Nathaël Alibaud

Del Félix

Jørgen Teso

Espen Robstad Endal

Jakobsen

Ë L Natha

F Alibaud

Del Élix

Espen R Endal

J Endal

E R Jakobsen

Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of R d

Keywords: 2010 Mathematics Subject Classification. 35B53, 35B10, 35J70, 35R09, 60G51, 65R20 Liouville theorem, Periodic solutions, Nonlocal diffusion operators, Propagation of maximum, Subgroups of R d, Irrational numbers N. ALIBAUD, F. DEL TESO

come   L'archive ouverte pluridisciplinaire

The classical Liouville theorem states that bounded functions u solving ∆u = 0 in the distributional sense are constant. We revisit this result for nonlocal (or anomalous) diffusion operators of the form (1)

L µ [u](x) := P.V.

ˆRd \{0} u(x + z) -u(x) dµ(z),
where µ ≥ 0 is symmetric and ´(|z| 2 ∧ 1) dµ(z) < +∞.

Our motivation goes back to the result of Courrège [START_REF] Courrège | Générateur infinitésimal d'un semi-groupe de convolution sur R n , et formule de Lévy-Khinchine[END_REF] on linear operators satisfying the positive maximum principle. These operators can have an integro-differential part which is precisely L µ . Our symmetry assumption on µ means that we are considering self-adjoint operators. Operators of the form (1) coincides with generators of symmetric pure-jump Lévy processes [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF][START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF]. A famous example is the fractional Laplacian -(-∆) α 2 which corresponds to α-stable processes. Other examples from biology, mechanics, etc., are convolution operators J * u -u (cf. [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF][START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF][START_REF] Brasseur | A counterexample to the Liouville property of some nonlocal problems[END_REF]), relativistic Schrödinger operators m α I -(m 2 I -∆) α 2 (cf. [START_REF] Fall | Unique continuation properties for relativistic Schrödinger operators with a singular potential[END_REF][START_REF] Fall | Liouville theorems for a general class of nonlocal operators[END_REF]), or finite difference discretization like (u(x + h) + u(x -h) -2u(x))/h 2 ≈ ∂ 2

x u(x), as well as other general similar discretizations in R d for local and nonlocal diffusion operators (cf. [START_REF] Del Teso | On distributional solutions of local and nonlocal problems of porous medium type[END_REF][START_REF] Del Teso | Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type[END_REF][START_REF] Del Teso | Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory[END_REF][START_REF] Del Teso | Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments[END_REF]).

There is a huge literature on the Liouville theorem and we focus our attention on nonlocal PDEs. Such a result is more or less understood for the fractional Laplacian or variants [START_REF] Landkof | Foundations of modern potential theory[END_REF][START_REF] Bogdan | Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes[END_REF][START_REF] Stinga | Harnack's inequality for fractional nonlocal equations[END_REF][START_REF] Cabré | Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates[END_REF][START_REF] Zhuo | A Liouville theorem for the fractional Laplacian[END_REF][START_REF] Chen | Some Liouville theorems for the fractional Laplacian[END_REF][START_REF] Fall | Entire s-harmonic functions are affine[END_REF][START_REF] Barrios | A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions[END_REF], certain Lévy operators [START_REF] Barlow | The Liouville property and a conjecture of De Giorgi[END_REF][START_REF] Priola | Liouville theorems for non-local operators[END_REF][START_REF] Serra | C σ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels[END_REF][START_REF] Serra | Regularity for fully nonlinear nonlocal parabolic equations with rough kernels[END_REF][START_REF] Ros-Oton | Boundary regularity for fully nonlinear integro-differential equations[END_REF][START_REF] Ros-Oton | Regularity theory for general stable operators[END_REF][START_REF] Del Teso | On distributional solutions of local and nonlocal problems of porous medium type[END_REF][START_REF] Del Teso | Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type[END_REF], relativistic Schrödinger operators [START_REF] Fall | Liouville theorems for a general class of nonlocal operators[END_REF], or convolution operators [START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF][START_REF] Brasseur | A counterexample to the Liouville property of some nonlocal problems[END_REF]. The techniques vary from Fourier analysis, potential theory, probabilistic methods, or classical PDEs arguments. Let us also refer to works on propagation of maximum which is a closely related topic (cf. [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF][START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF][START_REF] Shankar | A strong maximum principle for nonlinear nonlocal diffusion equations[END_REF][START_REF] Del Teso | On distributional solutions of local and nonlocal problems of porous medium type[END_REF][START_REF] Del Teso | Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type[END_REF][START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF][START_REF] Huang | Maximum principles for nonlocal parabolic Waldenfels operators[END_REF][START_REF] Brasseur | A counterexample to the Liouville property of some nonlocal problems[END_REF]).

As far we know, there is not yet any complete classification of nonlocal operators for which the Liouville property holds. This is precisely what we provide in this paper for operators of the form [START_REF] Alibaud | The Liouville theorem and linear operators satisfying the maximum principle. A complete characterization in the constant coefficient case[END_REF]. Our main theorem is a periodicity result for the bounded solutions of L µ [u] = 0 (cf. Theorem 4.11). It states that L µ [u] = 0 if and only if u is periodic with respect to the subgroup G(supp(µ)) generated by the support. In particular, L µ satisfies the Liouville property if and only if G(supp(µ)) is dense (cf. Theorem 4.13). If d = 1, we give an equivalent condition in terms of irrationality of the ratio between points in the support (cf. Theorem 3.7).

We also precisely identify the form of the operators L µ which do not satisfy the Liouville property. Roughly speaking, this property fails if and only if supp(µ) ⊆ R n × Z m , for some n < d, m ≤ d -n, and up to some change of coordinates. If d = 1 the corresponding measures µ are thus series of Dirac masses (cf. Proposition 4.15), and in the general case d ≥ 1 we get series of measures of dimension less or equal d -1 (cf. Theorem 4.17).

In particular, every one dimensional nonpurely discrete measure corresponds to a nonlocal operator satisfying the Liouville theorem. This includes measures whose supports contain intervals like absolutely continuous measures (e.g. the fractional Laplacian) or singular continuous Cantor measures. Certain atomic measures may have the Liouville property as well like e.g. µ(z) = ± (δ ±a (z) + δ ±b (z)) if b a / ∈ Q. To easily verify whether the Liouville theorem holds if d = 1, we provide an algorithmic procedure in relation with irrational numbers (cf. Corollary 3.8).

For general dimension d ≥ 1, measures containing balls in their supports like absolutely continuous measures enjoy the Liouville property. Other interesting examples are mean value operators with which we can recover the classical Liouville theorem (cf. Example 5.4), purely atomic measures with few points in the supports satisyfing a so-called Kronecker's approximation condition from group theory (cf. Corollary 5.7), or certain nonstandard discretizations of the Laplacian on nonuniform grids (cf. Example 5.10).

Our proofs are based on simple reasoning on irrational numbers, certain results on groups (cf. e.g. [START_REF] Effros | Dimension groups and their affine representations[END_REF][START_REF] Godement | of Collection Enseignement des Sciences [Collection: The Teaching of Science[END_REF][START_REF] Martinet | Perfect lattices in Euclidean spaces[END_REF][START_REF] Gonek | Kronecker's approximation theorem[END_REF] and the references therein), and pure PDEs arguments of propagation of maximum (cf. [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF][START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF][START_REF] Shankar | A strong maximum principle for nonlinear nonlocal diffusion equations[END_REF][START_REF] Del Teso | On distributional solutions of local and nonlocal problems of porous medium type[END_REF][START_REF] Del Teso | Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type[END_REF][START_REF] Huang | Maximum principles for nonlocal parabolic Waldenfels operators[END_REF][START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF][START_REF] Brasseur | A counterexample to the Liouville property of some nonlocal problems[END_REF]).

Note finally that our periodicity result Theorem 4.11 not only implies the Liouville theorem for sufficiently nice L µ , but also charaterizes the bounded solutions of L µ [u] = 0 in every case. It is thus a natural extension of the classical Liouville theorem for general nonlocal operators. Moreover, it is possible to obtain a similar result for general generators of Lévy processes, or equivalently general linear operators with constant coefficients satisfying the positive maximum principle a la Courrège. This wider class includes local and nonlocal operators possibly nonself-adjoint. In a work in progress [START_REF] Alibaud | The Liouville theorem and linear operators satisfying the maximum principle. A complete characterization in the constant coefficient case[END_REF], we are writing a complete characterization of such operators satisfying the Liouville theorem as a byproduct of an extended version of the periodicity result Theorem 4.11.

Outline of the paper. General assumptions are presented in Section 2. Section 3 concerns the case where d = 1 (1-d for short) whose main results are Theorem 3.7 and Corollary 3.8 followed by representative examples. These results are obtained independently of the case d ≥ 1 (multi-d for short). The analysis in the latter case starts in Section 4 with the general periodicity result Theorem 4.11. The classifications for the Liouville theorem are given in Theorems 4.13 and 4.17 General notation. The symbol ∧ denotes min. The symbol ⊆ denotes inclusion of sets and ⊆ is the negation. We define lim := lim sup. Let B r (x) denote the ball of center x and radius r.

A point x ∈ S ⊆ R is isolated in the set S if there exists r > 0 such that B r (x) ∩ S = {x}. If every point of S is isolated, S is discrete. The function spaces C ∞ b (R d ) and C ∞ c (R d )
µ d Ω \ D = 0.
The support of a discrete measure does not need to be discrete. Here is a standard result that will be needed (cf. e.g. [START_REF] Rudin | Real and complex analysis[END_REF]).

Standard assumptions and examples

We assume that L µ is an operator of the form (1) for some Radon measure

µ on R d \ {0} satisfying (A µ )
µ ≥ 0, µ is symmetric, and ˆ(|z| 2 ∧ 1) dµ(z) < +∞.

Remark 2.1. The symmetry means that µ(B) = µ(-B) for any Borel set.

In particular, L µ is self-adjoint and the principal value in (1) makes sense for u ∈ C 2 b (R d ) via the formula

L µ [u](x) = lim r→0 ˆ|z|>r u(x + z) -u(x) dµ(z) = ˆ0<|z|<r 0 u(x + z) -u(x) -z • ∇u(x) dµ(z) + ˆ|z|≥r 0 u(x + z) -u(x) dµ(z) valid for any r 0 > 0. For general u ∈ L ∞ (R d ) we say that L µ [u] = 0 in D (R d ) if (3) ˆRd u(x)L µ [ψ](x) dx = 0 ∀ψ ∈ C ∞ c (R d ).
Remark 2.2. By the Lebesgue decomposition theorem (cf. e.g. [START_REF] Rudin | Real and complex analysis[END_REF]), µ can be written in a unique way as

dµ(z) = f (z) dz + dµ s (z) + dµ d (z),
for some To identify the discrete measure, we have used Lemma 1.1 with a rewritting of the formula taking into account the symmetry.

|z| (d+α)/2 dz,
where K is a modified Bessel function (cf. [START_REF] Fall | Unique continuation properties for relativistic Schrödinger operators with a singular potential[END_REF]Remark 7.3]); (c) the convolution operator u → J * u -u with J ≥ 0, J(z) = J(-z), and ´J(z) dz = 1, corresponding to the absolutely continuous Lévy measure dµ(z) = J(z) dz (cf. e.g. [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF][START_REF] Andreu-Vaillo | Nonlocal diffusion problems[END_REF]) ;

(d) the 1-d discrete Laplacian ∆ h u(x) := u(x + h) + u(x -h) -2u(x) h 2 corresponding to the atomic Lévy measure µ(z) = 1 h 2 (δ h (z) + δ -h (z))
; (e) the general multi-d discrete and self-adjoint diffusion operator (cf. [START_REF] Del Teso | Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory[END_REF][START_REF] Del Teso | Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments[END_REF])

L h [u](x) := a∈D ω(a) u(x + a) -u(x -a) -2u(x)
for some D and ω(•) as in Remark 2.2, corresponding to the atomic Lévy measure µ(z) = a∈D ω(a)(δ a (z) + δ -a (z)); (f) or the mean value operator associated to the Laplacian

M[u](x) := 1 c d ˆ∂B 1 (0) u(x + z) -u(x) dσ(z),
where dσ(z) is the surface measure of the unit sphere and c d its area.

Characterization in R in terms of irrational numbers

This section is devoted to the dimension d = 1. The main results Theorem 3.7 and Corollary 3.8 are given in Section 3.3 and their proofs in Sections 3.4 and 3.5.

3.1.

Reminders on integers and irrational numbers. For any n, m ∈ Z\{0}, we let gcd(n, m) denote the greatest common positive divisor between n and m. The integers n and m are coprimes if gcd(n, m) = 1. Let us also precise that we use the notation N + := N \ {0}.

Let us recall the Bézout identity that will be needed later (cf. e.g. [START_REF] Godement | of Collection Enseignement des Sciences [Collection: The Teaching of Science[END_REF]). Here is another basic result that will be needed (cf. e.g. [START_REF] Godement | of Collection Enseignement des Sciences [Collection: The Teaching of Science[END_REF]). Lemma 3.2. For any a, b ∈ R \ {0}, aZ + bZ = R if and only if b a ∈ Q. Lemma 3.2 will be fundamental for our analysis in dimension d = 1, and for completeness we recall its proof in Appendix B.

3.

2. An explicit condition on the Lévy measure. Along with (A µ ), the following condition will turn out to be necessary and sufficient for L µ in R to satisfy the Liouville theorem:

(A L ) ∃a ∈ supp(µ) such that sup b∈supp(µ) Q(a, b) = +∞,
where Hence

Q : R 2 \ {(0, 0)} → N + ∪ {+∞} is defined as (4) Q(a, b) :=    +∞ if b a ∈ Q, q if b a = p q ∈ Q with |p|, q ∈ N + and gcd(p, q) = 1. Note that: Lemma 3.3. Let µ be a nonnegative Radon measure on R \ {0} such that (A L ) holds. Then sup b∈supp(µ) Q(a, b) = +∞ for all a ∈ supp(µ).
Q(a, b) = Q(a 0 , b) gcd(Q(a 0 , a), Q(a 0 , b)) ≥ Q(a 0 ,b) Q(a 0 ,a) |p a | gcd(|p a |, |p b |) ≥1 , and therefore since 1 ≤ Q(a 0 , a) < +∞, sup b∈supp(µ) Q(a, b) ≥ sup b∈supp(µ) Q(a 0 , b) Q(a 0 , a) = +∞.
The proof is complete.

It will be useful to refer to the negated version of (A L ):

(NA L ) ∀a ∈ supp(µ), sup b∈supp(µ) Q(a, b) < +∞.
Remark 3.4. By Lemma 3.3, it suffices that sup b∈supp(µ) Q(a, b) < +∞ for just one a ∈ supp(µ) to deduce that (NA L ) is satisfied.

Remark 3.5. As concerning (A L ), there are two ways of satisfying this assumption: (i) Either the ratio b a is irrational for some a, b ∈ supp(µ), or (ii) there are a ∈ supp(µ) and {b n } n ⊆ supp(µ) such that q n → +∞ as n → +∞ where bn a = pn qn for |p n |, q n ∈ N + with gcd(p n , q n ) = 1. Remark 3.6. Later, we will see that (A L ) (resp. (NA L )) is a necessary and sufficient condition on supp(µ) to generate a dense (resp. discrete) subgroup of R (cf. Remark 4.14(a)-(b)).

Main results and examples.

Our first main result in R is the following: Theorem 3.7. Assume d = 1, (A µ ), and let L µ be defined by [START_REF] Alibaud | The Liouville theorem and linear operators satisfying the maximum principle. A complete characterization in the constant coefficient case[END_REF]. The following statements are equivalent: (i) If supp(µ) contains only a finite number of points,

(a) If u ∈ L ∞ (R) satisfies L µ [u] = 0 in D (R), then u is constant. (b) µ satisfies the property (A L ).
supp(µ) = {-a N , . . . , -a 1 , a 1 , . . . , a N } with 0 < a 1 < • • • < a N ,
then we have the Liouville property if and only if an a 1 ∈ Q for some a n ; (ii) if supp(µ) has an infinite number of points,

supp(µ) = {-a n , a n } n≥1 with 0 < a 1 < a 2 < . . . ,
then we write an a 1 = pn qn in irreducible form and we have the Liouville property if and only if sup n≥1 q n = +∞.

Proof. Part (b) is an immediate consequence of Theorem 3.7 and Remarks 3.4 and 3.5. For (a), we need Corollary 3.13 that will be proved later. Corollary 3.13 implies that if (NA L ) holds then supp(µ) ⊆ gZ for some g > 0.

But such an inclusion is not possible if supp(µ)

R has an accumulation point.

This means that necessarily (A L ) holds and we conclude again by Theorem 3.7. Examples are the fractional Laplace measure, relativistic Schrödinger operators, convolution operators, or even purely Cantor measures.

Example 3.10 (Discrete measures satisfying Liouville). (a) A nice example of a discrete operator satisfying the Liouville theorem is the following nonstandard discretization of the Laplacian

L h [u] = u(x + h) + u(x -h) -2u(x) 2h 2 + u(x + πh) + u(x -πh) -2u(x) 2π 2 h 2 .
(

) 5
This follows from (bi) of Corollary 3.8 because πh h ∈ Q. We refer the reader to Section 5.

(µ(B 1 (0)) = +∞ but ´(|z| 2 ∧ 1) dµ(z) < +∞) like µ = n≥1 δ 1 n + δ -1 n here ˆ(|z| 2 ∧ 1) dµ(z) = n≥1 2 n 2 < +∞ .
The Liouville property follows from (a) of Corollary 3.8 (0 is an accumulation point of supp(µ) R ). See Section 5.1 for a similar discussion in multi-d. (c) A last illustrative example is given by a discrete measure with infinitely many rational and isolated points:

µ = n≥1 ω n δ an + δ -an
for {a n } n≥1 and {ω n } n≥1 satisfying

ω n > 0, n≥1 ω n < +∞, and a n = n 2 + 1 n .
Now the Liouville property follows from (bii) of Corollary 3.8. Indeed gcd(n 2 + 1, n) = 1 by the Bézout identity Lemma 3.1 and therefore since an

a 1 = n 2 +1 2n , we have Q(a 1 = 1, a n ) ≥ n → +∞ as n → +∞.
Example 3.11 (Counterexamples for standard finite difference operators). By Corollary 3.8(bi), the usual discretization of the 1-d Laplacian

∆ h u(x) = 1 h 2 u(x + h) + u(x -h) -2u(x)
does not have the Liouville theorem, neither will any other discretization of the form

L h [u](x) = n≥1 ω n u(x + nh) + u(x -nh) -2u(x) .
To preserve the Liouville property for a numerical scheme, one could consider nonuniform grids with irrational ratios between nodes as in (5).

3.4.

Proof of the necessary condition. Let us now prove that (a) ⇒ (b) in Theorem 3.7. We will proceed by the contrapositive statement. This means that if (NA L ) holds, we can find a nonconstant function U ∈ L ∞ (R) such that L µ [U ] = 0 in D (R). Let us start with elementary results.

The first lemma shows that 0 / ∈ supp(µ) R .

Lemma 3.12. Assume (A µ ), d = 1, and (NA L ). Then either µ = 0 (trivial measure) or

a 1 := inf {b > 0 : b ∈ supp(µ)} > 0.
Proof. Assume by contradiction that µ is not the trivial measure and a 1 = 0. Then, by definition there exists a sequence {b n } n ⊆ supp(µ) ∩ (0, +∞) such that b n → 0 as n → +∞. Now take any a ∈ supp(µ)∩(0, +∞). If

bn 0 a ∈ Q for some n 0 ∈ N + then Q(a, b n 0 ) = +∞ which contradicts (NA L ).
Otherwise, let q n ∈ N + be such that bn a = pn qn with p n ∈ N + and gcd(p n , q n ) = 1, and therefore

Q(a, b n ) = q n . Then, since p n ≥ 1 and b n → 0, we have sup b∈supp(µ) Q(a, b) ≥ Q(a, b n ) = q n = a p n b n ≥ a b n n→∞ -→ +∞,
and again we reach a contradiction with (NA L ).

Corollary 3.13. Assume (A µ ), d = 1, and (NA L ). Then there is g > 0 such that supp(µ) ⊆ gZ.

Proof. If µ = 0 then supp(µ) = ∅ and the result is trivial. In the other case, let a 1 > 0 be given by Lemma 3.12. By (NA L )

sup b∈supp(µ) Q(a 1 , b) < +∞,
where Q(a 1 , b) ∈ N + for any b ∈ supp(µ) by the definition (4). This supremum is therefore a maximum and there exists q ∈ N + such that

q = max b∈supp(µ) Q(a 1 , b).
For any b ∈ supp(µ), b a 1 ∈ Q by (NA L ) and there exists an integer

p b such that b = a 1 b a 1 = a 1 p b Q(a 1 , b) = a 1 q! q! p b Q(a 1 , b) .
Note that Q(a 1 , b) divides q! by the above definition of q. Defining g := a 1 q! > 0 we conclude that b ∈ gZ. Since b was arbitrary in supp(µ) and the choice of g did not depend on b, the proof is complete.

The following proposition completes the proof of (a) ⇒ (b) in Theorem 3.7. Proposition 3.14. Assume (A µ ), d = 1, and (NA L ). Then there exists a nonconstant function U ∈ C ∞ b (R) such that L µ [U ] = 0 in R. Proof. By Corollary 3.13, there exists g > 0 such that supp(µ) ⊆ gZ and thus µ is a discrete measure. By Lemma 1.1 and Remark 2.2,

L µ [U ](x) = n≥1 ω n U (x + ng) + U (x -ng) -2U (x) ,
with ω n ≥ 0 satisfying n≥1 ω n < +∞. Hence, every g-periodic U satisfy L µ [U ] = 0 like e.g. U (x) = cos( 2π g x).

3.5. Proof of the sufficient condition. We will now show the Liouville theorem given by the implication (b) ⇒ (a) in Theorem 3.7. Roughly speaking, we will use techniques from [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF][START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF][START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF] (cf. Theorem 3.18 and Lemma 3.20) to deduce that u is constant on some abstract set, that we will rigorously identify to be the whole domain R under (A L ). Later, we will ameliorate the techniques of [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF][START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF][START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF] to get a more general periodicity result valid for any dimension d ≥ 1 (cf. Theorem 4.11). This general result will imply the multi-d Liouville theorem as a byproduct (cf. Theorem 4.13). But let us first use more standard arguments and proceed in a serie of representative examples to shed light on the main ideas.

Example 1: Only two points with irrational ratio. Let us consider a prototypical operator satisfying (A L ) as in Remark 3.5(i). Let

(6) µ = ω 1 δ a + δ -a + ω 2 δ b + δ -b for a, b, ω 1 , ω 2 > 0 with b a ∈ Q.
Here supp(µ) = {-b, -a, a, b} since a = b.

Proposition 3.15. Assume µ is given by (6

) and u ∈ C ∞ b (R) solves L µ [u] = 0 in R. If u has a global maximum then u is constant.
Here the proof easily follows from the reasoning in [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF][START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF] and Lemma 3.2. Let us give detail for completeness.

Proof of Proposition 3.15. Without loss of generality assume that

M := u(0) = max u. First note that 0 = L µ [u](0) = ω 1 ± (u(±a) -M ) + ω 2 ± (u(±b) -M ),
and since each term in the sum is nonpositive by definition of w 1 , w 2 and M , we have that u(±a

) = u(±b) = M . Then if u(mb + ka) = M for some m, k ∈ Z, again 0 = L µ [u](mb + ka) = ω 1 ± (u(mb + (k ± 1)a) -M ) + ω 2 ± (u((m ± 1)b + ka) -M ),
and again by the nonpositivity of the terms we get

u(mb + (k ± 1)a) = u((m ± 1)b + ka) = M.
By induction, we find that

u(x) = M = max u ∀x ∈ aZ + bZ.
Since u is continuous the proof is complete by Lemma 3.2.

Example 2: Infinitely many points with asymptotic irrational ratio. Let us consider a prototypical operator satisfying (A L ) as in Remark 3.5(ii). Let ( 7)

µ(z) = n≥1 ω n δ an (z) + δ -an (z) with ω n > 0, n≥1 (a 2 n ∧ 1)ω n < +∞,
as well as

aZ + bZ = a Z + b a Z = a Z + p Q(a, b) Z = a Q(a, b) (Q(a, b)Z + pZ) = a Q(a, b) Z.
From that identity and (A L ) the rest of the proof is obvious.

Proof of Proposition 3.16. Without loss of generality assume that

M := u(0) = max u.
Let us again propagate this maximum as in [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF][START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF]. First note that

0 = L µ [u](0) = n≥1 ω n u(±a n ) -M ≤0 .
so that every term is 0, i.e u(±a n ) = M for all n ≥ 1. Now if u(ma n 1 + ka n 0 ) = M for some m, k ∈ Z, n 0 ≥ 1, and n 1 ≥ 1, then

0 = L µ [u](ma n 1 + ka n 0 ) = n≥1 ± ω n u(ma n 1 + ka n 0 ± a n ) -M ≤0 ,
and every term in the sum is again 0. In particular, for n = n 0 , n 1 , we get

u(ma n 1 + (k ± 1)a n 0 ) = u((m ± 1)a n 1 + ka n 0 ) = M.
By induction, this shows that for all n 0 ≥ 1 and n 1 ≥ 1, we have

u(x) = M = max u ∀x ∈ a n 0 Z + a n 1 Z.
Since ( 7)-( 8) imply (A L ), we can then apply Lemma 3.17 to conclude the proof.

Case of general measures satisfying (A L ). We are now ready to prove that (b) ⇒ (a) in Theorem 3.7. Let us first recall a general result from [START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF] (cf. [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF]) which will give us a way of computing the set on which we can propagate the maximum. Let us state it in multi-d for later use.

Theorem 3.18 (Thm. 2 in [START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF]).

Assume (A µ ), d ≥ 1, u ∈ C ∞ b (R d ) satisfies L µ [u] = 0 in R d ,
and u has a global maximum at x 0 . Then u = u(x 0 ) on x 0 + ∪ n≥0 A n where

A 0 = {0} and A n+1 = A n + supp(µ) ∀n ∈ N.
Remark 3.19. This result apply for fully nonlinear PDEs in the context of possibly irregular viscosity subsolutions (cf. [START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF] for more details). Theorem 3.18 is roughly speaking for the case where u reaches a global maximum as assumed in Propositions 3.15 and 3.16. For the general case, we need a technique developed in Step 1 of the proof of Lemma 7.2 in [START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF], that we recall under the form of a lemma. Lemma 3.20 (cf. [START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF]). Assume (A µ ) and

∪ k≥0 A k = R where ∪ k≥0 A k is defined in Theorem 3.18. If v ∈ C ∞ b (R) satisfies L µ [v] = 0 in R and {x n } n is a sequence such that lim n→+∞ v(x n ) = sup v, then for any R ≥ 0, lim n→+∞ min x∈B R (xn) v(x) = sup v.
Remark 3.21. This result works in R d but we will not need it. The lim is also in fact a true limit but we will not need it either.

Proof. Define v n (x) := v(x + x n ) where v ∈ C ∞ b (R).
Then by the Arzelà-Ascoli theorem, v n → v ∞ locally uniformly on R for some v ∞ along a subsequence (denoted as the whole sequence to simplify). Taking another subsequence if necessary, we can assume that the derivatives up to second order locally uniformly converge too. Hence, we can easily take the limit in the equation

L µ [v n ] = 0 to deduce that L µ [v ∞ ] = 0 in R. Moreover, v ∞ ≤ sup v =: M and v ∞ (0) = lim n→+∞ v(x n ) = M . By Theorem 3.18, we conclude that v ∞ = M on ∪ k≥0 A k and by assumption ∪ k≥0 A k = R. Hence for each R ≥ 0, max B R (xn) |v -M | = max B R (0) |v n -v ∞ | → 0 as n → +∞ by local uniform convergence of v n . In particular min B R (xn) v → M (along a subsequence).
Proof of Theorem 3.7. The implication (a) ⇒ (b) was already proved in Section 3.5 (cf. Proposition 3.14). Let us consider the reciprocal assertion. We thus assume that (b) holds, u is merely bounded and solves L µ [u] = 0 (in R), and we would like to prove that u is a.e. equaled to a constant. 1. Reduction of the proof to smooth functions.

Let ρ ε be a mollifier such that ρ ε ∈ C ∞ c (R) and ρ ε * u → u in D (R) as ε → 0. If all ρ ε * u are constant, it will follow that u is constant since u x will be zero as limit in

D (R) of (ρ ε * u) x = 0. But ρ ε * u ∈ C ∞ b (R)
and we can also see that L µ [ρ ε * u] = 0 by taking ψ(y) = ρ ε (x -y) in (3) for fixed x. Hence, if we can prove Theorem 3.7[(b) ⇒ (a)] for smooth functions like ρ ε * u, the result will follow for merely bounded functions like u. Let us thus continue by assuming without loss of generality that u

∈ C ∞ b (R) solves L µ [u] = 0 in R.

Uniform lower bounds on u x .

In the rest of the proof, we will prove that u x = 0 which implies that u is constant. We will only show that M := sup u x ≤ 0, since v = -u solves L µ [v] = 0 and the same reasoning will automatically give us that inf u x = -sup v x ≥ 0. Let us start from the equation for u x obtained by differentiating ∂

x (L µ [u]) = 0, L µ [u x ] = 0 in R. Let {x n } n ⊆ R and {ε n } n ⊆ (0, +∞) such that u x (x n ) = M -ε n ,
where ε n → 0 as n → +∞. Let now ∪ k≥0 A k be defined in Theorem 3.18 i.e.

A k = {0} + supp(µ) + • • • + supp(µ) k times . Since supp(µ) = -supp(µ), it is easy to see that aZ + bZ ⊆ ∪ k≥0 A k ∀a, b ∈ supp(µ).
When (A L ) holds, Lemma 3.17 then implies that R = a,b∈supp(µ) The following result justifies the introduction of the notion of lattices.

(aZ + bZ) ⊆ k≥0 A k .
Lemma 4.5. A subgroup G of R d is discrete if and only if it is a relative lattice.
This lemma is a consequence of the following more general result (cf. e.g. [27, Thm. 1.1.2]):

Theorem 4.6. A closed subgroup G of R d is of the form G = V ⊕Λ, for some vector space V ⊆ R d and relative lattice Λ ⊆ R d such that V ∩Span R Λ = {0}.
In particular:

Corollary 4.7. A subgroup G of R d is dense if and only if G ⊆ H + cZ for all vector spaces H ⊆ R d of codimension 1 and c ∈ R d .
Remark 4.8. In dimension d = 1, G is either dense or discrete i.e. of the form G = gZ for some g ≥ 0 (g = 0 corresponding to G = {0}).

Periodicity of the solutions of

L µ [u] = 0. Definition 4.9. Given S ⊆ R d , we say that u ∈ L ∞ (R d ) is S-periodic if ˆRd u(x + s) -u(x) ψ(x) dx = 0 ∀s ∈ S, ∀ψ ∈ C ∞ c (R d ).
Remark 4.10. If u is continuous, then u is S-periodic if and only if it is s-periodic for any s ∈ S.

Here is our general periodicity result for arbitrary Lévy measures satisfying only the standard assumption (A µ ).

Theorem 4.11. Assume (A µ ) and let u ∈ L ∞ (R d ). Then L µ [u] = 0 in D (R d ) if and only if u is G(supp(µ))-periodic almost everywhere.
Proof. For notational simplicity, denote G := G(supp(µ)).

Case u smooth. Let us first show the theorem for

u ∈ C ∞ b (R d ). If u is G-periodic then for all x ∈ R d we have that u(x) = u(x + z) for all z ∈ supp(µ) ⊆ G. Then L µ [u](x) = P.V. ˆRd \{0} u(x + z) -u(x) dµ(z) = P.V.
L µ [v n ] = 0 in R d . Since v ∈ C ∞ b (R d ),
Arzelà-Ascoli theorem implies that there exists v ∞ such that v n → v ∞ locally uniformly (up to a subsequence), and taking another subsequence if necessary, we can assume that the derivatives up to second order converge and pass to the limit in the equation

L µ [v n ] = 0 to deduce that L µ [v ∞ ] = 0 in R d . Moreover, v ∞ attains its maximum at x = 0 since v ∞ ≤ M and v ∞ (0) = lim n→+∞ v n (0) = lim n→+∞ v(x n ) = M.
Thus by Theorem 3.18 we have v ∞ = M in ∪ k≥0 A k , where

A k = {0} + supp(µ) + • • • + supp(µ) k times .
Since supp(µ) = -supp(µ), it is easy to see that ∪ k≥0 A k is a subgroup containing A 1 = supp(µ) and therefore

G = G(supp(µ)) ⊆ ∪ k≥0 A k . It follows that v ∞ = M in G.
Arguing with {u n } n as for {v n } n , there is some u ∞ such that u n → u ∞ as n → +∞ locally uniformly (up to a further subsequence if necessary to get the convergence of u n and v n along the same sequence). By construction

v ∞ (x) = u ∞ (x + g) -u ∞ (x). For any m ∈ Z, we take x = mg to get M = v ∞ (mg) = u ∞ ((m + 1)g) -u ∞ (mg). By iteration u ∞ ((m + 1)g) = u ∞ (mg) + M = . . . = u ∞ (0) + (m + 1)M.
But since u ∞ is bounded, then the only choice is M = 0 and thus v ≤ M = 0. A similar arguments shows that v ≥ 0 and completes the proof when

u ∈ C ∞ b (R d ).
2. Case u merely bounded.

We again consider the convolution ρ ε * u with a standard mollifier. Recall that

ρ ε * u → u in L 1 loc (R d ) as ε → 0 and that ρ ε * u ∞ ≤ u ∞ . Now if u is G-periodic, then every ρ ε * u is also G-periodic: (ρ ε * u)(x + g) = (ρ ε * u(• + g))(x) = (ρ ε * u)(x),
and thus L µ [ρ ε * u] = 0 in R d by the previous step. Taking any test function

ψ ∈ C ∞ c (R d ), it is easy to see that L µ [ψ] ∈ C ∞ b ∩ L 1 (R d )
, and this is enough to pass to the limit in

0 = ˆRd L µ [ρ ε * u]ψ dx = ˆRd L µ [ψ](ρ ε * u) dx → ˆRd L µ [ψ]u dx as ε → 0. This proves that L µ [u] = 0 in D (R d ). Conversely, if L µ [u] = 0 in D (R d ) then every ρ ε * u satisfy L µ [ρ ε * u] = 0 in R d .
By the previous step, every ρ ε * u is G-periodic and at the limit ε → 0 we obtain that for any g ∈ G and

ψ ∈ C ∞ c (R d ), 0 = ˆRd ρ ε * u(x + g) -ρ ε * u(x) ψ(x) dx → ˆRd u(x + g) -u(x) ψ(x) dx.
The proof is complete.

Remark 4.12. Notice that to show Theorem 4.11, we did not use at all any lower bound on balls as in Lemma 3.20! However, Theorem 4.11 is very general and will have many consequences on the Liouville theorem.

4.3.

The Liouville theorem as a byproduct of periodicity. We are now ready to give multi-d necessary and sufficient conditions to have the Liouville theorem. G(supp(µ)) ⊆ H + cZ, for some vector space H ⊆ R d of codimension 1 and c ∈ R d . We can assume that c / ∈ H because if not we will have [START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF] for any c / ∈ H. In particular,

R d = H ⊕ Span R {c} and for any x ∈ R d there exists a unique pair (x H , λ x ) ∈ H × R such that x = x H + λ x c.
Take now U (x) = cos(2πλ x ) and note that for any h ∈ H and k ∈ Z,

x + h + ck = (x H + h) ∈H + (λ x + k) ∈R c, so that U (x + h + ck) = cos(2π(λ x + k)) = cos(2πλ x ) = U (x).
This proves that U is H + cZ-periodic and thus G(supp(µ))-periodic. By Theorem 4.11, L µ [U ] = 0 and the proof is complete. 4.4. The class of operators not satisfying Liouville. We can have a clearer view of the operators which satisfy the following negated version of the Liouville theorem:

(10) There is a nonconstant u ∈ L ∞ (R d ) such that L µ [u] = 0 in D (R d ).
To understand the general result in mutli-d, it could be useful to first explain what happens if d = 1.

(11) L µ [u](x) = n≥1 ω n u(x + ng) + u(x -ng) -2u(x) ,
for some g > 0 and ω n ≥ 0 such that n≥1 ω n < +∞, and where (11) can be understood in D (R) or equivalently as a normally convergent serie in L ∞ (R).

Remark 4.16. The largest possible g > 0 for which we can write L µ in the form ( 11) is uniquely determined as the generator of G(supp(µ)). The weights are also entirely determined as ω n = µ({ng}) (cf. Lemma 1.1 and Remark 2.2).

We omit the detailed verification of Proposition 4.15 because it is in fact an immediate consequence of the arguments already used in the proof of Proposition 3.14. Let us focus instead on the general case d ≥ 1 where we may have series of more general measures than Dirac masses. Theorem 4.17. Assume d ≥ 1 and (A µ ). Then L µ satisfies (10) if and only if it is of the following form:

L µ [u](x) = P.V. ˆz∈V \{0} u(x + z) -u(x) dµ 0 (z) + a∈Λ\{0} ˆz∈V +a u(x + z) -u(x) dµ a (z), (12)
for some vector space V ⊆ R d , relative lattice Λ ⊆ R d , and family {µ a } a∈Λ of Radon measures on R d \ {0} such that

V = R d and V ∩ Span R Λ = {0}, (13)
µ a ≥ 0 and µ a (•) = µ -a (-•), (14) supp(µ a ) ⊆ V + a, and (15)
ˆ(|z| 2 ∧ 1) dµ 0 (z) + a∈Λ\{0} µ a (R d ) < +∞, (16)
where [START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF] has to be understood in D (R d ) and ( 14)-( 15) hold for all a ∈ Λ.

Remark 4.18. (a) A representative example with V = H of codimension 1 is given in Figure 2.

(b) In ( 12), the Lévy measure of L µ satisfies µ = a∈Λ µ a where Λ is a relative lattice thus a countable set (cf. Remark 4.4(a)). (c) By ( 13) and [START_REF] Del Teso | On distributional solutions of local and nonlocal problems of porous medium type[END_REF], supp(µ) ⊆ V ⊕ Λ = ∪ a∈Λ (V + a) which is a periodic superposition of parrallel affine subspaces V + a of dimension d V < d (cf. Figure 2). (d) Each µ a is supported by the affine space V + a ≈ R d V , and the theorem is written in such a way that some µ a could be the zero measure. In particular, supp(µ) may be just a subset of V ⊕ Λ. (e) The triplet (V, Λ, {µ a } a∈Λ ) in the decomposition ( 12) is however unique if we require also to have G(supp(µ)) = V ⊕Λ with V ⊥ Λ (cf. Corollary A.4 in Appendix A). (f) By ( 14), the measure µ 0 is symmetric and µ = a∈Λ µ a is as well although every other µ a may not be symmetric (cf. Figure 2). (g) By ( 16), every µ a are bounded except eventually µ 0 which is the only measure whose support may contain the singularity z = 0 (cf. Figure 2). (h) For further representative examples of operators than the one in Figure 2, see Section 5.

2.

Example of decomposition of L µ as in ( 12) with V = H of codimension 1, Λ = cZ, and µ ±1 = µ ±c .

Proof of Theorem 4.17. Assume first that [START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF] holds and let us prove [START_REF] Cabré | Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates[END_REF]. Recall that µ = a∈Λ µ a by [START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF] and supp(µ) ⊆ V ⊕ Λ by [START_REF] Del Teso | On distributional solutions of local and nonlocal problems of porous medium type[END_REF], where V ∩ Span R Λ = {0} by [START_REF] Courrège | Générateur infinitésimal d'un semi-groupe de convolution sur R n , et formule de Lévy-Khinchine[END_REF]. By Theorem 4.13(c), it suffices to have V = R d to get the negated version [START_REF] Cabré | Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates[END_REF] of the Liouville theorem, but this is assumed in [START_REF] Courrège | Générateur infinitésimal d'un semi-groupe de convolution sur R n , et formule de Lévy-Khinchine[END_REF] which completes the proof of the if part.

Assume now that [START_REF] Cabré | Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates[END_REF] holds and let us prove that L µ is of the form [START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF]. By Theorem 4.6, G(supp(µ)) = V ⊕ Λ for some vector subspace V ⊆ R d and relative lattice Λ ⊆ R d such that [START_REF] Del Teso | Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory[END_REF] V ∩ Span R Λ = {0}.

By Theorem 4.13 and (10), V ⊕ Λ = R d . It follows that V = R d and this proves [START_REF] Courrège | Générateur infinitésimal d'un semi-groupe de convolution sur R n , et formule de Lévy-Khinchine[END_REF]. It only remains to construct µ a satisfying ( 12) and We will need to show that for any a, b ∈ Λ such that a = b,

(18) (V + a) ∩ (V + b) = ∅.
To prove [START_REF] Del Teso | Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments[END_REF], assume that x ∈ (V + a) ∩ (V + b) and let us find a contradiction. In that case, x = v + a = w + b with v, w ∈ V and by [START_REF] Del Teso | Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory[END_REF],

v -w = a -b ∈ V ∩ Λ = {0}.
This gives us the contradiction that a = b and shows [START_REF] Del Teso | Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments[END_REF]. Now for any B,

µ(B) = µ B ∩ (V ⊕ Λ) (since supp(µ) ⊆ V ⊕ Λ) = µ B ∩ ∪ a∈Λ (V + a) = a∈Λ µ B ∩ (V + a) :=µa(B)
.

The measures µ a (•) = µ(• ∩ (V + a)) satisfy ( 12) and [START_REF] Del Teso | On distributional solutions of local and nonlocal problems of porous medium type[END_REF]. They are also nonnegative since µ ≥ 0, and by the symmetry of µ,

µ a (B) = µ B ∩ (V + a) = µ -(B ∩ (V + a)) = µ (-B) ∩ (-V -a) = µ -a (-B)
since V = -V . This proves [START_REF] Coville | Remarks on the strong maximum principle for nonlocal operators[END_REF]. Using finally (A µ ),

ˆ(|z| 2 ∧ 1) dµ a (z) = ˆ(|z| 2 ∧ 1) dµ(z) < +∞ (19) a∈Λ
where for any a = 0,

ˆ(|z| 2 ∧ 1) dµ a (z) ≥ (ε 2 a ∧ 1 µ k (R d )
with the distance ε a := dist(0, V + a). The affine space V + a does not contain 0 if a = 0 which implies that ε a > 0. To be more precise, we have

ε a = |0 -proj V +a (0)| = |proj V +a (0)|,
with the orthogonal projection onto V + a. Or equivalently,

ε a = |proj V +a (0)| = |a -proj V (a)|.
If we can prove that (20) inf a∈Λ\{0} ε a > 0,

we will obtain the last property ( 16) from [START_REF] Effros | Dimension groups and their affine representations[END_REF] and complete the proof.

But [START_REF] Fall | Entire s-harmonic functions are affine[END_REF] is rather standard (cf. e.g. [START_REF] Martinet | Perfect lattices in Euclidean spaces[END_REF]). Let us give details for completeness. We use that Λ is discrete by Lemma 4.5 to infer that there is some B ε (0) with ε > 0 such that B ε (0) ∩ Λ = {0}. Let us now assume by contradiction that [START_REF] Fall | Entire s-harmonic functions are affine[END_REF] does not hold. This will imply the existence of some {a n } n ⊆ Λ ∩ B ε (0) c such that |a n -proj V (a n )| → 0 as n → +∞. Note then that b n := a n /|a n | converges to some b = 0 along a subsequence which satisfies

|b n -proj V (b n )| = |a n -proj V (a n )| |a n | |a n -proj V (a n )| ε → 0
as n → +∞. The limit then satisfies

0 = b = proj V (b) ∈ V ∩ Span R Λ = {0},
which is a contradiction with [START_REF] Del Teso | Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory[END_REF] and completes the proof.

Representative examples of multi-d operators

Let us illustrate Theorems 4.13 and 4.17 with model operators having or not having the Liouville property. By changing coordinates, it is sufficient to understand this situation for ( 21)

µ(z) = δ c (z) + δ -c (z) + d i=1 (δ e i (z) + δ -e i (z)),
where c = 0 and (e 1 , . . . , e d ) is the canonical basis. In that case

G(supp(µ)) = cZ + Z d ,
and the Kronecker theorem (cf. e.g. page 507 in [START_REF] Gonek | Kronecker's approximation theorem[END_REF])) provides a necessary and sufficient condition to have cZ

+ Z d = R d .
h u(x) = d i=1 u(x + he i ) + u(x -he i ) -2u(x) h 2 , where µ(z) = d i=1 (δ he i (z) + δ -he i (z))h -2
. By Theorem 4.13(c), the Liouville theorem fails since G(supp(µ)) = hZ d is a lattice.

To get discrete operators satisfying Liouville, we need more nodes and they must be defined on nonuniform grids as in the following example: Example 5.10. Let h, ρ > 0. The following operator is a nonstandard discretization of the Laplacian

L h [u](x) := 1 2 ∆ h u(x) + 1 2 ∆ hρ u(x) with measure µ(z) = d i=1 1 2h 2 δ he i (z) + δ -he i (z) + d i=1 1 2h 2 ρ 2 δ hρe i (z) + δ -hρe i (z) . By Lemma 3.2, G(supp(µ)) = h(Z + ρZ) d is dense in R d if and only if ρ /
∈ Q. This is also a necessary and sufficient condition on L h to satisfy Liouville (cf. Figure 4). Indeed, if x ∈ G ∩ B ε (0) then x = v + a for some v ∈ V and a ∈ Λ such that |v + a| < ε and v ⊥ a. In particular, |v + a| = |v| 2 + |a| 2 and it follows that |a| < ε. By [START_REF] Fall | Liouville theorems for a general class of nonlocal operators[END_REF], we deduce that a = 0 and thus x = v ∈ V with |v| < ε. This shows that

G ∩ B ε (0) ⊆ V ∩ B ε (0),
and since the reverse inclusion is obvious we deduce the first part of [START_REF] Godement | of Collection Enseignement des Sciences [Collection: The Teaching of Science[END_REF]. We argue similarly for the part in Ṽ and complete the proof of our claim [START_REF] Godement | of Collection Enseignement des Sciences [Collection: The Teaching of Science[END_REF]. But V = Span R (V ∩ B ε (0)) and Ṽ = Span R ( Ṽ ∩ B ε (0)), and therefore

V = Ṽ . If now a ∈ Λ then a = v + ã with v ∈ V and ã ∈ Λ. It follows that v = a -ã ⊥ v
and the only possibility is that v = 0. Hence a = ã ∈ Λ and we conclude that Λ ⊆ Λ. We show the reverse inclusion in the same way and conclude the proof.

Here is now a general decomposition result for Radon measures. and similarly µ(B) = μa 0 (B). This shows that µ a 0 = μa 0 for any such arbitrarily given a 0 ∈ Λ and completes the proof.

As an immediate consequence, we get the following uniqueness result on the decomposition [START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF] in Theorem 4.17. The proof uses elementary arguments (cf. e.g. [START_REF] Godement | of Collection Enseignement des Sciences [Collection: The Teaching of Science[END_REF]). Let us give the details for completeness. To prove the result we assume by contradiction that inf G * = l > 0 for some l ∈ (0, a). Since l is strictly positive, we can find N ∈ N + such that N l < a ≤ (N + 1)l. We then take N ∈ N + such that N z 0 < z ≤ (N + 1)z 0 , and observe that |z -N z 0 | < z 0 < ε. Since z 0 = nb-n b a a for some n ∈ N + , z 0 obviously belong to aZ+bZ. We can do the same reasoning if z ∈ (-∞, 0] and we conclude that aZ + bZ is dense in R.

3 . 1 . 3 . 2 . 3 . 3 . 3 . 4 .

 31323334 in R in terms of irrational numbers 6 Reminders on integers and irrational numbers 6 An explicit condition on the Lévy measure 7 Main results and examples 8 Proof of the necessary condition 10 3.5. Proof of the sufficient condition 11 4. Characterization in R d in terms of groups and lattices 15 4.1. Reminders on groups and lattices 15 1. Introduction

  (cf. also Proposition 4.15 for further results in 1-d). Section 5 gives representative multi-d examples, and technical or complementary results are postponed in appendices.

  are the respective spaces of smooth functions with bounded derivatives of all orders and of smooth functions with compact supports. Measures on R d \ {0}. We only consider nonnegative Radon measures µ on Ω := R d \ {0}, i.e. Borel measures finite on compact sets. Their supports are defined as supp(µ) = z ∈ Ω : µ(B r (z)) > 0, ∀r > 0 . By definition, supp(µ) is a closed set of Ω = R d \ {0} and it does not contain zero. To avoid confusion, its closure in R d (which may contain zero) is denoted by supp(µ) R d. For any a ∈ R d , we denote by δ a (z) the Dirac measure at z = a. A measure µ d is discrete (or atomic) if there exists a countable set D ⊆ Ω such that[START_REF] Andreu-Vaillo | Nonlocal diffusion problems[END_REF]

Lemma 1 . 1 .

 11 If (2) holds then µ d (z) = a∈D ω(a)δ a (z) with ω(a) = µ({a}).

( a )

 a Borel measurable and locally integrable (on R d \ {0}) function f ≥ 0, (b) measure µ s ≥ 0 satisyfing µ s (• ∩ N ) = µ s (•) for some N of null Lebesgue measure, µ({z}) = 0 for all z, (c) and discrete or atomic measure µ d ≥ 0. The measure f (z) dz is the absolutely continuous part, and µ s the nonatomic or diffuse singular part. All the three parts satisfy (A µ ). This means that f (z) = f (-z) and ˆ(|z| 2 ∧ 1)f (z) dz < +∞, and µ d (z) = a∈D ω(a) δ a (z) + δ -a (z) for some countable set D ⊆ R d \ {0} and ω : D → R such that ω > 0 and a∈D (|a| 2 ∧ 1)ω(a) < +∞.

Example 2 . 3 .

 23 Here are some classical nonlocal Lévy operators: (a) The fractional Laplacian -(-∆) α 2 with α ∈ (0, 2) corresponding to an absolutely continuous Lévy measure (cf. e.g. [32]) of the form dµ(z) = c d,α |z| d+α dz; (b) the relativistic Schrödinger operator m α I -(m 2 I -∆) α 2 with m > 0 and α ∈ (0, 2) corresponding to an absolutely continuous Lévy measure of the form dµ(z) = c d,α,m K (d+α)/2 (m|z|)

Lemma 3 . 1 .

 31 Two integers n, m ∈ Z \ {0} are coprimes if and only if nZ + mZ = Z.

Proof.

  Assume sup b∈supp(µ) Q(a 0 , b) = +∞ for some a 0 and take any other fixed a ∈ supp(µ). If b a / ∈ Q for some b ∈ supp(µ), then Q(a, b) = +∞ and the proof is complete. In the other case, p a , p b = 0 such that gcd(p a , Q(a 0 , a)) = 1 = gcd(p b , Q(a 0 , b)).

  The parts (a) ⇒ (b) and (b) ⇒ (a) are proved separately in Sections 3.4 and 3.5. In practice, we can check algorithmically if a 1-d operator enjoys the Liouville property with the help of the following procedure: Corollary 3.8 (A practical method of verification in 1-d). Assume (A µ ) and d = 1. Then: (a) If supp(µ) R contains an interval or just one accumulation point, then L µ satisfies the Liouville theorem (in the sense of Theorem 3.7(a)). (b) If part (a) does not hold, then:

Example 3 . 9

 39 (1-d continuous Lévy measures satisfy Liouville). Any measure with either an absolutely continuous part or a nontrivial diffusive singular part (cf. Remark 2.2) satisfies the Liouville theorem by Corollary 3.8(a).

  3 for examples in multi-d. (b) Other typical examples are nonzero order discrete operators with unbounded measures at 0

( 8 )

 8 a n > 0 and sup n≥1 Q(a 1 , a n ) = +∞. Proposition 3.16. Assume (7)-(8) and u ∈ C ∞ b (R) solves L µ [u] = 0 in R. If u has a global maximum, then u is constant. Now we need the density result below to prove Proposition 3.16. Lemma 3.17. Assume (A µ ), d = 1, and (A L ). Then a,b∈supp(µ) (aZ + bZ) = R. Proof. If b a / ∈ Q then the result follows from Lemma 3.2. Assume now that any such ratio is rational. By the Bézout identity Lemma 3.1 and the definition of Q(•, •) in (4), we infer that for any a, b ∈ supp(µ) there exists p ∈ Z such that b a = p Q(a,b) with gcd(p, Q(a, b)) = 1, and

Remark 4 . 4 .

 44 (a) A typical example of (full) lattice is Λ = Z d . More generally, every (full) lattice are isomorphic to Z d , up to a change of coordinates from the basis (a 1 , . . . , a d ) into the canonical one. (b) By convention, Λ = {0} is a relative lattice.

  ˆz∈supp(µ) u(x + z) -u(x) dµ(z) = 0, which proves the if part. Now assume that L µ [u] = 0 in R d . Fix an arbitrary g ∈ G, and define v(x) := u(x + g) -u(x). If we show that v = 0 in R d , then the proof is complete. Let us first show that v ≤ 0. Let M := sup v and take a sequence {x n } n such that v(x n ) n→∞ -→ M. Let us define u n (x) := u(x+x n ) and v n (x) := v(x+x n ) similarly than in the proof of Lemma 3.20 but with the new function v, and note that

Theorem 4 . 13 .

 413 Assume d ≥ 1 and (A µ ). Then the following statements are equivalent: (a) If u ∈ L ∞ (R d ) satisfies L µ [u] = 0 in D (R d ), then u is constant. (b) The additive subgroup generated by supp(µ) is dense in R d . (c) For all vector space V ⊆ R d of dimension less or equal d -1 and relative latticeΛ ⊆ R d such that V ∩ Span R Λ = {0}, supp(µ) ⊆ V ⊕ Λ. (d) For all vector space H ⊆ R d of dimension d -1 and vector c ∈ R d , supp(µ) ⊆ H + cZ.Remark 4.14. (a) By Theorem 3.7, we note that if d = 1 then Liouville ⇐⇒ (A L ) ⇐⇒ G(supp(µ)) is dense. (b) By Remark 4.8, we have the following contrapositive version if d = 1: Not Liouville ⇐⇒ (NA L ) ⇐⇒ supp(µ) ⊆ gZ for some g > 0.This includes the trivial measure µ = 0 whose support is empty. (c) In practice, it might be easier to verify either (c) or (d) than (b). In Figure1, we explain the relations between (the negated version of) these properties. (d) If d > 1, we do not have any algorithmic method like Corollary 3.8 to verify (c) and (d), but we will identify more precisely the class of operators that do not satisfy these properties (cf. Section 4.4). (e) Representative examples are given in Section 5.

Figure 1 .

 1 Figure 1. An example where (c) and (d) of Theorem 4.13 fail. We have supp(µ) = S ∪ (-S) and Λ := G(supp(µ)) is already lattice so supp(µ) ⊆ V ⊕ Λ with V = {0} and supp(µ) ⊆ H + cZ with H and c as in the picture.

Proposition 4 . 15 .

 415 Assume d = 1 and (A µ ). Then the negated version (10) of the Liouville theorem holds if and only if

5. 1 .

 1 Nonatomic measures and Liouville. If d = 1 the Liouville theorem can fail only for atomic measures (cf. Proposition 4.15). In multi-d, this is slightly different.Example 5.1 (Absolutely continuous measures satisfy Liouville). If the support of µ contains a ball (cf. Figure 3(a)), then the Liouville theorem holds by Theorem 4.13(d). This is e.g. the case for absolutely continuous measures like the fractional Laplacian, etc. Example 5.2 (Diffuse measures: Case 1). Consider now the non absolutely continuous diffuse measure associated to Lµ = -(-∆ d-1 ) α 2 where ∆ d-1 = d-1 i=1 ∂ 2 x i . It is of the form dµ(z) = f (z) dz 1 . . . dz d-1 dδ 0 (z d ) with f (z) = c d,α 1 {z d =0} |z| d-1+α .Since supp(µ) = H + cZ with H = R d-1 and c = 0, the Liouville theorem fails by Theorem 4.13(d) (cf. Figure3(b)).

Remark 5 . 3 (Example 5 . 4 (

 5354 Unbounded measures at z = 0). In Example 5.2, µ is unbounded where as the Liouville theorem fails for L µ . This may occur only if d > 1. Indeed, by Proposition 4.15, every 1-d Lévy measure for which the Liouville theorem fails should be purely atomic and bounded up to z = 0. Diffuse measures: Case 2). Let us consider another non absolutely continuous diffuse measure given by the mean value operator M[u](x) = 1 c d ˆ∂B 1 (0) u(x + z) -u(x) dσ(z),

  (a) supp(µ) contains a ball. The Liouville property holds. (b) supp(µ) = R d-1 ×{0}. The Liouville property fails. (c) supp(σ) = ∂B1(0). The Liouville property holds.

Figure 3 . 5 . 2 .

 352 Figure 3. Supports satisfying or not satisfying Thm. 4.13(d).

Figure 4 .

 4 Figure 4. Graphical representation of supp(µ) for L h in Example 5.10 with d = 2, h = 1, and ρ = π. We have supp(µ) = S ∪ (-S) and G(supp(µ)) = R × R.

Lemma A. 3 .

 3 Let µ be a nonnegative Radon measure on R d \ {0}. Then there exists a unique triplet (V, Λ, {µ a } a∈Λ ) such that• V ⊆ R d is a vector space, • Λ ⊆ R d is a relative lattice,• µ a is a nonnegative Radon measure on R d \ {0} for any a ∈ Λ, and the following properties hold:G(supp(µ)) = V ⊕ ⊥ Λ and µ = a∈Λ µ a with supp(µ a ) ⊆ V + a. Proof. Since G(supp(µ)) is closed, Lemma A.1 implies that G(supp(µ)) = V ⊕ ⊥ Λwhere such a decomposition is unique. If V = R d then Λ = {0} and the only family {µ a } a∈Λ that will satisfy the theorem is given by the singleton{µ a = µ} a∈{0} . If now V = R d then V ⊕ ⊥ Λ = ∪ a∈Λ (V + a)where the union is disjoint, and we can argue as in the proof of Theorem 4.17 to show that µ = a∈Λ µ a with the trace measures µ a (•) = µ(• ∩ (V + a)) satisfying supp(µ a ) ⊆ V + a.Moreover, if µ = a∈Λ μa for other Radon measures satisfying supp(μ a ) ⊆ V + a then for any fixed a 0 ∈ Λ and all Borel sets B ⊆ V + a 0 , µ(B) = a∈Λ µ a (B) = µ a 0 (B)

Corollary A. 4 .

 4 Assume (A µ ) and the negated version (10) of the Liouville theorem. Then there exists a unique triplet (V, Λ, {µ a } a∈Λ ) satisfying (12)-[START_REF] Del Teso | Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type[END_REF] andG(supp(µ)) = V ⊕ ⊥ Λ.Appendix B. Proof of Lemma 3.2

Lemma B. 1 .

 1 Let a, b > 0 and define G* := {nb -n b a a : n ∈ N + }. If b a ∈ Q, then inf G * = 0. Proof of Lemma B.1. First note that G * ⊆ (0, a): Since b a ∈ Q, n b a < n b a < n b a + 1 and thus n b a a < nb < n b a a + a. Hence 0 < nb -n b a a < a.

  Define r := (N + 1)l -a and note that[START_REF] Gonek | Kronecker's approximation theorem[END_REF] r = N l -a + l < l < a.By definition of infimum, there exist sequences {ε k } k ⊂ [0, 1) and{n k } k ⊆ N + with ε k → 0 as k → +∞ such that n k b -n k b a a = l + ε k .If ε k = 0 then l a / ∈ Q and therefore r > 0. This proves that either r > 0 or {ε k } k ⊂ (0, 1).Using then that(N + 1)n k b = (N + 1) n k b a a + (N + 1)l + (N + 1)ε k = (N + 1) n k b a a + a + r + (N + 1)ε k , r < a by[START_REF] Gonek | Kronecker's approximation theorem[END_REF], we have r + (N + 1)ε k < a for k large enough, which by[START_REF] Huang | Maximum principles for nonlocal parabolic Waldenfels operators[END_REF] again leads to (N + 1)n k b < ((N + 1) n k b a + 1)a + a and then[START_REF] Martinet | Perfect lattices in Euclidean spaces[END_REF] (N + 1)n k b a < ((N + 1) n k b a + 1) + 1.By (26) and (27), we conclude that(N + 1) n k b a + 1 = (N + 1)n k b afor all sufficiently large k. Then since (N + 1)n k b -(N + 1)n k b a a ∈ G * , we can use[START_REF] Huang | Maximum principles for nonlocal parabolic Waldenfels operators[END_REF] with ε k < l-r N +1 (remember that l -r > 0 by (24)) to get(N + 1)n k b -(N + 1)n k b a a = r + (N + 1)ε k < r + (N + 1) l -r N + 1 = l.This contradicts the fact that inf G * = l and concludes the proof.Proof of Lemma 3.2. The only if part can be shown from e.g. Remark 4.8, but we omit the detail since it was not used during the proofs of our main results. Let us focus on the if part. It suffices to prove it for a, b > 0, which we assume together with b a / ∈ Q. Fix any z ∈ [0, +∞) and ε > 0. Consider the set G * = nb -n b a a : n ∈ N + as defined in Lemma B.1 and recall that G * ⊆ (aZ + bZ) ∩ (0, a) (cf. the proof of Lemma B.1). By Lemma B.1, there is z 0 ∈ G * such that 0 < z 0 < ε.

  ± (ω 1 δ ±a + ω 2 δ ±b ) satisfies the Liouville property if and only if b a / ∈ Q. (b) The condition that {1, c 1 , . . . , c d } is linearly independent over Q is related to Diophantine equations (cf.[START_REF] Gonek | Kronecker's approximation theorem[END_REF]).5.3. Finite difference operators.Let us now give some examples coming from numerical methods.Example 5.9. Let (e 1 , . . . , e d ) be the canonical basis in R d and h > 0.

	Consider the discrete Laplacian
	∆

Let us recall this result. Theorem 5.6 (Kronecker's approximation). For any c = (c 1 , . . . , c d ) ∈ R d , we have cZ + Z d = R d if and only if {1, c 1 , . . . , c d } is linearly independent over Q. Corollary 5.7. The measure µ in (21) satisfies the Liouville property if and only if {1, c 1 , . . . , c d } is linearly independent over Q. Proof. Use Theorems 4.13 and 5.6. Remark 5.8. (a) In 1-d, we recover that µ =

Acknowledgements J. Endal and E. R. Jakobsen were supported by the Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway. F. del Teso and J. Endal are grateful to Laboratoire de Mathématiques de Besançon (LMB, UBFC) and Ecole Nationale Supérieure de Mécanique et des Microtechniques (ENSMM) for providing accommodation during their visit in May in 2018.

(N. Alibaud) ENSMM, 26 Chemin de l'Epitaphe, 25030 Besanc ¸on cedex, France and, LMB, UMR CNRS 6623, Université de Bourgogne Franche-Comté (UBFC), France E-mail address: nathael.alibaud@ens2m.fr (F. del Teso) Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway E-mail address: felix.delteso@ntnu.no (J. Endal) Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway E-mail address: jorgen.endal@ntnu.no (E. R. Jakobsen) Department of Mathematics, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway E-mail address: espen.jakobsen@ntnu.no

An application of Lemma 3.20 with v = u x then shows that for all R ≥ 0 min B R (xn) u x → M as n → +∞ (up to a subsequence).

Conclusion.

If M > 0, then for any 0 < η < M and R > 2 u ∞ /(M -η), we can take a fixed n large enough such that min

We obtain the contradiction that u(x n ) > u ∞ for this fixed n. This completes the proof that u x = 0, and thus the proof of the theorem.

Characterization in R d in terms of groups and lattices

Let us now consider the dimension d ≥ 1. Our general periodicity result Theorem 4.11 is given in Section 4.2. Our characterization results for the Liouville theorem are given in Theorems 4.13 and 4.17 of Sections 4.3 and 4.4, respectively. Theorem 4.13 is a generalization of Theorem 3.7 expressed in terms of subgroups of R d , and Theorem 4.17 precises the general form of the operators for which Liouville fails (cf. also Proposition 4.15 for the case d = 1). All the proofs are given (almost) just after the statements of the results.

4.1. Reminders on groups and lattices. Let us first recall some basic facts about groups that will be needed (cf. e.g. [START_REF] Godement | of Collection Enseignement des Sciences [Collection: The Teaching of Science[END_REF], [7, Chapter VII] or [27, Section 1.1]).

The subgroup generated by a set S ⊆ R d , denoted G(S), is the smallest group containing S. In this appendix, we discuss the uniqueness of the decomposition [START_REF] Ciomaga | On the strong maximum principle for second-order nonlinear parabolic integro-differential equations[END_REF] in Theorem 4.17. We will need the elementary lemma below for the uniqueness of the decomposition in Theorem 4.6.

Lemma A.1. A closed subgroup G of R d can be written in a unique way in the form G = V ⊕ Λ for some vector space V ⊆ R d and relative lattice

Remark A.2. For an illustrative example, see Figure 5. We then claim that