ
fr (ψ,η). (15)
For each integer r > 0, let ηr ∈ Υr denote a minimizer of the right-hand side of (15). By
Claim 4, there exists a sequence rn → ∞ such that go( · ,ηrn ) converges pointwise to a function
h : V → [0,∞). Clearly h is a potential.
Fix m > 0. By Claim 7, there exists an integer ℓ such that Phµh
(
h(Yℓ) ≥ 2m
)≥ 1/2. We now apply
Lemma 10 and sum (9) over all paths γ= (γ1, . . . ,γℓ) ⊂ Sh such that h(γℓ) ≥ 2m. We infer that
lim
n→∞
∑
v
ηrn (v)Po
(
h(Xℓ) ≥ 2m
∣∣τv < τ+o )=Phµh (h(Yℓ) ≥ 2m)≥ 1/2.
Lemma 11 states that if ℓ< rn and v ∈ ∂B(o,rn), then
Po
(
h(Xℓ) ≥ 2m
∣∣τv < τ+o )≤ h(v)2m .
Combining the last two inequalities yields that
liminf
n→∞ frn (h,ηrn ) = liminfn→∞
∑
v
ηrn (v)h(v) ≥ m.
In conjunction with (15), this gives that liminfn M(rn) ≥ m for every m > 0. Since M(r ) is non-
decreasing (Remark 9), it must tend to infinity as r →∞. □
Acknowledgments
We thank Omer Angel, Ori Gurel-Gurevich and Fedja Nazarov for helpful discussions.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] N. Berestycki and D. van Engelenburg, “Harnack inequality and one-endedness of UST on
reversible random graphs”, Probab. Theory Relat. Fields 188 (2024), no. 1-2, pp. 487–548.
[2] D. van Engelenburg and T. Hutchcroft, “The number of ends in the uniform spanning tree
for recurrent unimodular random graphs”, Ann. Probab. 52 (2024), no. 6, pp. 2079–2103.
[3] G. C. Evans, “Potentials and positively infinite singularities of harmonic functions”,
Monatsh. Math. Phys. 43 (1936), no. 1, pp. 419–424.
[4] M. Nakai, “On Evans potential”, Proc. Japan Acad. 38 (1962), pp. 624–629.
[5] M. Sion, “On general minimax theorems”, Pac. J. Math. 8 (1958), pp. 171–176.
[6] F. Spitzer, Principles of random walk, Second edition, Graduate Texts in Mathematics,
Springer, 1976, pp. xiii+408.


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