Olivier Durieu
email: olivier.durieu@univ-tours.fr

Gennady Samorodnitsky

Yizao Wang
email: yizao.wang@uc.edu

FROM INFINITE URN SCHEMES TO SELF-SIMILAR STABLE PROCESSES

Keywords: 2010 Mathematics Subject Classification. Primary, 60F17; Secondary, 60G18, 60G52 Infinite urn scheme, regular variation, functional central limit theorem, selfsimilar process, stable process

We investigate the randomized Karlin model with parameter β ∈ (0, 1), which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index β/2 ∈ (0, 1/2). We show here that when the randomization is heavy-tailed with index α ∈ (0, 2), then the odd-occupancy process scales to a (β/α)-self-similar symmetric α-stable process with stationary increments.

Introduction and main results

Consider the following infinite urn scheme. Suppose there is an infinite number of urns labeled by N = {1, 2, . . . }, all initially empty. Balls are thrown into the urns randomly one after another. At each round, a ball is thrown independently into the urn with label k with probability p k , with k≥1 p k = 1. This random sampling strategy dates back to at least the 60s [START_REF] Bahadur | On the number of distinct values in a large sample from an infinite discrete distribution[END_REF][START_REF] Karlin | Central limit theorems for certain infinite urn schemes[END_REF]. The urns may represent different species in a population of interest, and in various applications an interesting question is to infer the population frequencies (p k ) k≥1 ; see [START_REF] Gnedin | Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws[END_REF] and references therein. This urn scheme has been extensively investigated in the literature on combinatorial stochastic processes as it induces the so-called paintbox partition of N, an infinite exchangeable random partition; see for example [START_REF] Pitman | Combinatorial stochastic processes[END_REF].

Asymptotic results for many statistics of this urn scheme, and in particular of the random partition it induces, have been investigated in the literature (e.g. [START_REF] Gnedin | Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws[END_REF] and references therein), including in particular the so-called odd-occupancy process. In the sequel, let Y n denote the label of the urn that the n-th ball falls into, so P(Y n = k) = p k . Then (Y n ) n∈N are i.i.d. random variables, and we let Y n,k := n i=1 1 {Yi=k} denote the number of balls in the urn with label k after first n rounds. The odd-occupancy process is then defined as

U * n = ∞ k=1 1 {Y n,k odd} .
This process counts the number of urns that contain an odd number of balls after the first n rounds. An interpretation of this process due to Spitzer [START_REF] Spitzer | Principles of random walk[END_REF] is as follows. One may associate to each urn a lightbulb, and start the sampling procedure with all lightbulbs off. Each time a ball falls in an urn, the corresponding lightbulb changes its status (from off to on or from on to off). The process U * n then represents the total number of lightbulbs that are on after the first n rounds.

Karlin [START_REF] Karlin | Central limit theorems for certain infinite urn schemes[END_REF] proposed and investigated the aforementioned model under the following assumptions on (p k ) k≥1 : p k is decreasing in k and [START_REF] Aaronson | An introduction to infinite ergodic theory[END_REF] max{k ≥ 1 | p k ≥ 1/t} = t β L(t), t ≥ 0, for some β ∈ (0, 1),

where L is a slowly varying function at infinity. Among many results, Karlin proved that

U * n -EU * n σ n ⇒ N (0, 1)
as n → ∞ with σ n = 2 β-1 (Γ(1 -β)n β L(n)) 1/2 . Here and in the sequel ⇒ denotes weak convergence and N (0, 1) stands for the standard normal distribution. We are interested in the randomized version of the odd-occupancy process, defined as

(2)

U n = ∞ k=1 ε k 1 {Y n,k odd} ,
where (ε k ) k∈N are i.i.d. symmetric random variables independent of (Y n ) n∈N . This randomization was recently introduced in [START_REF] Durieu | From infinite urn schemes to decompositions of selfsimilar Gaussian processes[END_REF], and it was shown in Corollary 2.8 therein that with (ε k ) k∈N being a sequence of independent Rademacher (±1-valued and symmetric) random variables, with the same normalization σ n ,

U nt σ n t∈[0,1] ⇒ 2 -β/2 B β/2 t t∈[0,1] in D([0, 1]) (3)
, where B H is a standard fractional Brownian motion with Hurst index H ∈ (0, 1), a centered Gaussian process with covariance function

Cov(B H s , B H t ) = 1 2 s 2H + t 2H -|t -s| 2H , s, t ≥ 0.
The following aspect of the randomization and the resulting functional central limit theorem is particularly interesting. For an arbitrary symmetric distribution of ε 1 , the randomized odd-occupancy process U n is the partial-sum process for a stationary sequence, (4)

U n = X 1 + • • • + X n with X i = -ε Yi (-1) Y i,Y i , i ∈ N, n ∈ N.
Therefore, the infinite urn scheme provides a specific way to generate a stationary sequence of random variables whose marginal distribution is the given symmetric law, and whose partial-sum process is the randomized odd-occupancy process. Moreover, at least for the Rademacher marginal distribution, this stationary sequence (X n ) n≥1 exhibits, in view of the limiting result (3), anomalous behavior. Namely, the normalization σ n has an order of magnitude different from the "usual" √ n normalization needed for partial sums of i.i.d. random variables with the same marginal distribution. Such a behavior indicates long-range dependence in the stationary sequence [START_REF] Beran | Long-memory processes[END_REF][START_REF] Pipiras | Long-range dependence and self-similarity[END_REF][START_REF] Samorodnitsky | Stochastic processes and long range dependence[END_REF]. The long-range dependence in this case is due to the underlying random partition. In particular, the covariance function of X is determined by the law of the random partition.

In this paper, we are interested in the randomized odd-occupancy process when ε 1 has a heavy-tailed distribution. Specifically, we will assume that ε 1 has infinite variance and, even more specifically, is in the domain of attraction of a non-Gaussian stable law. The stationary-process representation (4) is, clearly, still valid. However, in a functional central limit theorem one expects now a symmetric stable process that is self-similar with stationary increments (we abbreviate the latter two properties as the sssi property). The only sssi Gaussian process is the fractional Brownian motion; however there are many different sssi symmetric stable processes, which often arise in limit theorems for the partial sums of stationary sequences with long-range dependence, [START_REF] Pipiras | Stable non-Gaussian self-similar processes with stationary increments[END_REF][START_REF] Samorodnitsky | Stochastic processes and long range dependence[END_REF][START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]. While many sssi stable processes have been well investigated in the literature, new processes in this family are still being discovered [START_REF] Owada | Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows[END_REF]. One way to classify different sssi stable processes is via the flow representation, introduced by Rosiński [START_REF] Rosiński | On the structure of stationary stable processes[END_REF] and developed by Samorodnitsky [START_REF] Samorodnitsky | Null flows, positive flows and the structure of stationary symmetric stable processes[END_REF], of the corresponding increment processes which is, by necessity, stationary. Certain ergodictheoretical properties of these flows are invariants for each stationary stable process, and processes corresponding to different types of flows, namely positive, conservative null and dissipative, have drastically different properties.

The main result of this paper is to show that for the Karlin model with a heavy-tailed randomization, the scaling limit of the randomized odd-occupancy process is a new selfsimilar symmetric stable process with stationary increments. We continue to assume that (ε k ) are i.i.d. and symmetric. For simplicity we will also assume that they are in the normal domain of attraction of a symmetric stable law. That is, for some α ∈ (0, 2), [START_REF] Billingsley | Convergence of probability measures[END_REF] lim

x→∞ P(|ε 1 | > x) x -α = C ε ∈ (0, ∞).
We will define the limiting sssi symmetric α-stable (SαS) process in terms of stochastic integrals with respect to an SαS random measure as follows. Let (Ω , F , P ) be a probability space and N a standard Poisson process defined on this space. Let M α,β be a SαS random measure on R + × Ω with control measure βr -β-1 drP (dω ); we refer the reader to [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] for detailed information on SαS random measures and stochastic integrals with respect to these measures. The random measure M α,β is itself defined on a probability space (Ω, F, P), the same probability space on which the limiting process U α,β will be defined. This is accomplished by setting The process U α,β is, to the best of our knowledge, a new class of sssi SαS processes, with self-similarity index β/α. In particular, we shall show that its increment process is driven by a positive flow [START_REF] Rosiński | On the structure of stationary stable processes[END_REF][START_REF] Samorodnitsky | Null flows, positive flows and the structure of stationary symmetric stable processes[END_REF]. The following is the main result of the paper; we use the notation f.d.d.

→ for convergence in finite-dimensional distributions.

Theorem 1. Under the assumptions (1) and [START_REF] Billingsley | Convergence of probability measures[END_REF],

with b n = (n β L(n)) 1/α , U nt b n t∈[0,1] f.d.d. → σ ε U α,β t t∈[0,1]
,

where

σ α ε = C ε ∞ 0 x -α sin x dx.
If, in addition, α ∈ (0, 1), then convergence in distribution in the Skorohod J 1 topology on D([0, 1]) also holds.

We will prove this theorem by first conditioning on the urn sampling sequence (Y n ) n≥1 . It turns out that the characteristic function of finite-dimensional distributions of U can be expressed in terms of certain statistics of that sequence. The same idea can be used in the case of a bounded ε 1 (although the proof in [START_REF] Durieu | From infinite urn schemes to decompositions of selfsimilar Gaussian processes[END_REF] was different and actually more involved as the results are stronger; see also [START_REF] Durieu | From random partitions to fractional Brownian sheets[END_REF] for the same idea applied to a generalization of Karlin model). Indeed, in this case the limit process is Gaussian, so one addresses the convergence of the covariance function by essentially examining the joint even/oddoccupancies at two different time points. In the present case the limit is a stable process and, hence, no longer characterized by bivariate distributions. Therefore, as an intermediate step, we have to examine the joint even/odd-occupancies at multiple time points (n 1 , . . . , n d ).

For this purpose we investigate the following multiparameter process

M δ1,...,δ d n1,...,n d = k≥1 1 {Yn 1 ,k =δ1 mod 2,...,Y n d ,k =δ d mod 2} , with n 1 , . . . , n d ∈ N 0 = {0} ∪ N and (δ 1 , . . . , δ d ) ∈ {0, 1} d \ {(0, . . . , 0)}.
We refer to this process as the multiparameter even/odd-occupancy process. A weak law of large numbers for the process M will turn out to be sufficient to prove the first part of Theorem 1. However, we will establish a functional central limit theorem for the multiparameter even/odd-occupancy process (Theorem 2 below). The limit in that result can be viewed as a multiparameter generalization of the bi-fractional Brownian motion [START_REF] Houdré | An example of infinite dimensional quasi-helix[END_REF] and, hence, is of interest on its own. The paper is organized as follows. Section 2 reviews the background on flow representation of stationary stable processes and shows that the increment process of U α,β is driven by a positive flow. Section 3 establishes limit theorems for the multiparameter odd-occupancy process M . Section 4 presents the proof of main result.

A new class of self-similar stable processes with stationary increments

We start by verifying the self-similarity of the process U α,β introduced in ( 6). It will also follow from Theorem 1 and the Lamperti theorem (see e.g. [START_REF] Samorodnitsky | Stochastic processes and long range dependence[END_REF]), but a direct argument is simple.

Proposition 1. The process U α,β is (β/α)-self-similar. That is, U α,β λt t≥0 f.d.d. = λ β/α U α,β t t≥0
for all λ > 0.

Proof. Fix λ > 0. For any d ∈ N, t 1 , . . . , t d ≥ 0, a 1 , . . . , a d ∈ R,

E exp i d k=1 a k U α,β λt k = exp - ∞ 0 Ω d k=1 a k 1 {N (λt k r)(ω ) odd} α P (dω )βr -β-1 dr = exp - ∞ 0 Ω λ β d k=1 a k 1 {N (t k r)(ω ) odd} α P (dω )βr -β-1 dr = E exp -i d k=1 a k λ β/α U α,β t k , as required.
We now consider the increment process of U α,β . We will see that the increment process is stationary (which, of course, will follow from Theorem 1 as well). More importantly, we will classify the flow structure of this process in the spirit of Rosiński [START_REF] Rosiński | On the structure of stationary stable processes[END_REF] and Samorodnitsky [START_REF] Samorodnitsky | Null flows, positive flows and the structure of stationary symmetric stable processes[END_REF]. We start with a background on the flow structure of stationary SαS processes. It was shown by Rosiński [20] that, given a stationary SαS process X = (X t ) t∈R there exist a measurable space (S, S, µ), a non-singular flow (T t ) t∈R on it, and a function f ∈ L α (S, µ), such that

(7) (X t ) t∈R f.d.d. = S c t (s)f • T t (s) dµ • T t dµ (s) 1/α M α (ds) t∈R ,
where M α is an SαS random measure on (S, S) with control measure µ, and (c t ) t∈R is a ±1-valued cocycle with respect to (T t ) t∈R . Recall that a non-singular flow (T t ) t∈R is a group of measurable maps from S onto S such that for all t ∈ R the measure µ • T is equivalent to µ. A cocycle (c t ) t∈R is a family of measurable functions on (S, S) such that for all t 1 , t 2 ∈ R, c t1+t2 (s) = c t1 (s)c t2 • T t1 (s) µ-almost everywhere. See Krengel [START_REF] Krengel | Ergodic theorems[END_REF] and Aaronson [START_REF] Aaronson | An introduction to infinite ergodic theory[END_REF] for more information. In particular, T induces a unique decomposition of (S, S) modulo µ,

S = P ∪ CN ∪ D,
where P, CN, D are disjoint T -invariant measurable subsets of S and, restricted to each subset (if non-empty), T is positive, conservative null and dissipative, respectively. This decomposition generates a unique in law decomposition of the process X in ( 7) into a sum of 3 independent stationary SαS processes, X = X P + X CN + X D , where X P corresponds to a positive flow, X CN corresponds to a conservative null flow, and X D corresponds to a dissipative flow, with one or two of the components, possibly, vanishing. In fact, one can define all 3 processes as in [START_REF] Durieu | From infinite urn schemes to decompositions of selfsimilar Gaussian processes[END_REF], but integrating over P, CN, D correspondingly. It is known that X P is non-ergodic, X D is mixing, while X CN is ergodic, and can be either mixing or non-mixing; see [START_REF] Rosiński | On the structure of stationary stable processes[END_REF][START_REF] Samorodnitsky | Null flows, positive flows and the structure of stationary symmetric stable processes[END_REF][START_REF] Samorodnitsky | Stochastic processes and long range dependence[END_REF]. The processes generated by a dissipative flow have necessarily a mixed moving-average representation. The least understood family of processes are those generated by a conservative null flow. We will show that the increment process of U α,β is generated by a positive flow. In fact, a stationary SαS process X is generated by positive flow if and only if it can be represented as in [START_REF] Durieu | From infinite urn schemes to decompositions of selfsimilar Gaussian processes[END_REF], where now the control measure µ is a probability measure invariant under action of the operators (T t ) t∈R (so that the factor (dµ

• T t /dµ) 1/α disappears); see [21, Remark 2.6].
For this purpose, we first present a natural extension of U α,β to a stochastic process indexed by t ∈ R. We may and will assume that Ω is the space of Radon measures on R equipped with the Borel σ-field corresponding to the topology of vague convergence, P is the law of the unit rate Poisson point process on R and

N (t)(ω ) := ω ([0, t]) t ≥ 0 ω ([t, 0)) t < 0.
In this way, we now define

U α,β t := R+×Ω 1 {N (tr)(ω ) odd} M α,β (dr, dω ), t ∈ R.
This definition extends (6).

Proposition 2. The increment process of U α,β defined as

(8) X t := U α,β (t + 1) -U α,β (t), t ∈ R,
is stationary and driven by a positive flow.

Proof. Let m β denote the measure βr -β-1 dr on R + . It follows from the stochastic integral representation of U α,β that,

(9) (X t ) t∈R f.d.d. = R+×Ω 1 {N ((t+1)r)(ω ) odd} -1 {N (tr)(ω ) odd} M α,β (dr, dω ) t∈R ,
where M α,β is the SαS random measure described above. Let θ t be the standard left shift on the space of Radon measures on R, t ∈ R, and define a group of measurable operators on R + × Ω by

T t (r, ω ) := (r, θ tr (ω )), t ∈ R .
If ν is a probability measure on R + equivalent to m β then the probability measure µ = ν ×P on R + × Ω is preserved by the operators (T t ) t∈R . Denote h = dm β /dν and define

f (r, ω ) = h(r) 1/α 1 {N (r)(ω ) odd} , and
c t (r, ω ) = 1 {N (tr)(ω ) even} -1 {N (tr)(ω ) odd} , (r, ω ) ∈ R + × Ω .
If M α is an SαS random measure on R + × Ω with control measure µ, then, in law, ( 9) is the same as

(X t ) t∈R f.d.d. = R+×Ω c t (r, ω )f • T t (r, ω )M α (dr, dω ) t∈R .
Since (c t ) t∈R is, clearly, a ±1-valued cocycle, this will establish both stationarity of the increment process and the fact that it is driven by a positive flow once we check that the function f ∈ L α (µ). However, Let δ ∈ Λ d and consider the multiparameter odd-occupancy process

|f | α dµ = 1 {N (r)(ω ) odd} m β (dr)P (dω ) = ∞ 0 βr -β-1 P (N (r) odd) dr = ∞ 0 βr -β-1 1 2 1 -e -2r dr = Γ(1 -β)2 β-1 < ∞.
M δ nt := ∞ k=1 d j=1 1 {Yn j ,k =δj mod 2} , n ∈ N, t ∈ [0, 1] d . Let M δ = (M δ t ) t∈[0,1] d be a centered Gaussian random field with covariance function Cov(M δ t , M δ s ) = ∞ 0 Cov 1 { N (rt)=δ mod 2} , 1 { N (rs)=δ mod 2} βr -β-1 dr,
where N is a standard Poisson process on R + and ( 11) N (nt) = δ mod 2 ≡ {N (nt j ) = δ j mod 2 for all j = 1, . . . , d} .

The next result is a limit theorem for M δ nt . It uses the normalization

d n = b α n = n β L(n)
, where b n is as in Theorem 1. We use the new notation to emphasize the fact that the normalization in Theorem 2 does not depend on α.

m δ t := lim n→∞ EM δ nt d n = ∞ 0 P N (rt) = δ mod 2 βr -β-1 dr . (12)
Moreover,

M δ nt -EM δ nt √ d n t∈[0,1] d ⇒ M δ t t∈[0,1] d in D([0, 1] d )
with respect to the J 1 -topology, and the limiting random field has a version with continuous sample paths.

Remark 1. For the weak convergence in Theorem 2 (and in Proposition 3 below), we shall prove that it holds with respect to any topology generated by a complete separable metric on D([0, 1] d ) weaker than the uniform metric.

Only the first part of Theorem 2 is needed for Theorem 1. However, the Gaussian random field M δ t is of interest on its own. In fact, it was shown in [7, Theorem 2.3] that, when d = 1, the process M 1 n (which is simply the odd-occupancy process) satisfies the weak convergence in Theorem 2 with the limiting process M 1 being, up to a multiplicative constant, the bifractional Brownian motion [START_REF] Houdré | An example of infinite dimensional quasi-helix[END_REF][START_REF] Lei | A decomposition of the bifractional Brownian motion and some applications[END_REF] with parameters H = 1/2, K = β. This is a centered Gaussian process with covariance function

Cov(M 1 t , M 1 s ) = Γ(1 -β)2 β-2 (s + t) β -|s -t| β , s, t ≥ 0.
Therefore, the limit obtained in Theorem 2 can be viewed as a random field generalization of the bi-fractional Brownian motion. In order to analyze the multiparameter odd-occupancy process M δ nt , we introduce a Poissonization of the underlying urn sampling sequence (Y n ) n∈N . Let N be a standard Poisson process independent of (Y n ) n∈N and (ε n ) n∈N . Set

N k (t) := N (t) i=1 1 {Yi=k} , k ∈ N, t ≥ 0.
Clearly (N k ) k∈N are independent Poisson processes with respective parameters (p k ) k∈N . We use the notation { N k (t) = δ mod 2} whose meaning is analogous to [START_REF] Houdré | An example of infinite dimensional quasi-helix[END_REF], and the Poissonized version of the multiparameter odd-occupancy process is

M δ t := ∞ k=1 1 { N k (t)=δ mod 2} , t ∈ R d + .
Lemma 1. Under the assumption (1), for all

d ∈ N, t ∈ [0, 1] d , δ ∈ Λ d , (13) lim
n→∞ M δ nt d n = m δ t in L 2 .
Proof. Consider a Radon measure ν on R + defined by ν := ∞ k=1 δ 1/p k . By the assumption (1), we have ν(x) := ν([0, x]) = x β L(x), for all x > 0. Observe that

E M δ nt = ∞ k=1 E1 { N k (nt)=δ mod 2} = ∞ k=1 P N (np k t) = δ mod 2 = ∞ 0 P N (nt/x) = δ mod 2 ν(dx) = ∞ 0 ϕ t n x ν(dx)
with ϕ t (s) := P( N (st) = δ mod 2). It is easy to see that ϕ t is differentiable and vanishes at zero. Integrating by parts gives us ( 14)

∞ 0 ϕ t n x ν(dx) = ∞ 0 1 x 2 ϕ t 1 x ν(nx)dx.
Assume, without loss of generality, that t 1 < • • • < t d and δ 1 = 1. We have the following explicit expression for ϕ t , ϕ t (s) = P(N (st 1 ) odd)

d k=2 P(N (s(t k -t k-1 )) = |δ k -δ k-1 | mod 2) = 1 -e -2st1 2 d d k=2 1 + (-1) |δ k -δ k-1 | e -2s(t k -t k+1 ) ,
from which we can easily deduce that for t fixed, ϕ t (s) is bounded and there exist T > 0 and C > 0 such that |ϕ t (s)| ≤ Ce -sT for all s > 0. A standard argument using the Potter bounds ([22, Corollary 10.5.8]) tells us that

lim n→∞ 1 ν(n) ∞ 0 ϕ t n x ν(dx) = ∞ 0 1 x 2 ϕ t 1 x lim n→∞ ν(nx) ν(n) dx = ∞ 0 ϕ t 1 x x β-2 dx = ∞ 0 ϕ t 1 x βx β-1 dx = 0 P N (tr) = δ mod 2 βr -β-1 dr. Since Var( M δ nt ) = ∞ k=1 Var 1 { N k (nt)=δ mod 2} ≤ ∞ k=1 P N k (nt) = δ mod 2 = E M δ nt ,
the L 2 convergence follows.

Proposition 3. Under the assumption (1), for all

d ∈ N, δ ∈ Λ d , M δ nt -E M δ nt √ d n t∈[0,1] d ⇒ M δ t t∈[0,1] d in D([0, 1] d )
with respect to the J 1 -topology, and the limiting random field has a version with continuous sample paths.

Proof. We continue to use the notation in the proof of Lemma 1. For s, t

∈ [0, 1] d , Cov( M δ nt , M δ ns ) = ∞ k=1 Cov 1 { N k (nt)=δ mod 2} , 1 { N k (ns)=δ mod 2} = ∞ 0 Cov 1 { N (nt/x)=δ mod 2} , 1 { N (ns/x)=δ mod 2} ν(dx).
Since d n = ν(n), we can use the same argument as in the proof of Lemma 1 to show that

lim n→∞ Cov( M δ nt , M δ ns ) d n = ∞ 0 Cov 1 { N (rt)=δ mod 2} , 1 { N (rs)=δ mod 2} βr -β-1 dr = Cov(M δ t , M δ s ). Write M δ nt = M δ nt -E M δ nt .
For each n, M δ nt is the sum of independent random variables that are centered and uniformly bounded (by 2). Further, √ d n → ∞ as n → ∞. Therefore, the Lindeberg-Feller condition holds. Together with the Cramér-Wold's device, this shows the convergence of the finitedimensional distributions in the statement of the proposition.

It remains to prove the tightness. Letting

A η := (t, t ) ∈ [0, 1] d × [0, 1] d : max j=1,...,d |t j -t j | ≤ η ,
it is enough to prove that for all > 0, (15) lim

η↓0 lim sup n→∞ P sup (t,t )∈Aη |M δ nt -M δ nt | √ d n > = 0.
It is easy to see that it suffices to show [START_REF] Owada | Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows[END_REF] with A η replaced by

A (i) η := (t, t ) ∈ [0, 1] d × [0, 1] d : 0 ≤ t i ≤ t i ≤ t i + η, t j = t j , j = i ,
for all i = 1, . . . , d, and we prove [START_REF] Owada | Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows[END_REF] for A

η . By [START_REF] Durieu | From infinite urn schemes to decompositions of selfsimilar Gaussian processes[END_REF]Lemma 3.7] which is due to a chaining argument, it suffices to establish the following: for all (t, t ) ∈ A

M δ nt -M δ nt ≤ N (n(t 1 + η)) -N (nt 1 ) + nη almost surely, (1) η , (16)
and for all p ∈ N and γ ∈ (0, β), there exists C p,γ > 0 such that for all (t, t ) ∈ A

E M δ nt -M δ nt 2p ≤ C p,γ (|t 1 -t 1 | γp ν(n) p + |t 1 -t 1 | γ ν(n)) . (1) η , (17)

Fix (t, t ) ∈ A

(1) η and to simplify the notation introduce

B k = {N k (nt) = δ mod 2} and B k = {N k (nt ) = δ mod 2}. We have B k ∆B k ⊂ {N k (nt 1 ) -N k (nt 1 ) = 0} .
To show [START_REF] Pipiras | Long-range dependence and self-similarity[END_REF], it suffices to observe that

M δ nt -M δ nt = ∞ k=1 1 B k -1 B k -P(B k ) + P(B k ) ≤ ∞ k=1 1 B k ∆B k + ∞ k=1 P(B k ∆B k ) ≤ N (nt 1 ) -N (nt 1 ) + EN (n(t 1 -t 1 )) ≤ N (nt 1 ) -N (nt 1 ) + nη.
To show [START_REF] Pipiras | Stable non-Gaussian self-similar processes with stationary increments[END_REF], by Rosenthal's inequality [START_REF] Rosenthal | On the subspaces of L p (p > 2) spanned by sequences of independent random variables[END_REF], for some constant C depending on p ∈ N only,

E M δ nt -M δ nt 2p ≤ C ∞ k=1 E 1 B k -1 B k -P(B k ) + P(B k ) 2p + ∞ k=1 E 1 B k -1 B k -P(B k ) + P(B k ) 2 p . Since E|1 A -1 B -P(A) + P(B)| 2p ≤ 2 2p-1 Var(1 A -1 B ) ≤ 2 2p-1 P(A∆B) , the above expression does not exceed C ∞ k=1 P (B k ∆B k ) + ∞ k=1 P (B k ∆B k ) p ≤ C [V (n(t 1 -t 1 )) + V (n(t 1 -t 1 )) p ] with V (t) = ∞ k=1 P(N k (t) = 0) = ∞ k=1 1 -e -p k t .
By [7, Lemma 3.1], for each γ ∈ (0, β), there exists C γ such that

V (nt) ≤ C γ t γ ν(n) for all t ∈ [0, 1], n ∈ N.
Therefore, [START_REF] Pipiras | Stable non-Gaussian self-similar processes with stationary increments[END_REF] follows. Therefore, we have proved [START_REF] Owada | Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows[END_REF], which also implies that the limit process has a version with continuous sample path [START_REF] Straf | Weak convergence of stochastic processes with several parameters[END_REF]Theorem 5.6]. This completes the proof.

Proof of Theorem 2. We will prove the second part of the theorem using a multivariate version of the change-of-time lemma from [5, p. 151] (the proof of which is the same as that of the univariate version). Since the limiting random field is continuous, the first part will then follow.

Let (τ n ) n∈N0 be the arrival times of the Poisson process N (with

τ 0 = 0). For t ∈ R d + , set τ nt = (τ nt1 , . . . , τ nt d ) ∈ R d + .
By the strong law of large numbers and monotonicity,

λ n ≡ τ nt n ∧ 2 t∈[0,2] d → id almost surely,
as n → ∞, where id is the identity function from [0, 2] d to [0, 2] d . By the multivariate change-of-time lemma,

  M δ n((τ nt /n)∧2) √ d n   t∈[0,2] d ⇒ M δ t t∈[0,2] d .
In particular, the convergence also holds if the random fields are restricted to t

∈ [0, 1] d . Since M δ nt -EM δ nt √ d n t∈[0,1] d =   M δ n(τ nt /n) √ d n   t∈[0,1] d , and
lim n→∞ P τ nt n ∧ 2 t∈[0,1] d = τ nt n t∈[0,1] d = 1,
the desired result follows.

= exp -

R+×Ω δ∈Λ d | a, δ | α 1 { N (rt)=δ mod 2} βr -β-1 dr dP = exp - δ∈Λ d | a, δ | α m δ t .
Similarly, with the notation n j = nt j ,

E exp   i d j=1 a j U nj b n   = E exp   i d j=1 a j b n ∞ k=1 ε k 1 {Yn j ,k odd}   = E exp   i δ∈Λ d a, δ b n ∞ k=1 ε k d j=1 1 {Y n j ,k =δj mod 2}   .
Let Y denote the σ-algebra generated by (Y i ) i∈N . Then, the above expression becomes

E    δ∈Λ d E   exp   i a, δ b n M δ nt =1 ε   Y      = E δ∈Λ d φ a, δ b n M δ nt ,
with φ(θ) = E exp(iθε 1 ) being the characteristic function of ε 1 . Therefore,

E exp   i d j=1 a j U nj b n   = E δ∈Λ d φ a, δ b n M δ nt = E exp δ∈Λ d M δ nt log φ a, δ b n = E exp - δ∈Λ d M δ nt b α n σ α ε | a, δ | α log φ(c n ) -σ α ε |c n | α , (19)
with c n := a, δ /b n . Recall that assumption [START_REF] Billingsley | Convergence of probability measures[END_REF] implies that (see e.g. [6, Theorem 8.1.10]) log φ(θ) ∼ φ(θ) -1 ∼ -σ α ε |θ| α as θ → 0. By Theorem 2 and dominated convergence theorem the expression in [START_REF] Rosenthal | On the subspaces of L p (p > 2) spanned by sequences of independent random variables[END_REF] converges to the characteristic function in [START_REF] Pitman | Combinatorial stochastic processes[END_REF].

In the remainder of this section we consider the case α ∈ (0, 1) and prove the tightness of the sequence of processes (U nt /b n ) t∈[0,1] , n ≥ 1 in the J 1 -topology on the space D([0, 1]). For κ > 0 we decompose

U n = U +,κ n + U -,κ n with U +,κ n = k≥1 ε k 1 {|ε k |>κbn} 1 {Y n,k odd} U -,κ n = k≥1 ε k 1 {|ε k |≤κbn} 1 {Y n,k odd} .
We start by showing that for any α ∈ (0, 2) and any κ > 0, the sequence of the laws of the processes (U +,κ nt /b n ) t∈[0,1] , n ≥ 1, is tight in the J 1 -topology. To this end define

T (κ) (n) = {i ∈ {1, . . . , n} : Y i = k for k such that |ε k | > κb n } .
Lemma 2. For any κ > 0, lim δ↓0 lim sup

n→∞ P ∃ i 1 , i 2 ∈ T (κ) (n) such that |i 1 -i 2 | < nδ = 0.
Proof. Fix κ > 0 and let > 0. Let Θ n = k≥1 1 {|ε k |>κbn} p k , n ≥ 1. We first prove that ( 20)

∃ C = C( , κ) < +∞ such that lim sup n→∞ P(Θ n > C/n) ≤ .
To see this, choose c > 0. Recalling the notation ν(x) = x β L(x) we consider Θ (c) n = k>ν(cn) 1 {|ε k |>κbn} p k . Note that by [START_REF] Billingsley | Convergence of probability measures[END_REF],

P(Θ (c) n = Θ n ) ≤ P(∃ k ≤ ν(cn) such that |ε k | > κb n ) = 1 -(1 -P(|ε 1 | > κb n )) [ν(cn)] → 1 -e -c β Cεκ -α , as n → ∞,
and so we can choose c = c( , κ) such that

(21) lim sup n→∞ P(Θ (c) n = Θ n ) ≤ /2.
Further, k>ν(cn)

p k = k≥1 p k 1 {p k <1/cn} = (cn,∞) 1 x ν(dx) ∼ β 1 -β (cn) β-1 L(n), as n → ∞,
where the equivalence is due to integration by parts and an application of a Karamata Theorem (see [9, Theorem 1 p. 281]). Thus, [START_REF] Rosiński | On the structure of stationary stable processes[END_REF] follows from the Markov inequality and [START_REF] Samorodnitsky | Null flows, positive flows and the structure of stationary symmetric stable processes[END_REF]. By [START_REF] Rosiński | On the structure of stationary stable processes[END_REF],

EΘ (c) n ∼ β 1 -β (cn) β-1 L(n)P(|ε 1 | > κb n ) ∼ β 1 -β c β-1 C ε κ -α n -1 , as n → ∞. Now (
lim sup n→∞ P ∃ i 1 , i 2 ∈ T (κ) (n) such that |i 1 -i 2 | < nδ ≤ lim sup n→∞ P ∃ i 1 , i 2 ∈ T (κ) (n) such that |i 1 -i 2 | < nδ Θ n ≤ C/n + . Letting (B (
P ∃ i 1 , i 2 ∈ {1, . . . , n} such that |i 1 -i 2 | < nδ and B (n) i1 = B (n) i2 = 1 .
The latter probability has a limit, equal to the probability that a Poisson point process with intensity C over [0, 1] has two points less than δ apart. This probability goes to zero as δ ↓ 0. This completes the proof. Proposition 4. For all α ∈ (0, 2), κ > 0, the sequence of processes (U +,κ ( nt

)/b n ) t∈[0,1] , n ≥ 1, is tight in the J 1 -topology on D([0, 1]).
Proof. Since sup s∈[r,t] |U +,κ nr -U +,κ ns | = 0 as soon as T (κ) ( nt ) \ T (κ) ( nr ) = ∅, from the preceding lemma we infer that for all η > 0,

lim δ↓0 lim sup n→∞ P    sup 0≤r≤s≤t≤1 |r-t|<δ U +,κ nr -U +,κ ns ∧ U +,κ ns -U +,κ nt > η    = 0,
which yields the tightness of (U +,κ n• ) n≥1 (see [START_REF] Billingsley | Convergence of probability measures[END_REF]).

Proof of tightness of (U nt /b n ) t∈[0,1] when α ∈ (0, 1). Let α ∈ (0, 1). In view of Proposition 4, it is sufficient to show that for any η > 0,

κ→0 lim sup n→∞ P sup t∈[0,1] U -,κ nt b n > η = 0. (22) lim
Note that

P sup t∈[0,1] U -,κ nt b n > η ≤ P   k≥1 ε k b n 1 {|ε k |≤κbn} 1 {Y n,k >0} > η   ≤ P Kn k=1 ε k b n 1 {|ε k |≤κbn} > η ,
where K n is the number of nonempty boxes at time n in the infinite urn scheme. Since for large n,

E Kn k=1 ε k b n 1 {|ε k |≤κbn} = E(K n )E ε 1 b n 1 {|ε1|≤κbn} ≤ Cb α n b -1 n κb n P (|ε 1 | > κb n ) → CC ε κ 1-α (
for a finite constant C) by [10, Proposition 2] and Karamata's theorem, [START_REF] Samorodnitsky | Stochastic processes and long range dependence[END_REF] follows by Markov's inequality.

Remark 2. Whether or not the full weak convergence in Theorem 1 holds when α ∈ [START_REF] Aaronson | An introduction to infinite ergodic theory[END_REF][START_REF] Bahadur | On the number of distinct values in a large sample from an infinite discrete distribution[END_REF] remains an open question. In this case, it is not even clear to us whether U α,β has a càdlàg modification: sufficient conditions are given, for example, in [START_REF] Basse-O'connor | On the uniform convergence of random series in Skorohod space and representations of càdlàg infinitely divisible processes[END_REF]Theorem 4.3], but they are not satisfied here.

Discussions

There are a few limit theorems for other statistics in [START_REF] Durieu | From infinite urn schemes to decompositions of selfsimilar Gaussian processes[END_REF] that we have not addressed yet. We provide a brief discussions here focusing on other processes that appear in the limit. As for the proofs, they do not require new ideas (if one ignores the tightness issues).

For the odd-occupancy process U n in (2), one can write, for β < α,

nt , U where here and below M α,β and N are as before. We need the constraint β < α so that U

(1)

n , U (2)
n , U α,β,(1) t and U α,β,(2) t are well defined. Such a weak convergence, for the original randomized Karlin model (ε k ∈ {±1} and α = 2), has been proved in [START_REF] Durieu | From infinite urn schemes to decompositions of selfsimilar Gaussian processes[END_REF]. An appealing feature is that the corresponding decomposition of recovers a decomposition of fractional Brownian motion by a bi-fractional Brownian motion and another smooth self-similar Gaussian process discovered in [START_REF] Lei | A decomposition of the bifractional Brownian motion and some applications[END_REF], and in particular, in this case the two processes are independent. For α ∈ (0, 2), the convergence of finite-dimensional distributions to the decomposition still holds, although U α,β, (1) and U α,β, (2) are no longer independent. The convergence in ( 23) could be established by computing characteristic functions and applying the same conditioning trick.

Another statistics considered in [START_REF] Durieu | From infinite urn schemes to decompositions of selfsimilar Gaussian processes[END_REF][START_REF] Karlin | Central limit theorems for certain infinite urn schemes[END_REF] is the occupancy process

Z n := ∞ k=1 ε k 1 {Y n,k >0} .
Correspondingly, the limit process is = Z α (t β ) t≥0 , where (Z α (t)) t≥0 is an SαS Lévy process (Ee iθZ α (1) = e -|θ| α , θ ∈ R). A similar decomposition for Z α,β , and the corresponding limit theorem as in [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] can also be established, again by computing characteristic functions. See [START_REF] Durieu | From infinite urn schemes to decompositions of selfsimilar Gaussian processes[END_REF]Theorem 2.1] for results in the case α = 2.

1 {N

 1 (tr)(ω ) odd} M α,β (dr, dω ), t ≥ 0.

3 .

 3 The multiparameter odd-occupancy process Throughout, for d ∈ N, t = (t 1 , . . . , t d ) ∈ [0, 1] d , we write (10) nt = ( nt 1 , . . . , nt d ) = (n 1 , . . . , n d ) and denote Λ d = {0, 1} d \ {(0, . . . , 0)}.

Theorem 2 .

 2 Under the assumption (1), for all d ∈ N, t ∈ [0, 1] d , δ ∈ Λ d ,

4 .a j 1

 41 Proof of Theorem 1 Proof of convergence of finite-dimensional distributions. Let d ≥ 1 and fix t = (t 1 , . . . , t d ) ∈ [0, 1] d and a = (a 1 , . . . , a d ) ∈ R d . Note that {N (rtj ) odd} α m β (dr) P (dω )  

U n = ∞ k=1 ε k 1 ε k 1

 11 {Y n,k odd} = ∞ k=1 {Y n,k odd} -P(Y n,k odd) + ∞ k=1 ε k P(Y n,k odd) =: U (1)n + U(2) n , and one could eventually prove that[START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] 1 b n U nt , U

( 2 ) 1 {N

 21 nt t∈[0,1] f.d.d. → σ ε U α,β t , U α,β,(1) t , U α,β,(2) tas n → ∞, with U (tr)(ω ) odd} -P (N (tr) odd) M α,β (dr, dω ) tr) odd)M α,β (dr, dω ),

Z α,β t = R+×Ω 1 {N

 1 (tr)>0} M α,β (dr, dω ), t ≥ 0.At the same time, this is nothing but a time-changed SαS Lévy process, as one can verify by computing the characteristic functions that Z α,β t t≥0 f.d.d.

Acknowledgments. The first author would like to thank the hospitality and financial support from Taft Research Center and Department of Mathematical Sciences at University of Cincinnati, for his visits in 2016 and 2017. The second author's research was partially supported by NSF grant DMS-1506783 and the ARO grant W911NF-12-10385 at Cornell University. The third author's research was partially supported by the NSA grants H98230-14-1-0318 and H98230-16-1-0322, the ARO grant W911NF-17-1-0006, and Charles Phelps Taft Research Center at University of Cincinnati.


Bernard Helffer

AND Hynek Kovařík

Mikael Persson Sundqvist

Mikael Persson

Sundqvist On

ON THE SEMI-CLASSICAL ANALYSIS OF THE GROUNDSTATE ENERGY OF THE DIRICHLET PAULI OPERATOR III: MAGNETIC FIELDS THAT CHANGE SIGN

Keywords: 2010 Mathematics Subject Classification. 35P15; 81Q05, 81Q20 Pauli operator, Dirichlet, semi-classical, flux effects

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

in L 2 (Ω, C 2 ). Here, the spin-up component P + (B, h) and spin-down component P -(B, h) are defined by

P ± (B, h) := (hD x 1 -A 1 ) 2 + (hD x 2 -A 2 ) 2 ± hB(x), (2)
D x j = -i∂ x j for j = 1, 2, and the vector potential A = (A 1 , A 2 ) satisfies

B(x) = ∂ x 1 A 2 -∂ x 2 A 1 , ∀ x ∈ Ω. (3)
The reference to A is not necessary when Ω is simply connected, in which case it will be omitted, but it could play an important role if the domain is not simply connected. In the sequel we will write λ D ± (h, A, B, Ω) = inf σ(P ± (B, h)) [START_REF] Freidlin | Random perturbations of dynamical systems[END_REF] and skip the reference to A when not needed. The smallest eigenvalue Λ D (B, h, Ω) of P (B, h) is given by

Λ D (B, h, Ω) = min{λ D -(h, B, Ω), λ D + (h, B, Ω)}. ( 5
)
The Pauli operator is non-negative (this follows from an integration by parts, or from the view-point that the Pauli operator is the square of a Dirac operator) and, as a consequence, the bottom of the spectrum is non-negative,

λ D ± (h, A, B, Ω) ≥ 0.
Moreover, if Γ is defined by Γu = ū, then

P + (B, h)Γ = ΓP -(-B, h).
It immediately follows that

λ D + (h, B, Ω) = λ D -(h, -B, Ω). (6)
Hence, to understand the properties of Λ D (B, h, Ω) it suffices to study λ D -(h, A, B, Ω), and we will mostly do so.

Let us now specify the amount of regularity we assume about our domain Ω and magnetic field B. To do so, we introduce the notation C p,+ to mean the Hölder class C p,α , for some unspecified α > 0.

Assumption 1. The boundary of Ω is continuous and piecewise in the Hölder class C 2,+ . We allow the boundary to have at most a finite number of corners, each with aperture less than π. The magnetic field B is assumed to be in class C 0 (Ω).

For later reference, we introduce the notation C p,+ pw for the piecewise C p,+ condition from the assumption. From now we will always work under Assumption 1.

1.2. The state of art. Given a magnetic field B, we introduce the sets Ω + B and Ω - B as the subsets of Ω where the magnetic field B is positive and negative, respectively,

Ω + B = {x ∈ Ω : B(x) > 0} and Ω - B = {x ∈ Ω : B(x) < 0}. ( 7
)
Assuming that Ω + B is non-empty, we know from [START_REF] Ekholm | Estimates for the lowest eigenvalue of magnetic Laplacians[END_REF][START_REF] Helffer | On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains[END_REF] that λ D -(h, A, B, Ω) is exponentially small as the semi-classical parameter h > 0 tends to zero. More precisely, we have Theorem 1.1 ([3, 8]). Let Ω be a connected domain in R 2 . If B does not vanish identically in Ω there exists > 0 such that, for all h > 0 and for all A such that curl

A = B, λ D -(h, A, B, Ω) ≥ λ D (Ω) h 2 exp(-/h) . ( 8
)
Here, λ D (Ω) denotes the ground state energy of the Dirichlet Laplacian in Ω.

In [START_REF] Ekholm | Estimates for the lowest eigenvalue of magnetic Laplacians[END_REF], the statement in the theorem is proved under the assumption that Ω is simply connected. The generalization to non-simply connected domains, given in [START_REF] Helffer | On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains[END_REF], was relatively straight forward using domain monotonicity of the ground state energy in the case of the Dirichlet problem,

λ D -(h, A, B, Ω) ≥ λ D -(h, B, Ω) ,
as soon as extensions of A and B to the simply connected envelope Ω of Ω were constructed. The proof in [START_REF] Ekholm | Estimates for the lowest eigenvalue of magnetic Laplacians[END_REF] gives a way of computing a lower bound for , by considering the oscillation of the scalar potential ψ, i.e. of any solution of ∆ψ = B, and optimizing over ψ. For future reference, we introduce a specific choice of the scalar potential by letting the function ψ 0 be a solution of

∆ψ 0 = B(x) in Ω , ψ 0 = 0 on ∂Ω. ( 9
)
The regularity conditions in Assumption 1 guarantee that ψ 0 belongs to C 0 (Ω) ∩ C 2 (Ω) and that it has C 1 regularity up to the boundary, away from the corners. With this particular choice, proposed for simply connected domains in [START_REF] Helffer | On the semi-classical analysis of the Dirichlet Pauli operator[END_REF], we have Theorem 1.2 ([3]). Assume that Ω is a simply connected domain in R 2 and that ψ 0 satisfies [START_REF] Helffer | A proof of the Bott inequalities[END_REF]. Then

λ D -(h, A, B, Ω) ≥ λ D (Ω) h 2 exp -2 Osc Ω ψ 0 /h . ( 10
)
For positive magnetic fields the following theorem was proved in [START_REF] Helffer | On the semi-classical analysis of the Dirichlet Pauli operator[END_REF]:

Theorem 1.3 ([7]).
Let Ω be a simply connected domain in R 2 . If B > 0, and if ψ 0 satisfies (9), then, for any h > 0,

λ D -(h, B, Ω) ≥ λ D (Ω) h 2 exp 2 inf Ω ψ 0 /h .
In the semi-classical limit,

lim h→0 h log λ D -(h, B, Ω) = 2 inf Ω ψ 0 .
In the non-simply connected case, effects from the the circulations of the magnetic potential along different components of the boundary could in principle introduce a different constant than inf Ω ψ 0 inside the exponential function. Thus, a modified scalar potential, taking the circulation along the holes into account, was used in [START_REF] Helffer | On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains[END_REF]. It turns out, however, that in the semi-classical limit, such effects disappear, and it is again ψ 0 that gives the correct asymptotics.

Theorem 1.4 ([8]

). Assume that Ω is a connected domain in R 2 , and that B > 0. If ψ 0 is the solution of (9), then, for any A such that curl A = B,

lim h→0 h log λ D -(h, A, B, Ω) = 2 inf Ω ψ 0 .

Main results

The aim of this paper is to extend the above mentioned results to Pauli operators with sign changing magnetic fields.

2.1. The ground state energy of P ± (B, h). It turns out that the scalar potential ψ 0 , the solution of (9), still plays a main role for the asymptotic of the bottom of the spectrum of the Pauli operator. If B > 0 then, by the maximum principle ψ 0 < 0 in Ω, and similarly, if B < 0 then ψ 0 > 0 in Ω. For B with varying sign, it might still be the case that ψ 0 is of constant sign in Ω, but that will depend on B, and the situation is delicate (we study some examples in the sections 5 and 6).

With [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF] in mind we focus again on the eigenvalue λ D -(h, B, Ω). Our first result concerns the case when ψ 0 attains negative values in Ω.

Theorem 2.1. Assume that Ω is a simply connected domain in R 2 , and that inf Ω ψ 0 < 0, where ψ 0 is satisfying [START_REF] Helffer | A proof of the Bott inequalities[END_REF]. Then

lim sup h→0 h log λ D -(h, B, Ω) ≤ 2 inf Ω ψ 0 .
We recall from [START_REF] Helffer | On the semi-classical analysis of the Dirichlet Pauli operator[END_REF] the definitions of Ω + B and Ω - B , and will now turn to the case when both of them are non-void.

We assume in addition that Γ := B -1 (0) ⊂ Ω is of class C 2,+ and that Γ ∩ ∂Ω is either empty or, if non empty, that the intersection is a finite set, avoiding the corner points, with transversal intersection.

Under this assumption Ω ± B satisfies the same condition as Ω from Assumption 1, and we will denote by ψ0 the solution of

∆ ψ0 = B(x) in Ω + B , ψ0 = 0 on ∂Ω + B . ( 11
)
By domain monotonicity, with Ω + B ⊂ Ω, we can apply Theorem 2.1 with Ω replaced by Ω + B and get Corollary 2.2. Assume that Ω is a connected domain in R 2 . Assume further that ψ0 satisfies [START_REF] Luttrell | [END_REF]. Then

lim sup h→0 h log λ D -(h, B, Ω) ≤ 2 inf Ω + B ψ0 . ( 12
)
Now, the main problem is to determine if one of the bounds above, i.e. ( 10) and ( 12), is optimal. We have two possibly enlightening statements on this question. The first one gives a simple criterion under which the upper bound given in Corollary 2.2 is not optimal. Theorem 2.3. Assume that Ω is a simply connected domain in R 2 , Ω + B = ∅, and that ψ0 satisfies [START_REF] Luttrell | [END_REF].

If B -1 (0) either is a compact C 2,+ closed curve in Ω or a C 2,+ line crossing ∂Ω transversally away from the corners, then lim sup h→0 h log λ D -(h, B, Ω) < 2 inf Ω + B ψ0 .
Two examples where this condition is satisfied is when Ω is a disk, and the magnetic field is either radial, vanishing on a circle, or affine, vanishing on a line. We will return to them later.

We mentioned earlier that even though B changes sign, it might happen that the scalar potential ψ 0 does not. Our second statement says that in this case we actually have the optimal result. Theorem 2.4. Suppose that Ω is a simply connected domain in R 2 . If ψ 0 < 0 in Ω, where ψ 0 is the solution of (9), then

lim h→0 h log λ D -(h, B, Ω) = 2 inf Ω ψ 0 .
2.2. The ground state energy of the Pauli operator. We have already mentioned, that from (6) it follows that to understand the lowest eigenvalue of each of the components of P (B, h), it suffices to study one of them, with the extra price that we must do it both for B and -B. To discuss the lowest eigenvalue Λ D (B, h, Ω) of the Pauli operator P (B, h), we will compare the eigenvalues for the spin-up and spin-down components.

If the scalar potential ψ 0 does not changes sign in Ω, we can transfer the earlier results to Λ D (B, h, Ω). Theorem 2.5. Let Ω be a simply connected domain in R 2 , and let ψ 0 be given by [START_REF] Helffer | A proof of the Bott inequalities[END_REF]. If ψ 0 does not change sign in Ω, then

lim h→0 h log Λ D (h, B, Ω) = -2 Osc Ω ψ 0 . ( 13
)
Our final result concerns a case when ψ 0 changes sign.

Theorem 2.6. Let Ω be a simply connected domain in R 2 , and let ψ 0 be given by [START_REF] Helffer | A proof of the Bott inequalities[END_REF]. Assume that

ψ min = inf Ω ψ 0 < 0 < sup Ω ψ 0 = ψ max .
Assume further that ψ -1 0 (ψ max ) contains a C 2,+ closed curve enclosing a non-empty part of ψ -1 0 (ψ min ) or that ψ -1 0 (ψ min ) contains a C 2,+ closed curve enclosing a non-empty part of ψ -1 0 (ψ max ). Then lim

h→0 h log Λ D (h, B, Ω) = -2 Osc ψ 0 . ( 14
)
Remark 2.7. Since ψ 0 can attain its minimum only in Ω + B and its maximum only in Ω - B , see equation [START_REF] Helffer | On the semi-classical analysis of the Dirichlet Pauli operator[END_REF], a necessary condition for the hypothesis of Theorem 2.6 is that B -1 (0) contain a closed curve. A radial magnetic field which changes sign on a disc is a typical example in which this condition is satisfied, see Section 5.

3. Proof of Theorems 2.1, 2.4, 2.5, and 2.6

We assume in this section that Ω is simply connected. As it is standard, we can assume up to a gauge transform that div A = 0 in Ω , A • ν on ∂Ω .

In this case, the solution ψ 0 of (9) satisfies:

A = ∇ ⊥ ψ 0 = (-∂ x 2 ψ 0 , ∂ x 1 ψ 0 ).
We let ψ min = inf Ω ψ 0 and ψ max = sup Ω ψ 0 , and assume that

ψ min < ψ max = 0 . ( 15
)
We note that this is the case when B > 0 in Ω and that this condition implies that Ω B(x) dx > 0. An example of such a magnetic field will be given in Section 6.

Proof of Theorem 2.4. Under Assumption (15), the proofs of Section 4 in [START_REF] Helffer | On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains[END_REF] go through. One can first consider trial states in the form v η exp(-ψ 0 /h) with v η compactly supported in Ω and v η being equal to 1 outside a sufficiently small neighborhood of the boundary, or in the form (this equality defines v)

exp(-ψ 0 /h) -exp(ψ 0 /h) := exp(-ψ 0 /h)v.
The only change consists in replacing [8, equation (4.4)] by

h 2 Ω exp - 2ψ h (∂ x 1 + i∂ x 2 )v 2 dx ≤ 4h Ω |B(x)| dx. ( 16
)
Hence all the statements are unchanged under the condition of replacing Φ = B(x)dx by 2 |B(x)|dx. In particular, the statement in Theorem 2.4 follows.

Proof of Theorem 2.1. If we only assume ψ min < 0, we get by the same calculation as above that 2(ψ min -ψ max ) ≤ lim inf

h→0 h log λ D -(h, A, B, Ω) ≤ lim sup h→0 h log λ D -(h, A, B, Ω) ≤ 2ψ min . ( 17
)
This proves Theorem 2.1.

Proof of Theorem 2.5. We first note that, since Osc Ω ψ 0 = Osc Ω (-ψ 0 ), Theorem 1.2 in combination with [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF] gives

lim inf h→0 h log λ D ± (h, B, Ω) ≥ -2 Osc Ω ψ 0 . ( 18
)
If ψ 0 < 0 in Ω, the claim follows from Theorem 2.4 and (18). If ψ 0 > 0 in Ω, then Theorem 2.4 together with equation ( 6) implies that lim

h→0 h log λ D + (h, B, Ω) = 2 inf Ω (-ψ 0 ) = -2 Osc Ω ψ 0 . ( 19
)
The lower bound (18) then again completes the proof.

Proof of Theorem 2.6. Suppose first that ψ -1 0 (ψ max ) contains a C 2,+ closed curve γ 1 enclosing a non-empty part of ψ -1 0 (ψ min ). Let Ω 1 ⊂ Ω be the region enclosed by γ 1 and define on Ω 1 the function ψ 1 = ψ 0 -ψ max . Then ∆ψ 1 = B, ψ 1 < 0 and ψ 1 = 0 on ∂Ω 1 = γ 1 . The domain monotonicity and Theorem 2.1 thus imply that lim sup

h→0 h log λ D -(h, B, Ω) ≤ lim sup h→0 h log λ D -(h, B, Ω 1 ) ≤ -2 inf Ω 1 ψ 1 = -2 Osc Ω ψ 0 .
On the other hand, if ψ -1 0 (ψ min ) contains a C 2,+ closed curve γ 2 enclosing a non-empty part of ψ -1 0 (ψ max ), then we denote by Ω 2 ⊂ Ω the region enclosed by γ 2 and define ψ 2 = -ψ 0 + ψ min . Hence ∆ψ 2 = -B, ψ 2 < 0 and ψ 2 = 0 on ∂Ω 2 = γ 2 . In view of equation ( 6), Theorem 2.1 and the domain monotonicity we then get lim sup

h→0 h log λ D + (h, B, Ω) ≤ lim sup h→0 h log λ D + (h, B, Ω 2 ) = lim sup h→0 h log λ D -(h, -B, Ω 2 ) ≤ -2 inf Ω 2 ψ 2 = -2 Osc Ω ψ 0 .
In either case, an application of inequality (18) completes the proof.

4. Proof of Theorem 2.3

A deformation argument.

We have seen that we can have ψ < 0 in Ω without to assume B > 0 and that once this property is satisfied we can obtain an upper bound by restricting to the subset of Ω where ψ is negative instead. Hence a natural idea is to consider the family of subdomains of Ω defined by

F = {ω ⊂ Ω, ∂ω ∈ C 2,+ pw : ∆ψ = B in ω and ψ| ∂ω = 0 ⇒ ψ < 0 in ω} . We know that Ω + B ∈ F.
The idea behind the proof of Theorem 2.3 is to show that there exists ω ∈ F such that Ω + B ⊂ ω with strict inclusion. More precisely, we have Proposition 4.1. Let ω ∈ F and let ψ ω be the solution of ∆ψ = B in ω such that ψ ω = 0 on ∂ω. If ∂ ν ψ ω > 0 at some point M ω of ∂ω ∩ Ω, there exists ψ ω attached to ω, with ω ⊂ ω such that

inf ω ψ ω < inf ω ψ ω . ( 20
)
Proof. First we deform ω into ω (smooth and small perturbation). We refer to [START_REF] Henrot | Variation et optimisation de formes -une analyse géométrique-Mathématiques et Applications[END_REF]Chapter 5] for different ways to do this. We can for example extend the outward normal vector field to a vector field defined in a tubular neighborhood of ∂ω near M ω . We call this vector field X 0 , and take a function θ in C ∞ 0 (R 2 ) with compact support near M ω and equal to 1 near M ω . We then consider the vector field X := θX 0 , which is naturally defined in R 2 . If we consider the associated flow Φ t of the vector field X, t → Φ t (ω) defines he desired deformation for t small t. We then define ω = Φ t 0 (ω) for some t 0 > 0, and construct the corresponding scalar potential ψ ω. We claim that comparing ψ ω and ψ ω, which can be done by comparing ψ ω and ψ ω • Φ -1 t 0 which are defined on ω and arbitrarily close (for t 0 small enough) in H 2 (ω), hence in C 0 (ω), we get, noting that ψ ω -ψ ω is harmonic in ω, non positive on ∂ω, strictly negative near M ω and using the maximum principle, that ψ ω < ψ ω in ω. In particular, we get (20).

Proof of Theorem 2.3. With the deformation argument at hand, we are now ready to give a

Proof of Theorem 2.3. This is now an immediate application of Proposition 4.1 if we can show that at some point of B -1 (0), ∂ ν ψ0 = 0. However, from the Hopf boundary lemma it follows that at every regular point of ∂(Ω + B ∩ Ω) we have ∂ ν ψ0 > 0. Hence the claim.

Radial magnetic fields

A typical application of the general theorems concerns radial magnetic fields in Ω = D(0, R), the disk of radius R centered at 0. The following result is a combination of the theorems appearing in Section 2.

Theorem 5.1. Assume that Ω = D(0, R), and that the magnetic field B is radial and continuous. Then,

lim h→0 h log Λ D (h, B, Ω) = -2 Osc ψ 0 .
Proof. We observe that the solution ψ 0 of ( 9) is radial, and write r = x 2 1 + x 2 2 1/2 and φ 0 (r) = ψ 0 (x 1 , x 2 ). In view of Theorem 2.5 we may assume without loss of generality that φ 0 changes sign in (0, R). The claim then follows from Theorem 2.6.

Example.

As an example of a radial field we consider the function

B β (x 1 , x 2 ) = β 2 -r 2
on the unit disc D(0, 1). An explicit solution of ∆ψ 0 = B β in D(0, 1), ψ 0 = 0 on ∂D(0, 1), is radial, and given by

ψ 0 (x 1 , x 2 ) = 1 16 (r 2 + 1 -4β 2 )(1 -r 2 ).
Hence for β ∈ (0, 1 2 ) ∪ ( 1 √ 2 , 1) we can apply Theorem 2.5, while the case

β ∈ [ 1 2 , 1 √ 2
] is covered by Theorem 5.1. A simple calculation then shows that

lim h→0 h log Λ D (h, B β , D(0, 1)) = -2 Osc ψ 0 = -        1 8 (2β 2 -1) 2 β ∈ (0, 1 2 ) 1 2 β 4 β ∈ [ 1 2 , 1 √ 2 ] 1 8 (4β 2 -1) β ∈ ( 1 √ 2 , 1). Remark 5.2. Since Λ D (h, B β , D(0, 1)) = Λ D (h, -B β , D(0, 1)
) the example above also covers the magnetic field -B β , i.e. the case when the magnetic field is negative inside the domain delimited by the circle B -1 β (0).

6.

A magnetic field vanishing on a line joining two points of the boundary 6.1. Preliminaries. We present a numerical study for the case when the zero-set of the magnetic field, B -1 (0), is a line joining two points of the boundary. We consider again Ω = D(0, 1), the disk of radius 1, and assume that

B β (x 1 , x 2 ) = β -x 1 , -1 < β < 1. This means that B -1
β (0) is given by the line x 1 = β. The solution of ∆ψ β = (β -x 1 ) in D(0, 1) , ψ β = 0 on ∂D(0, 1) , is given by

ψ β (x 1 , x 2 ) = 1 8 (x 1 -2β) 1 -x 2 1 -x 2 2 . (21)
A straightforward calculation shows that

max ψ β =    ψ β 2β+ √ 3+4β 2 3 , 0 for -1 < β < 1/2 , 0 for 1/2 ≤ β < 1 , (22)
and

min ψ β =    0 for -1 < β ≤ -1/2 , ψ β 2β- √ 3+4β 2 3 , 0 for -1/2 < β < 1 . ( 23
)

It follows that

Osc

ψ β =              ψ β 2β+ √ 3+4β 2 3 , 0 -1 < β ≤ -1/2 , ψ β 2β+ √ 3+4β 2 3 , 0 -ψ β 2β- √ 3+4β 2 3 , 0 -1/2 < β < 1/2 , -ψ β 2β- √ 3+4β 2 3 , 0 1/2 ≤ β < 1 .
(24) Next, we introduce B + β as the subset of D(0, 1) where B β is positive, i.e.

B + β = {(x 1 , x 2 ) ∈ D(0, 1) | x 1 < β}. The solution ψβ of ∆ ψβ = (β -x 1 ) in Ω + β , ψβ = 0 on ∂Ω + β , ( 25
)
is not explicit, except in the case β = 0, where we have ψ0 = ψ 0 in Ω + 0 . The oscillation of ψβ can be calculated numerically. When β ∈ ( 1 2 , 1), the oscillation of ψβ is strictly smaller than the oscillation of ψ β . Indeed, the function Ψ β = ψ β -ψβ will in this case satisfy ∆Ψ β = 0 in Ω β and Ψ β = 0 on the circular part of the boundary of Ω β and Ψ β = ψ β ≤ 0 on the line x = β. The Maximum principle gives Ψ β < 0 in Ω β , and so ψ β < ψβ in Ω β . See also Figure 1. 6.2. Application. We assume that

- 1 2 < β < 1 2 .
Using the restriction to Ω + 2β , we get by (17) an upper-bound (using ( 23)) involving -ψ β (2β -3 + 4β 2 )/3 . On the other hand we have a lower bound by the general theorem involving the oscillation of ψ β (see (24) for the computation). Here we will discuss the application of Proposition 4.1 when applied with ω = Ω 2β . What we need is to compute

∂ x 1 ψ β (2β, x 2 ) = 1 8 (1 -4β 2 -x 2 2 ) ,
with ψ β defined in (21), and to observe that it does not vanish. As a consequence the upper bound of lim sup h→0 h log λ 1 given by Corollary 2.2 with ω = Ω 2β + is not optimal. Pushing the boundary will indeed improve the upper bound. The question of the optimality of the lower bound remains open.

6.3. Researching a better upper bound. Given β it is interesting to find the largest possible domain Ω max ⊂ D(0, 1) such that the solution to

∆ψ = β -x 1 , in Ω max , ψ = 0, on ∂Ω max (26)
is strictly negative in Ω max . The oscillation of that solution could contribute to a candidate for the optimal constant in the asymptotics of the Pauli eigenvalue. We are only able to consider this problem numerically.

To find Ω max numerically, we follow (a slightly modified version of) an iterative procedure that was kindly suggested by Stephen Luttrell [START_REF] Luttrell | [END_REF], described below. We start by numerically solving the problem on a regular polygon with (many) corners, positioned on the unit disk. Then we look at the sign of the solution close to each corner of the polygon, and move the corresponding points to make the new domain smaller if the calculated value is positive, and larger if it is negative. We also make sure that no point moves out from the disk. This gives us a new set of points. We build a new polygon, and repeat the procedure until the Euclidean distance between the corners of two iterations becomes as small as we wish (see Figure 2 for an example). We denote by ψ β,opt the function we end up with after the iterative procedure (ideally a solution of (26)). In Figure 3 we have made a comparison of the oscillation of ψ β,opt and ψβ , and we find that the oscillation of ψ β,opt is slightly larger than that of ψβ .

Conclusion

We have initially obtained from the previous papers [START_REF] Ekholm | Estimates for the lowest eigenvalue of magnetic Laplacians[END_REF][START_REF] Helffer | On the semi-classical analysis of the Dirichlet Pauli operator[END_REF][START_REF] Helffer | On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains[END_REF] two natural upper bounds and a natural lower bound. We have shown that in general these two initial upper bounds cannot be true. We have also presented particular cases where the results are optimal. In all these cases, the oscillation of ψ 0 is shown to be optimal. If ψ -1 0 (0, +∞) ∩ Ω = ∅ , we have obtained the optimality by constructing an open set ω in Ω for which the solution ψ ω of ∆ψ = B in ω and ψ = 0 on ∂ω satisfies in addition ∂ ν ψ = 0 on ∂ω ∩ Ω.

Numerically it could be interesting to see how to "push the boundary" in Proposition 4.1 in order to get a maximal domain.

Finally, as already observed in [START_REF] Helffer | On the semi-classical analysis of the Dirichlet Pauli operator[END_REF], one can also expect to get upperbounds by using previous results devoted to the asymptotics of the ground state energy of the Witten Laplacian (see [START_REF] Bovier | Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times[END_REF][START_REF] Bovier | Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues[END_REF][START_REF] Freidlin | Random perturbations of dynamical systems[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF]13] and the quite recent note of B. Nectoux [12]). This unfortunately does not lead to any improvment in the cases we have considered.

1 . Introduction 1 . 1 .

 111 The Pauli operator. Let Ω be a bounded, open, and connected domain in R 2 , let B : Ω → R be a bounded magnetic field and h > 0 a semi-classical parameter. We are interested in the analysis of the ground state energy Λ D (h, B, Ω) of the Dirichlet realization of the Pauli operator P (B, h) = P + (B, h) 0 0 P -(B, h) ,

Figure 1 .

 1 Figure 1. Here we have plotted Osc ψ β -Osc ψβ as a function of β ∈ (-1, 1).

Figure 2 .

 2 Figure 2. The set of domains converge quickly. In this example we start with 200 vertices, β = 0.2, and we exit the loop when the square norm of the difference between two consecutive iterations become less than 0.005 (after five steps). The red dots show the vertices used in the domain of the current step, and the black dots the ones that are calculated for the next step.

1 Figure 3 .

 13 Figure 3. The oscillation of ψ β,opt is slightly larger than that of ψβ . Here we show -2 Osc ψ β,opt for -0.5 ≤ β ≤ 1 (orange) and -2 Osc ψβ for -1 ≤ β ≤ 1 (black).

Acknowledgements

B. H. would like to thank D. Le Peutrec for discussions around the work of B. Nectoux. H. K. has been partially supported by Gruppo Nazionale per Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The support of MIUR-PRIN2010-11 grant for the project "Calcolo delle variazioni" (H. K.), is also gratefully acknowledged.

[12] B. Nectoux. Sharp estimate of the mean exit time of a bounded domain in the zero white noise limit. Personal communication.

[13] E. Witten. Supersymmetry and Morse inequalities. J. Diff. Geom. 17, 661-692 (1982).

(Bernard Helffer) Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 rue de la Houssinière, 44322 Nantes, France and Laboratoire de Mathématiques, Université Paris-Sud, France.

E-mail address: bernard.helffer@univ-nantes.fr (Hynek Kovařík) DICATAM, Sezione di Matematica, Università degli studi di Brescia, Italy.

E-mail address: hynek.kovarik@unibs.it (Mikael P. Sundqvist) Lund University, Department of Mathematical Sciences, Box 118, 221 00 Lund, Sweden.

E-mail address: mikael.persson_sundqvist@math.lth.se