Jean-François Bony
email: bony@math.u-bordeaux.fr

Laurent Michel
email: lamichel@math.u-bordeaux.fr

Real diffusion with complex spectral gap

come

Introduction

The generator of a diffusion process is generally a differential operator of order two with real coefficients. In the last decades, the asymptotic of its low-lying eigenvalues has been obtained [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF][START_REF] Bovier | Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues[END_REF][START_REF] Di Gesù | Sharp asymptotics of the first exit point density[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach[END_REF][START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF][START_REF] Hérau | Tunnel effect and symmetries for Kramers-Fokker-Planck type operators[END_REF][START_REF] Landim | Dirichlet's and Thomson's principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes[END_REF][START_REF] Peutrec | Sharp asymptotics for non-reversible diffusion processes[END_REF] in the low temperature regime (Eyring-Kramers law), see [START_REF] Berglund | Kramers' law: validity, derivations and generalisations, Markov Process[END_REF] for a general presentation. These results provide sharp informations on metastability or on return to equilibrium. For reversible processes, the generator is a self-adjoint operator on an appropriate Hilbert space and then its spectrum is always real. For irreversible processes, the generator is no longer self-adjoint on the natural Hilbert space and one can hope to observe non-real eigenvalues or Jordan's blocks. But, as recalled at the end of this part, there are strong constrains on the low-lying spectrum of generators which make such phenomena unlikely and explain why non-real spectra have not been obtained up to now. The goal of this paper is to construct generators with pathologic spectral gap.

We first discuss spectral properties of generators in the general setting of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF] and send the reader to this paper for precise statements and to the references of the previous paragraph for slightly different settings. In [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF], we consider the operator on L 2 (R d ) (1.1)

P = -h div •A • h∇ + 1 2 b • h∇ + h div •b + c,
where the symmetric matrix A = (a j,k (x, h)) j,k , the vector field b = (b j (x, h)) j and the function c(x, h) are smooth and real-valued. Moreover, these functions are symbols and have an asymptotic expansion in power of the parameter h, which is proportional to the temperature. We assume that P has an invariant distribution which has a Gibbs form. More precisely, there exists a confining smooth Morse function f such that P (e -f /h ) = P * (e -f /h ) = 0.

Let 1 ≤ n 0 < +∞ denote the number of minima of f . Hypoelliptic and hypocoercive assumptions are also made. Under these assumptions, P is maximal accretive and has domain

D(P ) = {u ∈ L 2 (R d ); P u ∈ L 2 (R d )},
as proved in Section 3 of [START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF]. The evolution equation naturally associated to P is the heat (or Fokker-Planck) equation

(1.2) h∂ t u(t, x) = -P u(t, x), u(0, x) = u 0 (x), where u 0 (x) ∈ L 2 (R 2 ) is the initial data. The low-lying spectrum of P is given by the following result (see Theorem 3 of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF]).

Theorem 1.1 (Eyring-Kramers law). There exists λ * > 0 such that, for h small enough, P has exactly n 0 eigenvalues counted with their algebraic multiplicity λ 1 (h), . . . , λ n 0 (h) in {z ∈ C; Re z ≤ λ * h}. Moreover, λ 1 (h) = 0 is simple with Ker P = e -f /h C. For n = 2, . . . , n 0 , the eigenvalue λ n (h) satisfies the asymptotic λ n (h) = a n (h)he -2Sn/h with a n (h) ≃ j≥0 a j n h j , S n = f (s n )f (m n ) > 0 for some particular saddle point s n and minimum m n , a 0 n = 0 explicitly known and a j n ∈ R for all j = 0.

Note that the first eigenvalue λ 1 = 0 is always real. Since all the coefficients a j n are real, it is not possible to use the Eyring-Kramers law to construct an operator with non-real small eigenvalues. Moreover, the imaginary part of λ n is always extremely small. More precisely, Remark 1.2. For all n = 1, . . . , n 0 , we have

(1.3) | Im λ n | = O(h ∞ ) Re λ n .
On the other hand, the particular form (1.1) of the generator P induces symmetries on its spectrum, as remarked on page 15 of [START_REF] Peutrec | Sharp asymptotics for non-reversible diffusion processes[END_REF]. More precisely, since the coefficients of P are real-valued and the domain of P is stable by complex conjugation, we get (1.4) (Pλ)u = (Pλ)u, for all λ ∈ C and u ∈ D(P ). This implies the following property which is also satisfied for PT -symmetric operators (see for instance [START_REF] Benbernou | Non-real eigenvalues for PT -symmetric double wells[END_REF] for the bifurcation of eigenvalues from the real axis to the complex plane).

Remark 1.3. The spectrum of P is invariant by complex conjugation.

In particular, when f has exactly two minima, P has two small eigenvalues λ 1 = 0 and λ 2 by Theorem 1.1. Since λ 2 = λ 2 by Remark 1.3, these two small eigenvalues are always real and simple for h small enough (see Remark 1.10 of [START_REF] Peutrec | Sharp asymptotics for non-reversible diffusion processes[END_REF]).

More generally, if the asymptotic expansion of λ n given by the Eyring-Kramers law is different from that of the other eigenvalues, then λ n is real and simple for h small enough. As an example, if the (Arrhenius) exponential factors S n are all different, then all the small eigenvalues λ n are real and simple for h small enough. This shows that the exponentially small eigenvalues of the generator of a diffusion as in (1.1) are generically real.

We now construct operators of the form (1.1) with non-real small eigenvalues or Jordan blocks. From the two previous paragraphs, the associated Morse function f must have at least 3 minima and some of exponential factors

S n = f (s n ) -f (m n ) must coincide.

Statement of the results

On R 2 , we consider a smooth Morse function f with f (x) = x 2 outside a compact set and which is invariant under R, the rotation of angle 2π/3 around 0. Moreover, we assume that the set of critical points of f consists of 3 (global) minima m 1 , m 2 , m 3 , 3 saddle points s 1 , s 2 , s 3 and 1 (local) maximum M as in Figure 1. Let P 0 be the Witten Laplacian associated to the function f , that is (2.1)

P 0 = d * f • d f with d f = e -f /h • h∇ • e f /h = h∂ x 1 + ∂ x 1 f h∂ x 2 + ∂ x 2 f .
A classical computation shows that this operator has the form

P 0 = -h 2 ∆ + |∇f | 2 -h∆f.
Since f is a compactly supported perturbation of x 2 , P 0 is self-adjoint on the domain of the harmonic oscillator

D(P 0 ) = H 2 (R 2 ) ∩ x -2 L 2 (R 2
), has a compact resolvent, P 0 ≥ 0 and

Ker P 0 = e -f /h C.
The spectrum of Witten Laplacians in such a geometric configuration has been studied in [START_REF] Hérau | Tunnel effect and symmetries for Kramers-Fokker-Planck type operators[END_REF]Section 7.4], [START_REF] Michel | About small eigenvalues of the Witten Laplacian[END_REF]Section 7C3] and [START_REF] Le Peutrec | Bar codes of persistent cohomology and arrhenius law for p-forms[END_REF]Section 9.3]. We send the reader to [START_REF] Helffer | Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten[END_REF] or to the second edition of the book [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF] for details on Witten Laplacians.

Throughout the paper, we set S = f (s)f (m) > 0 and µ(s) < 0 denotes the unique negative eigenvalue of Hess f (s). Since f is invariant by rotation, these quantities do not depend on the minimum m and the saddle point s where they are computed. The bottom of the spectrum of P 0 is given by the following result. Proposition 2.1 (low eigenvalues of P 0 ). There exists λ * > 0 such that, for h small enough, P 0 has exactly three eigenvalues counted with multiplicity λ

1 (h), λ 2 (h), λ 3 (h) in ] -∞, λ * h]. Moreover, λ 1 = 0, λ 2 = λ 3 and λ 2 ∼ 3|µ(s)|| det Hess f (m)| 1/2 π| det Hess f (s)| 1/2 he -2S/h .
This Proposition is mainly a consequence of previous results (see [START_REF] Hérau | Tunnel effect and symmetries for Kramers-Fokker-Planck type operators[END_REF][START_REF] Michel | About small eigenvalues of the Witten Laplacian[END_REF]). The unique novelty is that λ 2 has multiplicity two. This point and other spectral properties of P 0 are proved in Section 3. Since f is invariant under R, so are P 0 and all its eigenspaces.

µ1 = λ1 = 0 µ2 µ3 λ2 = λ3 λ * h/2 R Figure 2.
The low-lying eigenvalues of P 0 and P com .

We now construct an operator having a non-real spectral gap. For that, we perturb the operator P 0 by an anti-adjoint differential operator of order one. More precisely, we consider the operator (2.2)

P com = P 0 + B with B = 1 2 b • h∇ + h div •b .
We require that the vector-valued function b(x, h) ∈ C ∞ 0 (R 2 ; R 2 ) is a compactly supported real symbol of class S(h ∞ ), where

S(r) = b(x, h) ∈ C ∞ (R 2 ); ∀α ∈ N 2 , ∃C α > 0, ∀x ∈ R 2 , ∀h ∈]0, 1], |∂ α x b(x, h)| ≤ C α r(h)}, and b ∈ S(h ∞ ) means that b ∈ S(h j )
for all j ∈ N. In particular, P com is closed on the domain D(P 0 ). We also assume that (2.3) B(e -f /h ) = 0.

Then, the operator P com enters into the setting of (1.1).

Theorem 2.2 (Non-real eigenvalues). Let r(h) = O(h ∞ ) be a positive function. There exists a function b(x, h) ∈ C ∞ 0 (R 2 ; R 2 ) ∩ S(r) with (2.3) such that the spectrum of P com satisfies σ(P com ) ∩ {z ∈ C; Re z < λ * h/2} = µ 1 (h), µ 2 (h), µ 3 (h) , for h small enough, with µ 1 = 0, µ 2 = λ 2 + O(r), µ 3 = µ 2 and

(2.4) Im µ 2 = 0.

Here and in the sequel, σ(T ) denotes the spectrum of the operator T and the eigenvalues µ • (h) are simple for h small enough. The setting of Theorem 2.2 is illustrated in Figure 2. The symbol b(x, h) is only partially explicit (see Lemma 4.1, (4.6) and (5.3)). In particular, its size may be way more smaller than r. Then, the imaginary part of µ 2 and µ 3 is very small. But, as explained in (1.3), it is always the case in the general setting. Theorem 2.2 is proved using the perturbation theory at fixed h small enough. In particular, its proof shows that operators as in (2.2) with a small enough anti-adjoint part B have a nonreal spectral gap as soon as the leading term coming from the perturbation theory does not vanish (see Lemma 4.4). In this sense, the situation of Theorem 2.2 is generic.

For h small enough, let Π µ j denote the spectral projection of P com associated to the eigenvalue µ j . Using the Cauchy formula, it can be written

Π µ j = 1 2iπ γ (z -P com ) -1 dz,
where γ is a sufficiently small loop around µ j positively oriented. The relations (1.1) and (2.3) give

(2.5) Π µ 1 = e -f (x)/h e -f /h 2 e -f /h , • and Π µ 2 u = Π µ 3 u.
Let u(x, h) be an eigenvector of P com associated to the eigenvalue µ 2 . From (2.5), u is an eigenvector associated to the eigenvalue µ 3 . Then, (Re u, Im u) is a basis of Im Π µ 2 ⊕Im Π µ 3 = Ker(P comµ 2 ) ⊕ Ker(P comµ 3 ). In particular, u cannot be a real (or purely imaginary) function. From Corollary 1.5 of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF] and Theorem 2.2, the solution of the evolution equation (1.2) associated to P com satisfies the following metastable behavior.

Corollary 2.3. Consider P com as in Theorem 2.2 with h small enough. For all u 0 ∈ L 2 (R 2 ), the solution u = e -tPcom/h u 0 of (1.2) can be written

e -tPcom/h u 0 = u 1 + e -tµ 2 /h u 2 + e -tµ 3 /h u 3 + ε(t) = u 1 + e -t
with u j = Π µ j u 0 for j = 1, 2, 3, u c = u 2 + u 3 , u s = iu 3 -iu 2 and ε(t) L 2 (R 2 ) ≤ Ce -t/C u 0 L 2 (R 2 ) ,
for some constant C > 0 independent of t, h, u 0 .

If the function u 0 is real-valued, (2.5) implies that u 2 = u 3 and then u c and u s are also real-valued. If in addition u 2 , u 3 , u c or u s does not vanish identically, the discussion below

(2.5) shows that (u c , u s ) is a basis of Im Π µ 2 ⊕ Im Π µ 3 . In that case, t -→ cos(t Im µ 2 /h)u c + sin(t Im µ 2 /h)u s ,
is a non-vanishing periodic function of period 2πh| Im µ 2 | -1 which reaches all the directions of Im Π µ 2 ⊕ Im Π µ 3 . Then, the subprincipal term in (2.6), which measures the return to equilibrium, is oscillating. Nevertheless, this phenomenon may be difficult to see in the applications since (1.3) implies that this subprincipal term decays more quickly than it oscillates.

We now construct an operator having a spectral gap with a Jordan block. For that, we consider perturbations of P 0 of the form

P Jor = d * f • 1 + χ(x, h) Id • d f + B where B = 1 2 b • h∇ + h div •b , Id denotes the 2 × 2 identity matrix, χ ∈ C ∞ 0 (R 2 ; R) ∩ S(h ∞ ) and b ∈ C ∞ 0 (R 2 ; R 2 ) ∩ S(h ∞ ).
For h small enough, such an operator falls within the general framework of (1.1).

Theorem 2.4 (Jordan block). Let r(h) = O(h ∞ ) be a positive function. There exist functions χ(x, h) ∈ C ∞ 0 (R 2 ; R) ∩ S(r) and b(x, h) ∈ C ∞ 0 (R 2 ; R 2 ) ∩ S(r) with (2.
3) such that, for h small enough,

σ(P Jor ) ∩ {z ∈ C; Re z < λ * h/2} = λ 1 , λ 2 of
multiplicity 1 and 2 respectively, and P Jor has a non-trivial Jordan block associated with the eigenvalue λ 2 .

Let Π λ 1 and Π λ 2 be the spectral projectors of P Jor associated to λ 1 and λ 2 respectively. From Theorem 2.4 and (1.4), there exists an orthonormal basis of real-valued functions, denoted (e 1 , e 2 ), of Im Π λ 2 such that Π λ 2 P Jor Π λ 2 expressed in the basis (e 1 , e 2 ) writes

(2.7) λ 2 ρ 0 λ 2 ,
for some ρ(h) ∈ R \ {0} (see (6.26)). Note that e 1 and e 2 are unique modulo multiplication by ±1. By construction, the constant ρ is very small. More precisely,

(2.8) |ρ(h)| = O(h ∞ λ 2 ) = O(h ∞ e -2S/h ).
But, as for the imaginary part of the eigenvalues (1.3), this is a general fact: any Jordan block associated with a small eigenvalue of an operator of the form (1.1) satisfies an estimate similar to (2.8). Indeed, all the terms in the asymptotic expansion of the interaction matrices are self-adjoint (see Section 6 of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF]).

It is difficult to construct by perturbation theory an operator of the form (2.2) satisfying Theorem 2.4. Indeed, Lemma 4.4 shows that such operators enter into the setting of Theorem 2.2 as soon as the leading term in the perturbation theory does not vanish. This is why we consider here more general perturbations which allow to "generate all the possible" leading terms (see Section 4 of [START_REF] Sjöstrand | Semiclassical resonances generated by nondegenerate critical points[END_REF] for similar ideas in resonances theory).

Contrary to Theorem 2.2, the spectral situation of Theorem 2.4 is unstable. Generically, a small perturbation (in the setting of (1.1)) splits the double eigenvalue λ 2 into two nonreal conjugate eigenvalues. This is general fact concerning the Jordan blocks. Moreover, the second eigenvalue of P 0 and P Jor is the same. The proof of Theorem 2.4 allows to change slightly the second eigenvalue of P Jor , but the actual statement simplifies the result.

Combining with Corollary 1.5 of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF], the time evolution equation associated to P Jor satisfies Corollary 2.5. Consider P Jor as in Theorem 2.4 with h small enough. For all u 0 ∈ L 2 (R 2 ), the solution u = e -tP Jor /h u 0 of (1.2) can be written (2.9) e -tP Jor /h u 0 = u 1 + te -tλ 2 /h u 2 + e -tλ 2 /h u 3 + ε(t),

with u 1 = Π λ 1 u 0 , u 2 = -ρ e 2 , Π λ 2 u 0 e 1 , u 3 = Π λ 2 u 0 and ε(t) L 2 (R 2 ) ≤ Ce -t/C u 0 L 2 (R 2 ) ,
for some constant C > 0 independent of t, h, u 0 .

In particular, we have the sharp return to equilibrium result (2.10) e -tP Jor /h -Π λ 1 ∼ αte -tλ 2 /h , in the limit t → +∞ for h small enough and some positive constant α(h) > 0. This estimate shows that the return to equilibrium is not purely exponentially decreasing in general and that some powers of t may appear.

Until now, we have only considered the spectral gap given by exponentially small eigenvalues, corresponding to several minima. But, if we study higher eigenvalues, it is more simple to have non-real spectrum. For ε ∈ R, consider the operator

P = -h 2 ∆ + x 2 -2h + ε x 1 h∂ x 2 -x 2 h∂ x 1 .
It enters in the setting of (1.1) with the Morse function f (x) = x 2 /2 which has a unique minimum at x = 0. The bottom of its spectrum is given by Proposition 2.6. For h > 0 and ε = 0, we have σ(P) ∩ {z ∈ C; Re z < 4h} = 0, 2h + iεh, 2hiεh , and these eigenvalues are simple.

Then, this operator has a non-real spectral gap. Nevertheless, it is not given by exponentially small eigenvalues responsible of metastable dynamics. In this simple well situation, the Eyring-Kramers law only provides the asymptotic of 0, the first eigenvalue of P. Note also that, for ε = 0 fixed, these eigenvalues do no longer satisfy (1.3).

The rest of the paper is organized as follows. In the next section, we collect some properties of the reference operator P 0 used in the sequel. Section 4 is devoted to the construction of the anti-adjoint perturbation B based on properties of nodal sets. This construction allows to prove Theorem 2.2 (resp. Theorem 2.4) in Section 5 (resp. Section 6) combining the perturbation theory and previous results of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF]. Lastly, Proposition 2.6 is obtained in Section 7 by direct computations.

Spectral properties of P 0

This part is devoted to the proof of Proposition 2.1 and to other technical results on P 0 . From Theorem 1.1, there exists λ * > 0 such that, for h small enough, P 0 has exactly three eigenvalues counted with multiplicity 0

= λ 1 (h) < λ 2 (h) ≤ λ 3 (h) in ] -∞, λ * h].
Moreover, λ 2 and λ 3 are exponentially small. Eventually, the asymptotic

λ 2 , λ 3 ∼ 3|µ(s)|| det Hess f (m)| 1/2 π| det Hess f (s)| 1/2 he -2S/h ,
is a direct consequence of Section 7C3 of [START_REF] Michel | About small eigenvalues of the Witten Laplacian[END_REF] (see also [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF][START_REF] Hérau | Tunnel effect and symmetries for Kramers-Fokker-Planck type operators[END_REF]). We denote

Π = 1 [λ 1 ,λ 3 ] (P 0 ), Π 1 = 1 {λ 1 } (P 0 ) and Π 23 = 1 [λ 2 ,λ 3 ] (P 0 ),
the spectral projectors of P 0 . They satisfy Π = Π 1 + Π 23 ,

(3.1) Π • u = Π • u and RΠ • = Π • R,
since P 0 commutes with R and the complex conjugation. Here, R is viewed as the rotation acting on functions (i.e. R(f

) = f • R for f ∈ L 2 (R d )). Let χ ∈ C ∞ 0 (R 2 ; [0, 1]) be supported near m 1 with χ = 1 near m 1 . We set (3.2) ψ 1 = χ(x)e -f (x)/h χe -f /h , ψ 2 = ψ 1 • R and ψ 3 = ψ 1 • R 2 ,
with the estimates

(3.3) χe -f /h ∼ √ πh(det Hess f (m)) -1/4 and e -f /h ∼ √ 3πh(det Hess f (m)) -1/4 .
Since f is invariant by rotation, these quantities do not depend on the minimum m where they are computed. The function ψ j is localized near m j from Figure 1, and the family (ψ j ) j is orthonormal. We then set ϕ j = Πψ j . We have ψ j ∈ C ∞ 0 (R 2 ) and ϕ j ∈ S(R 2 ). A classical result (see the proof of Proposition 2.5 of [START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF]) yields (3.4)

ϕ j = ψ j + O(e -δ/h ),
showing that the family (ϕ j ) j is an almost orthonormal basis of Im Π. Furthermore, (3.1) implies that (3.5) the function ϕ j is real for all j and ϕ j = ϕ k R j-k for all j, k.

Eventually (3.2), (3.3) and (3.4) give

(3.6) √ 3 e -f /h e -f /h = ϕ 1 + ϕ 2 + ϕ 3 + O(e -δ/h ).
We can now show that λ 2 = λ 3 . Lemma 3.1. For h small enough, the second eigenvalue λ 2 of P 0 has multiplicity 2.

Proof. We prove this result by contradiction. Assume that λ 2 has multiplicity one and let u be a normalized eigenvector. From (1.4), we can always choose u real-valued. In the basis (ϕ j ) j of Im Π, this function can be written

(3.7) u = u 1 ϕ 1 + u 2 ϕ 2 + u 3 ϕ 3 ,
for some u j ∈ R. Since (ϕ j ) j is almost orthonormal,

(3.8) 1 = u 2 = u 2 1 + u 2 2 + u 2 3 + O(e -δ/h
). Applying the rotation R, (3.5) and (3.7) gives

u • R = u 1 ϕ 2 + u 2 ϕ 3 + u 3 ϕ 1 .
On the other hand, we have

P (u • R) = (P u) • R = λ 2 u • R. Since λ 2 is simple, there exists α ∈ C such that u • R = αu, that is (3.9) u 1 = αu 2 , u 2 = αu 3 and u 3 = αu 1 .
Since u and u • R are real valued, we necessarily have α ∈ R. The relation (3.9) implies u j = α 3 u j for j = 1, 2, 3. Since at least one of the u j is non-zero from (3.8), we get α 3 = 1 and then α = 1. Thus, u 1 = u 2 = u 3 and |u 1 | = 3 -1/2 + O(e -δ/h ). On the other hand, u and e -f /h are orthogonal since they belong to two different eigenspaces of the self-adjoint operator P 0 . Combining the previous properties with (3.6), it comes

0 = √ 3 e -f /h -1 e -f /h , u = |u 1 | ϕ 1 + ϕ 2 + ϕ 3 , ϕ 1 + ϕ 2 + ϕ 3 + O(e -δ/h ) = √ 3 + O(e -δ/h ),
which provides a contradiction for h small enough. We have just proved that λ 2 has multiplicity at least two. Since this multiplicity can not be larger than two, we get the lemma.

Construction of the anti-adjoint perturbation B

The anti-adjoint part of P is chosen of the form B = εB with (4.1)

B = d * f • G • d f with G = 0 g -g 0 ,
with d f defined in (2.1), for some constant ε(h) ∈]0, +∞[ and some function g(x, h) ∈ C ∞ 0 (R 2 ; R) fixed in the sequel.

Lemma 4.1. The operator B is formally anti-adjoint, Be -f /h = 0 and

B = 1 2 b • h∇ + h div •b with b(x, h) = h∂ x 2 g -2g∂ x 2 f -h∂ x 1 g + 2g∂ x 1 f ∈ C ∞ 0 (R 2 ; R 2 ).
Proof. The definition (4.1) of B immediately implies that B is formally anti-adjoint and that Be -f /h = 0. Moreover, a direct computation gives

B = -h∂ x 1 + ∂ x 1 f, -h∂ x 2 + ∂ x 2 f 0 g -g 0 h∂ x 1 + ∂ x 1 f h∂ x 2 + ∂ x 2 f = (-h∂ x 1 + ∂ x 1 f )g(h∂ x 2 + ∂ x 2 f ) -(-h∂ x 2 + ∂ x 2 f )g(h∂ x 1 + ∂ x 1 f ) = (∂ x 1 f )g(h∂ x 2 ) + (h∂ x 2 )g(∂ x 1 f ) -(∂ x 2 f )g(h∂ x 1 ) -(h∂ x 1 )g(∂ x 2 f ) -(h∂ x 1 )g(h∂ x 2 ) + (h∂ x 2 )g(h∂ x 1 ) = 1 2 b • h∇ + h div b ,
and the lemma follows.

Since we see B as a perturbation of P 0 and want to use the Kato's theory, we seek the function g ∈ C ∞ 0 (R 2 ; R) such that Π 23 BΠ 23 = 0. For that, let (u, v) be a real-valued orthonormal basis of Im Π 23 . From (4.1), we have

Bu, v = R 2 Ge -f /h h∇e f /h u • e -f /h h∇e f /h v dx = R 2 g ∂ x 2 u∂ x 1 v -∂ x 1 u∂ x 2 v dx, (4.2)

with (4.3)

u = e f /h u, v = e f /h v and g = h 2 e -2f /h g.

Thus, if ∂ x 2 u∂ x 1 v -∂ x 1 u∂ x 2 v
does not vanish identically, it is possible to find g such that Bu, v = 0. This justify the next intermediate result.

Lemma 4.2. For h small enough, we have

∂ x 2 u∂ x 1 v -∂ x 1 u∂ x 2 v ≡ / 0.
Proof. We prove this lemma by contradiction. If it does not hold true, we have

(4.4) ∂ x 2 u∂ x 1 v -∂ x 1 u∂ x 2 v ≡ 0,
for a sequence of positive h which goes to 0. Roughly speaking, this equation means that the level sets of u and v are the same. This leads to consider the nodal sets of u and v whose we recall now the general properties.

Proposition 4.3. Let w be a real-valued eigenvector of P 0 associated to the eigenvalue λ 2 (in particular, w ∈ S(R 2 ; R) and w ≡ / 0). Then, 1) the open set R 2 \ w -1 (0) has precisely two connected components Ω w ± on which ±w > 0, 2) the nodal set w -1 (0) is a (unique) smooth curve without crossing on which ∇w = 0, 3) if w 1 and w 2 are two of such eigenvectors, then w -1

1 (0) ∩ w -1 2 (0) = ∅.
Proof of Proposition 4.3. This result collects classical properties of nodal sets and we send the reader to the corresponding papers for the proofs. First, Section VI.6 of [START_REF] Courant | Methods of mathematical physics[END_REF] (see also [START_REF] Courant | Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke[END_REF]) shows that R 2 \ w -1 (0) has at most two connected components. This result, originally stated in domains, extends to our setting since the potential V is confining. Moreover, if R 2 \w -1 (0) has only one connected component, this function has a constant sign and cannot be orthogonal to the positive function e -f /h , an eigenvector of P 0 associated to its first eigenvalue λ 1 . Summing up, R 2 \ w -1 (0) has precisely two connected components.

The structure of the nodal set w -1 (0) is described in Theorem 2.5 of [START_REF] Cheng | Eigenfunctions and nodal sets[END_REF] in the present two dimensional case (see also Theorem 2.2 in the general case). Outside of isolated critical points, w -1 (0) is the reunion of smooth curves without crossing on which ∇w = 0. At the critical points, a finite number of nodal curves cross and form an equiangular system. If such a critical point exists, then there will be more than two connected components in R 2 \ w -1 (0). Thus, there is no critical point and w -1 (0) is the reunion of smooth curves without crossing. Since such a curve is either periodical or goes to infinity, each curve in w -1 (0) generates a connected component in R 2 \ w -1 (0). Since this set has precisely two connected components, w -1 (0) must be composed of an unique curve on which ∇w = 0. Since w changes sign across w -1 (0), the connected components of R 2 \ w -1 (0) can be labeled Ω w ± in a such way that ±w > 0 on Ω w ± . This proves 1) and 2). It remains to show 3). For that, we follow the proof of Lemma 4.2 of [START_REF] Cheng | Eigenfunctions and nodal sets[END_REF]. Assume that

w -1 1 (0) ∩ w -1 2 (0) = ∅. Since w -1 1 (0) is a single curve, we have w -1 1 (0) ⊂ Ω w 2 -or w -1 1 (0) ⊂ Ω w 2 + . We can suppose that w -1 1 (0) ⊂ Ω w 2 -. Then, Ω w 1 - Ω w 2 -or Ω w 1 + Ω w 2 -. We can suppose that Ω w 1 - Ω w 2
-. Eventually, by Courant's minimum principle, the first eigenvalue of the operator P 0 restricted to Ω w 1 -with Dirichlet boundary condition is greater than the first eigenvalue of the operator P 0 restricted to Ω w 2 -with Dirichlet boundary condition, whereas these two quantity are equal to λ 2 . This is a contradiction and 3) follows.

We now come back to the proof of Lemma 4.2. From (4.3), the zeros of u (resp. v) are those of u (resp. v). Moreover, Lemma 4.3 2) shows that (4.5)

∇ u = e f /h ∇u + u∇e f /h = e f /h ∇u = 0, on u -1 (0). Let x 0 be a point of u -1 (0) ∩ v -1 (0) which is not empty from Lemma 4.3 3), and consider the curve

x(t) ∈ R 2 solution of    ∂ t x(t) = ∂ x 2 u(x(t)) -∂ x 1 u(x(t)) , x(0) = x 0 . The definition of x(t) gives ∂ t u(x(t)) = (∂ x 1 u∂ x 2 u -∂ x 2 u∂ x 1 u)(x(t)) = 0, showing that u(x(t)) = 0 for all t ∈ R.
Combined with Lemma 4.3 2) and (4.5), it implies that x(t) is a parametrization of u -1 (0). On the other hand, (4.4) yields

∂ t v(x(t)) = (∂ x 1 v∂ x 2 u -∂ x 2 v∂ x 1 u)(x(t)) = 0,
showing as before that v(x(t)) = v(x 0 ) = 0 for all t ∈ R. This proves u -1 (0) = v -1 (0) from Proposition 4.3 2). Using Proposition 4.3 1), we deduce

Ω u ± = Ω v ± or Ω u ± = Ω v ∓ .
It implies u, v > 0 or u, v < 0 respectively. On the other hand, u, v = 0 since (u, v) is orthogonal. This contradiction finishes the proof of Lemma 4.2.

Let χ ∈ C ∞ 0 (R 2 ; [0, 1]) with supp χ ⊂ B(0, 1) and χ = 1 on B(0, 1/2). From Lemma 4.2, there exists x 0 = x 0 (h) ∈ R 2 for h small enough such that (∂

x 2 u∂ x 1 v -∂ x 1 u∂ x 2 v)(x 0 ) = 0. By continuity ( u, v ∈ C ∞ (R 2 )), there exists ν = ν(h) ∈]0, 1] such that ∂ x 2 u∂ x 1 v -∂ x 1 u∂ x 2 v
does not change it sign in B(x 0 , ν). We then set

(4.6) g(x, h) = x 0 -1 e -1/ν χ x -x 0 ν ∈ C ∞ 0 (R 2 ; R),
which satisfies, for h small enough and α ∈ N 2 , (4.7)

∀x ∈ R 2 , |∂ α x g(x, h)| = x 0 -1 ν -|α| e -1/ν χ (α) x -x 0 ν ≤ M α x 0 -1 ,
for some constant M α > 0. Combining with Lemma 4.1 and f = x 2 outside a compact set, it shows that b(x, h) is a symbol of class S(1). Moreover, using g = h 2 e -2f /h g and (4.2), this construction yields

(4.8) Bu, v = β,
for h small enough and some constant β(h) = 0.

Lemma 4.4. In any real-valued orthonormal basis of Im Π 23 , the operator Π 23 BΠ 23 writes

Π 23 BΠ 23 = 0 γ -γ 0 ,
for h small enough and some constant γ(h) ∈ R \ {0}.

Proof. In a real-valued orthonormal basis (e 1 , e 2 ) of Im Π 23 , we have

Π 23 BΠ 23 = e 1 , Be 1 e 1 , Be 2 e 2 , Be 1 e 2 , Be 2 .
Since e 1 , e 2 are real-valued, (4.1) gives e 1 , Be 1 = e 2 , Be 2 = 0 and e 2 , Be 1 =e 1 , Be 2 . Let us assume that e 1 , Be 2 = 0 for a sequence of positive h which goes to 0. In that case, the previous relations imply e j , Be k = 0 for all j, k ∈ {1, 2}. Since (e 1 , e 2 ) is a basis of Im Π 23 , it yields Bu, v = 0 in contradiction with (4.8). Summing up, γ(h) := e 1 , Be 2 = 0 for h small enough.

Proof of Theorem 2.2

We now apply the perturbation theory for all h fixed small enough. Let P com = P 0 + B with (5.1)

B = εB,
where B has been constructed in Section 4.

Proposition 5.1. The operator P com is closed on the domain of P 0 . Moreover, for h small enough, there exist ε 0 (h) > 0 and three analytic functions

ε -→ λ 1 (ε, h), λ 2 (ε, h), λ 3 (ε, h) defined for ε ∈ [-ε 0 , ε 0 ] with λ 1 (ε, h) = 0, λ 2 (ε, h) = λ 2 (h) + iγ(h)ε + O h (ε 2 ), λ 3 (ε, h) = λ 2 (h) -iγ(h)ε + O h (ε 2 ), such that σ(P com ) ∩ {z ∈ C; Re z < hλ * /2} = {λ 1 , λ 2 , λ 3 }, for h small enough and ε ∈ [-ε 0 , ε 0 ].
In this statement, the notation O h (1) designs a function bounded by a constant which may depend on h, and the eigenvalues are counted with multiplicity. The constant γ(h) ∈ R \ {0} is whose of Lemma 4.4.

Proof of Proposition 5.1. Since B is a relatively compact perturbation of P 0 from Lemma 4.1, the operator P com is well-defined and closed on the domain of P 0 (see Theorem IV.1.11 of [START_REF] Kato | Perturbation theory for linear operators[END_REF]). Moreover, ε -→ P 0 + εB is a holomorphic family of unbounded operators in the sense of Section VII of [START_REF] Kato | Perturbation theory for linear operators[END_REF]. Recall that λ 1 = 0 is a simple eigenvalue of P 0 . From Lemma 3.1, λ 2 is a double eigenvalue which is semisimple since P 0 is self-adjoint. On the other hand, Lemma 4.4 shows that the eigenvalues of Π 23 BΠ 23 are ±iγ. Since γ = 0, these eigenvalues are different. Then, by the perturbation theory of spectrum, more precisely the perturbation theory of finite systems of eigenvalues (see Section VII.1.3 of [START_REF] Kato | Perturbation theory for linear operators[END_REF]) and the reduction process for semisimple eigenvalues (see Section II.2.3 of [START_REF] Kato | Perturbation theory for linear operators[END_REF]), there exist analytic functions

ε -→ λ 1 (ε, h), λ 2 (ε, h), λ 3 (ε, h) defined for ε ∈ [-ε 0 , ε 0 ] with ε 0 (h) > 0 such that      λ 1 (0, h) = 0, λ 2 (ε, h) = λ 2 (h) + iγ(h)ε + O h (ε 2 ), λ 3 (ε, h) = λ 2 (h) -iγ(h)ε + O h (ε 2 ),
and

σ(P com ) ∩ {z ∈ C; Re z < hλ * /2} = {λ 1 , λ 2 , λ 3 }.
Since 0 is always an eigenvalue of P com by Lemma 4.1, we have λ 1 (ε, h) = 0 for all ε ∈ [-ε 0 , ε 0 ] after a possible shrinking of ε 0 .

The asymptotic expansions in Proposition 5.1 and γ(h

) ∈ R \ {0} yield that (5.2) Im λ 2 (ε, h) = 0 and Im λ 3 (ε, h) = 0, for all ε ∈ [-ε 1 , ε 1 ] \ {0}
with ε 1 (h) > 0 small enough. We eventually choose

(5.3) ε(h) = min ε 0 (h), ε 1 (h), r(h) .
Thus, b(x, h) is a symbol of order at most r(h) from Lemma 4.1 and (4.7). In the domain {z ∈ C; Re z < λ * h/2}, P com has three eigenvalues µ 1 (h) = 0, µ 2 (h) = λ 2 (ε(h), h) and µ 3 (h) = λ 3 (ε(h), h) with Im µ 2 = 0 and Im µ 3 = 0. From (1.4), we automatically have µ 3 = µ 2 . Finally, we can write

P com -z = (B(P 0 -z) -1 -1)(P 0 -z),
for z ∈ B(0, 1) \ σ(P 0 ) with

B(P 0 -z) -1 = B(P 0 + i) -1 1 + (z + i)(P 0 -z) -1 = O r dist(z, σ(P 0 )) -1 ,
from Lemma 4.1. By a classical argument, it implies that µ 2 = λ 2 + O(r) and finishes the proof of Theorem 2.2.

Proof of Theorem 2.4

To find a setting with a Jordan block, we consider operators of the form (6.1)

P τ = P 0 + τ 1 P 1 + τ 2 P 2 + τ 3 P 3 + τ 4 B,
where τ = (τ 1 , τ 2 , τ 3 , τ 4 ) ∈ R 4 , B has been constructed in Section 4 and P ν is as follows. For ν = 1, 2, 3, let χ ν ∈ C ∞ 0 (R 2 ; R) be supported near s ν and equal to 1 in a neighborhood of s ν . We also assume that χ 2 = χ 1 • R and χ 3 = χ 1 • R 2 (see Figure 1). Then, P ν is defined by (6.2)

P ν = d * f • χ ν • d f .
Summing up, P τ can be written

P τ = d * f • χ • d f + τ 4 B, with χ = 1 + τ 1 χ 1 + τ 2 χ 2 + τ 3 χ 3 .
Thus, for τ small enough, P τ enters into the setting of (1.1).

For j = 1, 2, 3, let φ j (x) denote the global quasimode of P 0 supported near the connected component of {f < f (s)} containing m j constructed in Section 4 of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF] (see also [START_REF] Peutrec | Sharp asymptotics for non-reversible diffusion processes[END_REF]). More precisely, this real-valued function can be written

(6.3) φ j (x) = θ j (x) v j (x) + 1 e -f (x)/h = v j (x)e -f (x)/h , where θ j ∈ C ∞ 0 (R 2
) is a plateau function near the connected component of {f < f (s)} containing m j and v j ∈ C ∞ (R 2 ) is given near the support of θ j by (6.4) The function ℓ j s (x, h) ≃ ℓ j s,0 (x) + ℓ j s,1 (x)h + • • • is smooth with ℓ j s,0 (s) = 0 and ∇ℓ j s,0 (s) = 0. As in (3.2), we can make these constructions so that (6.5)

v j (x) =      C -1 0 ℓ j s (x,
φ 2 = φ 1 • R and φ 3 = φ 1 • R 2 .
We choose χ ν in (6.2) such that χ ν = 1 near the support of θ j ∇v j if s ν is close to m j . By comparison with Section 3, we have φ j = 2ψ j + O(e -δ/h ) for some δ > 0, but φ j is a better quasimode than ψ j (see Lemma 6.1 below).

We define the geometric quantities S = f (s)f (m) and

C 1 = 2|µ(s)| | det Hess f (s)| 1/2 ,
where µ(s) is given above Proposition 2.1. The quasimodes φ j 's satisfy Lemma 6.1. For all ν, j, k ∈ {1, 2, 3}, we have

P ν φ j , φ k ∼      C 1 h 2 e -2S/h if s ν is close to m j = m k , -C 1 h 2 e -2S/h if s ν is close to m j = m k , 0 otherwise, (6.6)
P 0 φ j 2 = O(h ∞ )e -2S/h and P ν φ j 2 = O(h ∞ )e -2S/h . (6.7)
Here, the notation "s ν is close to m j = m k " means that j = k and that s ν is close to m j and m k . Roughly speaking, it means that s ν is between m j and m k (see Figure 1).

Proof. This result is similar to Proposition 5.1 of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF] (see also Section 4B of [START_REF] Peutrec | Sharp asymptotics for non-reversible diffusion processes[END_REF]). We only explain here the ideas of the proof and the necessary changes, and we send the reader to [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF] for the details.

Combining (6.2) and (6.3) leads to (6.8)

P ν φ j , φ k = χ ν d f φ j , d f φ k = χ ν e -f /h h∇ v j , e -f /h h∇ v k .
Using ∇ v j = (v j + 1)∇θ j + θ j ∇v j and e -f /h = O(e -(S+δ)/h ) on the support of (v j + 1)∇θ j , the previous equation becomes (6.9)

P ν φ j , φ k = h 2 χ ν θ j θ k ∇v j • ∇v k e -2f /h dx + O e -2(S+δ)/h .
From (6.4), we have on the support of θ j

∇v j = s close to m j C -1 0 ζ(ℓ j s )e -(ℓ j s ) 2 /2h ∇ℓ j s .
If s ν is close to m j = m k , (6.9) writes

P ν φ j , φ k = h 2 C -2 0 θ 2 j ζ(ℓ j sν ) 2 |∇ℓ j sν | 2 e -2 f + (ℓ j sν ) 2 2 /h dx + O e -2(S+δ)/h .
The asymptotic of such an integral has been obtained in Equation (5.4) of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF] using the Laplace method. This computation gives (6.10)

P ν φ j , φ k ∼ C 1 h 2 e -2S/h ,
when s ν is close to m j = m k . Assume now that s ν is close to m j and m k with m j = m k .

In that case, we have ℓ j sν = -ℓ k sν (see the discussion below (4.6) of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF]). Then, (6.9) and the parity of ζ give

P ν φ j , φ k = -h 2 C -2 0 θ j θ k ζ(ℓ j sν ) 2 |∇ℓ j sν | 2 e -2 f + (ℓ j sν ) 2 2 /h dx + O e -2(S+δ)/h .
As before, the Laplace method implies (6.11) P ν φ j , φ k ∼ -C 1 h 2 e -2S/h , when s ν is close to m j = m k . Finally, if s ν is not close to m j or m k , we directly get from (6.8) and the support properties of χ ν , θ j and θ k that (6.12) P ν φ j , φ k = 0, in that case. Summing up, (6.6) follows from (6.10), (6.11) and (6.12).

It remains to show (6.7). The first estimate is a direct consequence of Proposition 5.1 ii) and iii) of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF]. On the other hand, using (6.3) and P ν e -f /h = 0, we deduce P ν φ j = [P ν , θ j ](v j + 1)e -f /h + θ j P ν v j e -f /h . Since e -f /h = O(e -(S+δ)/h ) on the support of (v j + 1)∇θ j , the first term is O(e -(S+δ)/h ) in L 2 norm. Concerning the second term, we remark that χ ν is constant (equal to 0 or 1) near each connected component of the support of θ j ∇v j if the support of θ j has been chosen sufficiently close to the connected component of {f < f (s)} containing m j . Then, (6.2) and (6.4) lead to

θ j P ν v j e -f /h = θ j d * f χ ν d f v j e -f /h = θ j d * f χ ν e -f /h h∇v j = χ ν θ j d * f e -f /h h∇v j = χ ν θ j P 0 v j e -f /h .
It is proved below (5.5) of [START_REF] Bony | Eyring-Kramers law for Kramers-Fokker-Planck type differential operators[END_REF] that θ j P 0 (v j e -f /h ) = O(h ∞ )e -S/h . Summing up, P ν φ j = O(h ∞ )e -S/h , and (6.7) follows.

We construct a basis of the 2-dimensional spectral space of P τ associated to the eigenvalues close to λ 2 . We set (6.13)

         e 1 (x) = 1 √ 8 e -f /h 2φ 1 (x) -φ 2 (x) -φ 3 (x) , e 2 (x) = √ 3 √ 8 e -f /h φ 2 (x) -φ 3 (x) ,
with e -f /h estimated in (3.3). The idea behind this choice of functions is that

(6.14) 1 √ 3   1 1 1   , 1 √ 6   2 -1 -1   , 1 √ 2   0 1 -1   ,
form an orthonormal basis of R 3 (the first vector corresponding to e -f /h -1 e -f /h ). For ν ∈ {1, 2, 3}, let P ν ∈ M 2×2 (R) be the matrix of coefficients

( P ν ) j,k = C -1 2 h -1 e 2S/h P ν e j , e k with C 2 = |µ(s)|| det Hess f (m)| 1/2 4π| det Hess f (s)| 1/2 .
The asymptotic of these matrices are provided by the next result. Lemma 6.2. For all j, k ∈ {1, 2}, we have (6.15) e j , e k = δ j,k + O(e -δ/h ).

Moreover, the matrices P ν satisfy modulo o(1) terms (6.16)

P 1 = 3 - √ 3 - √ 3 1 , P 2 = 0 0 0 4 and P 3 = 3 √ 3 √ 3 1 .
Proof. From (6.3), we have φ ν = 2e -f /h near m ν and φ ν = O(e -δ/h ) outside. It implies

e -f /h -2 φ j , φ k = 4 3 δ j,k + O(e -δ/h ),
thanks to (3.3). Combining this relation with (6.13), we get (6.15).

To show (6.16), it is enough to combine (6.6) and (6.13). For instance

P 3 e 1 , e 2 = √ 3 8 e -f /h 2 P 3 (2φ 1 -φ 2 -φ 3 ), (φ 2 -φ 3 ) = √ 3 8 e -f /h 2 2 P 3 φ 1 , φ 2 -2 P 3 φ 1 , φ 3 -P 3 φ 2 , φ 2 + P 3 φ 2 , φ 3 -P 3 φ 3 , φ 2 + P 3 φ 3 , φ 3 = √ 3C 1 h 2 e -2S/h 8 e -f /h 2 (0 + 2 + 0 + 0 + 0 + 1) + o he -2S/h = √ 3C 2 he -2S/h + o he -2S/h , thanks to (3.
3) and

C 2 = C 1 | det Hess f (m)| 1/2 8π .
This provides the desired asymptotic of ( P 3 ) 1,2 . The other coefficients can be computed the same way.

We now apply the perturbation theory for h fixed small enough. For τ small enough (depending on h), P τ has 3 small eigenvalues counted with multiplicity: λ τ 1 = 0 associated to the eigenspace e f /h C and λ τ 2 , λ τ 3 such that λ τ j → λ 2 as τ → 0 for j = 2, 3. Mimicking Section 3, we introduce the spectral projectors

Π τ = 1 {λ 1 ,λ 2 ,λ 3 } (P τ ), Π τ 1 = 1 {λ 1 } (P τ ) and Π τ 23 = 1 {λ 2 ,λ 3 } (P τ ), which satisfy Π τ = Π τ 1 + Π τ 23 , Π τ 1 = Π 1 and Π τ • u = Π τ
• u as in (3.1). Moreover, τ → Π τ • is analytic in a real neighborhood of 0 which may depend on h. We define e τ 1 = Π τ 23 e 1 and e τ 2 = Π τ 23 e 2 . for j = 1, 2. These real-valued functions satisfy Lemma 6.3. For j = 1, 2 and τ small enough (depending on h), we have e τ j = e j + O(h ∞ )e -S/h , in H 2 (R 2 ).

Proof. Using the Cauchy formula, we can write e τ j = Π 23 e j -1 2iπ γ (P τz) -1 dz e j + 1 2iπ γ (P 0z) -1 dz e j = Π 23 e j + 1 2iπ γ (P τz) -1 (P τ -P 0 )(P 0z) -1 dz e j , where γ is a simple loop around λ 2 , oriented counterclockwise, which depends on h but not on τ . This implies (6.17) e τ j = Π 23 e j + O h (τ ) = Π 23 e j + O(h ∞ )e -S/h , in H 2 (R 2 ) for τ small enough.

Since Π 23 = Π -Π 1 , (6.17) gives e τ j = Π e j -Π 1 e j + O(h ∞ )e -S/h . Using (6.5) and that f is invariant under R, we get Π 1 e 1 = e -f /h 1 √ 8 e -f /h 3 e -f /h , 2φ 1φ 2φ 3 = e -f /h 1 √ 8 e -f /h 3 e -f /h , 2φ 1 -

φ 1 R -φ 1 R 2 = e -f /h 1 √ 8 e -f /h 3 e -f /h , 2φ 1 -φ 1 -φ 1 = 0.
The same way, Π 1 e 2 = 0. These relations follow in fact from (6.14). Summing up, (6.18) e τ j = Π e j + O(h ∞ )e -S/h , in H 2 (R 2 ) for τ small enough.

Finally, we work as in [START_REF] Peutrec | Sharp asymptotics for non-reversible diffusion processes[END_REF]Lemma 4.9] to remove the projector Π (see also [3, (5.12)]). The Cauchy formula gives Π e j = e j -1 2iπ ∂B(0,λ * h/2) (P 0z) -1 dz e j -1 2iπ ∂B(0,λ * h/2) z -1 dz e j = e j -1 2iπ ∂B(0,λ * h/2) z(P 0z) -1 dz P 0 e j .

Combining with (6.7), (6.13) and (6.18), the lemma follows.

From (6.13), (6.14) and Lemma 6.

In particular, (e τ

1 , e τ 2 ) is a orthonormal basis of Im Π τ 23 , e τ j is real-valued and τ → e τ j is analytic for j = 1, 2 and τ near 0. We now define the interaction matrix Q(τ ). More precisely, let Q(τ ) be the matrix of the operator Π τ 23 P τ Π τ 23 expressed in the basis (e τ 1 , e τ 2 ). More prosaically, it means that Q j,k (τ ) = e τ j , P τ e τ k . From the previous discussion, Q(τ ) is well-defined for τ small, (6.20)

Q ∈ C ∞ (R 4 ; M 2×2 (R)),
and Q(0) = λ 2 Id. Moreover, its partial derivates satisfy Lemma 6.4. We have To get the spectral gap of P, it remains to compute σ(B 1 ). In the basis (x 1 e -f /h , x 2 e -f /h ) of E 1 , the matrix of B 1 takes the form

B 1 = 0 h -h 0 .
Thus, σ(B 1 ) = {ih, -ih} and Proposition 2.6 follows from (7.1).

Figure 1 .

 1 Figure 1. The structure of the critical points of f and an example of such a Morse function.

1 +

 1 h) 0 ζ(r)e -r 2 /2h dr near s, one of the two saddle points close to m j , 1 outside,Here, we say that "a saddle point s is close to a minimum m" iff s is in the closure of the connected component of {f < f (s)} containing m (see Figure1). The functionζ ∈ C ∞ 0 (R; [0, 1]) is even and satisfies ζ(r) = 1 for r near 0, )e -r 2 /2h dr = πh 2 O(e -δ/h ) .

3 , ( e τ 1 , e τ 2 ) 1 = e τ 1 -1 e τ 1 and e τ 2 = e τ 2 -e τ 1 , e τ 2 e τ 1 -1 e τ 2 -e τ 1 , e τ 2 e τ 1 .

 3121112211211 is almost orthonormal and then is a basis of Im Π τ 23 . We orthonormalize ( e τ 1 , e τ 2 ) into (e τ 1 , e τ 2 ) by the Gram-Schmidt process. It means (6.19) e τ

( 6 ., e 0 2 ) 2 and [B, a * 2 ] = ha * 1 ,

 62221 21)∂ τν Q(0) = C 2 he -2S/h P ν + o(1) ,where the P ν 's are defined in Lemma 6.2 and γ(h) ∈ R \ {0} is the constant given by Lemma 4.4 and associated to the basis (e 0 1 of Im Π 23 . A direct computation gives [B, a * 1 ] = -ha * showing that E n is stable by B. Let B n denote the restriction of B to E n . Summing up the previous arguments, we deduce (7.1) σ(P) = +∞ n=0 2nh + εσ(B n ), where σ(B n ) ⊂ iR since B n is anti-adjoint as B.

  Re µ 2 /h cos(t Im µ 2 /h)u c + sin(t Im µ 2 /h)u s + ε(t), (2.6)

Proof. We compute these derivates using the classical trick in the reduction process (see Section II.2.3 of [START_REF] Kato | Perturbation theory for linear operators[END_REF]). We can write ∂ τν Q j,k (0) = ∂ τν Q j,kδ j,k λ 2 (0) = ∂ τν e τ j , (P τλ 2 )e τ k (0) = (∂ τν e 0 j ), (P 0λ 2 )e 0 k + e 0 j , (∂ τν P τ )(0)e 0 k + e 0 j , (P 0λ 2 )(∂ τν e 0 k ) = e 0 j , (∂ τν P τ )(0)e 0 k , (6.23) since (P 0λ 2 )e 0 j = (P 0λ 2 )e 0 k = 0. For ν = 1, 2, 3, (6.23) gives (6.24) ∂ τν Q j,k (0) = e 0 j , P ν e 0 k , with P ν defined in (6.2).

From (6.15) and Lemma 6.3, we have

) and e τ 2e τ 1 , e τ 2 e τ 1 = 1 + O(e -δ/h ). Using again Lemma 6.3, (6.19) becomes (6.25)

, where the O(e -δ/h )'s are constants. Then, (6.21) follows from (6.16), (6.24) and (6.25).

On the other hand, (6.24) gives ∂ τ 4 Q j,k (0) = e 0 j , Be 0 k . In other words, ∂ τ 4 Q(0) is the operator Π 23 BΠ 23 expressed in the basis (e 0 1 , e 0 2 ) of Im Π 23 . Then, (6.22) follows directly from Lemma 4.4.

Proof of Theorem 2.4. From Lemma 6.4 and (6.16), (∂ τν Q(0)) ν=1,2,3,4 is a basis of M 2×2 (R). Thus,

, is an isomorphism. By the inverse function theorem, τ → Q(τ ) is a local diffeomorphism from a neighborhood of 0 to a neighborhood of λ 2 Id, for h small enough. Note that the neighborhoods may depend on h. Then, there exists τ (h) ∈ R 4 with |τ | < r such that (6.26)

for some ρ(h) = 0. Since Q is the operator P τ restricted to its stable eigenspace Im Π τ 23 in the basis (e τ 1 , e τ 2 ), (6.26) shows that P Jor := P τ has a non-trivial Jordan block associated with the eigenvalue λ 2 and Theorem 2.4 follows.

Proof of Proposition 2.6

We write P = P 0 + εB with

The operator P 0 is the harmonic oscillator

, with the creation operators a * j = -h∂ x j + x j and annihilation operators a j = h∂ x j + x j . The spectrum of P 0 is 2hN and the eigenspace associated to 2nh is the (n + 1)-dimensional space


Alessandro Stanziani

Vulnerability beyond Revolutions: Rural Workers, Former Slaves and Indentured Migrants in the French Empire

In 1834, a society for the abolition of slavery, connected to the British associations, was created in Paris. Led by Duke Victor de Broglie, the French society encouraged legislative reforms, manumission and limited recognition of civil rights for gens de couleurs. 1 Several pamphlets and studies produced by parliamentary commissions summarised the longstanding debate over the comparative efficiency of wage labour versus slavery. 2 Contrasting estimates were produced, with some claiming the profitability of slavery and others challenging it. At the same time, new considerations arose: the profitability of slavery was said to be demonstrable only for the individual planter, but not for the "nation." Two main positions emerged; for some, slavery was an ethical and religious affair more than an economic issue. In response to this argument, anti-abolitionists replied that Africans would suffer famine if left in Africa; slavery was openly declared to be a humanitarian project.

 In 1843, the De Broglie Commission finally published three volumes of its minutes and criticised the idea of ownership of slaves as a natural right. This was acceptable for other kinds of property, but not for human beings. But it also added a new insight: the need for the strict regulation of free labour: once freed, former slaves had to be put under surveillance and strictly regulated.

In the end, the Commission opposed immediate abolition and suggested two alternative, gradual forms of emancipation, one that would take place over ten years, and another that would take place over twenty years. Opinions were also divided over the amount of compensation owed to slave owners and who should pay it. 5 In response, Schoelcher and the other abolitionist leaders intensified their action. In 1844-1845, they gained the support 1 "Gens de couleurs" was the term used for all free non-whites, that is, former slaves or non-whites born as free.

of workers' associations, which sent several massive petitions to the government demanding the immediate abolition of slavery. A cautious law was, therefore, adopted in 1845, encouraging manumission and the gradual purchase of freedom by slaves themselves. The slave was granted legal capacities and could, therefore, own movable and fixed property. However, this approach soon ran up against a swiftly evolving situation in the colonies as well as in France. In 1846-1847, in the West Indies, several thousands of slaves escaped from French to British islands (five thousand from Guyana to Trinidad, two thousand of whom reached their destination, and one thousand from Guadeloupe to Dominica). Planters spoke of on-going transformation through voluntary manumission under the law of 1845 and the introduction of machines and free indentured immigrants. Such an evolution would take time and legislators concluded that immediate abolition would put an end to this positive process. Nevertheless, strong state support was required to promote these dynamics. 6 Social tensions also escalated in France itself: artisans complained of increasing difficulties and urban workers protested against low wages, long working days and unstable employment. In 1847, strikes intensified in Paris and France.

We are thus confronted with a relatively well-known historical dynamic: elites were keen to support the abolition of slavery, but only on the long term and with due compensation to slave owners. Meanwhile, social unrest took the lead and pushed towards more radical politics. What was the outcome of these tensions?

Immediately after the Parisian uprising in February 1848, commissions on the abolition of slavery were formed in March and April. Led by Schoelcher, the commission advocated the immediate abolition of slavery, entitling all manumitted slaves to become French citizens. There were discussions concerning compensation for slave owners and a possible period of apprenticeship -the majority of the commission was favourable to both. 7 Meanwhile, another commission was settled to discuss the reform of labour in France. The connections between the parliamentary commissions on slavery and on labour were crucial. Several topics were common to both, including the organisation of legal courts to handle labour disputes, pensions for the infirm and the elderly, national workshops (ateliers), the control and punishment of vagrancy, and the length of working days. Strangely enough, at first sight, the decrees established a relation between rural workers in France, former slaves and new indentured immigrants in France's colonies. In both rural areas and the colonies, prud'hommes (industrial or labour tribunals) were not allowed, and justices of peace (rural low courts) were supposed to replace them. 8 Second, the worker's booklet (livret ouvrier), an instrument to control workers' mobility in France but which was unevenly enforced on the mainland, was introduced in the colonies in order to control "vagrancy." At the same time, a bill suggested extending the booklet to agricultural labourers; however, the measure failed to pass.

Here, a number of interesting questions arise: why did revolutionary France abolish slavery but keep former slaves and new immigrants under strong coercion and unfair rights, while criticising the British for doing so? And why were emancipated slaves and rural workers on the mainland regulated by similar rules? These questions are tied to broader questions about the abolitionist process and revolution: countless comparisons have been made between French and British abolitionism, 9 mostly stressing the hesitant and ambiguous attitude of the French. This chapter offers a complementary view, by arguing that the limits of the revolutionary movement as regards slavery were strongly related to the tensions on labour, and rural labour in France itself in particular. To this end, it will follow the links between indentured migrants, slaves and labourers from the late seventeenth century to the second half of the nineteenth century. This means that it will touch upon not only the revolution of 1848, but it will trace the roots of the ambiguities of this revolution back to the revolution of 1789 and its aftermath. The chapter will not discuss the entire French Empire, but will focus on Réunion Island. Even if some conclusions are common to the Antilles and the Atlantic world, we consider it worth focusing on the Indian Ocean world, where the transmutations of labour in the European colonies were confronted with the extremely blurred boundary between free and unfree labour in the whole area extending from Eastern Africa, to India and Southeast Asia. 10 1. Beyond 1789: Institutions and rural labour in the colonies, from the seventeenth to the mid-nineteenth century

Like in the British Empire, in the French colonies the contract of engagement, or indentured service was developed in the seventeenth century. It was initially intended for white settlers whose transport expenses were advanced by employers or their middlemen in exchange for special commitment to work for several years. The engagés were subject to criminal penalties and could be transferred to other masters along with their contract. Owing to the close resemblance between wage earners and domestic servants and the survival of forms of domestic service into the nineteenth century, the contract of engagement should not be understood in opposition to these other labour relationships but, rather, as an extension and of them in the colonial situation. If we look at the contracts, then we see that, in France, the agricultural day labourer was supposed to sell exclusive ownership of his time and services to his employer; the sailor's contract extended the duration of this sale with special clauses related to voyage expenses. All these clauses were reproduced in the contract of engagement. 11 Moreover, the engagé owed his labour to his master who, in exchange, was to teach him about colonial farming.

As in nearby English colonies, during the second half of the seventeenth century an increasing number of African slaves reached the Antilles as well as Mauritius and Réunion 9 Robin Blackburn, The Making of New World Slavery: From the Baroque to the Modern, 1492-1800 (London: Verso, 1998 Island.12 In all those cases, however, indentured migrants and slaves coexisted. In the Atlantic, in 1654, two-thirds of the indentured immigrants in Guadeloupe were domestics. In 1671, one-third of the households had no slaves or servants. Half of the households had both slaves and domestic servants. It was only during the second half of the seventeenth century that the proportion of the white population sharply declined. In Guadeloupe, it fell from 80 per cent in 1654 to 50 per cent in 1664 and 39 per cent in 1684. In Martinique, the white population was 51 per cent in 1664 and dropped to 29 per cent in 1678. 13 In Réunion and Mauritius Islands this was all the more relevant that the use of "engagés de couleur" (as they were designated, to stress the importance of race) developed in the eighteenth and even more in the nineteenth century. This immigration was partly linked to the need for artisans (Indian carpenters and masons) but, above all, to the demand for additional labourers at a time when, under pressure from the English, the price of slaves was constantly rising. 14 In the Indian Ocean, the French traded between 334,000 and 384,000 slaves to the Mascarene Islands between 1500 and 1850. 15 In the eighteenth and nineteenth centuries, about 200,000 slaves were imported to Réunion Island for the production of sugar, mostly from Madagascar and East Africa. Indians were also present: in 1708, the total count of adult slaves was 268 -197 were men and 71 were women; 20 per cent of the men and 36 per cent of the women were Indians. [START_REF]Fixation des salaires agricoles," an II, an III; N C 1157-61[END_REF] A century later, they comprised about 3 per cent of the total number of 54,000 slaves.

Race played an important although changing role over time; at the end of the seventeenth century, mixed marriages were extremely common: 67 per cent in 1690. The Royal Declaration of 1698 acknowledged children of mixed couples as French if they were baptised. In 1735, in Saint-Louis and Saint-Pierre, 86 per cent of estate owners were métis.17 This encouraged the rate of manumission, which was relatively high on Réunion Island, in particular in the presence of kinship and family relationships between the master and the slave. As a result, unlike the American colonies, in Réunion "freed coloured people" were not counted until 1767, when most métis were classified as white. Here, because of the long tradition of mixed marriages, a person's social condition seemed to be more important than the colour of his or her skin.

Despite this attitude, the Revolution of 1789 and the abolition of slavery in 1794 had a limited impact on Réunion Island. Initially, the implementation of the Revolution encountered serious problems in the colonies where the "freed coloured" requested their full integration in terms of rights, while whites -not only masters and elites but also those from other countries whose rights had just been acknowledged -resisted them. The outcome differed according to place: civil rights were granted to the "freed coloured" in Mauritius, but not in Réunion Island, where they did not win their political rights until 1793. 18 At the end of the day, the abolitionist decree of 1794 was never implemented in Réunion island, where Napoleon had no difficulties in officially proclaiming the restoration of slavery.

In the 1820s as well as under the July Monarchy, two contrasting trends emerged: while in most French colonies manumission and the social conditions of freed slaves improved (in Martinique, freed slaves owned slaves and had trades 19 ), there was an increase in illegal slave trafficking to the French West Indies and Réunion Island, partly in reaction to this trend and the mounting difficulties in French colonial production. An estimated forty-five thousand illicit slaves were imported to Réunion Island between 1817 and 1835. 20 A total of about three hundred thousand slaves were imported to the Mascarene archipelago between the eighteenth century and the first half of the nineteenth. Unlike the eighteenth century, East Africa and Mozambique were the main source of supply (60 per cent), with the rest coming from Madagascar (31 per cent) and the countries of southern Asia (9 per cent). These networks were to remain in place after the abolition of slavery. 21 In Réunion Island, slaves continued to be imported until 1848, when the practice was finally abolished. However, disguised forms of slavery persisted: ships' captains transporting Indians often resorted to fraud, and contracts of engagement to Singapore were signed, but the engagés ended up being sent to Réunion Island. 22 Moreover, the French colonial administration encouraged the Indian engagés to migrate and tried to establish rules that were sufficiently clear to avoid trouble, but they also worried about their actual enforcement. 23 Translating those principles into action remained difficult. During the first half of the 1830s, Indian engagés numbered about three thousand. 24 The legal rules in force provided that the engagés should receive food, lodging and wages. In practice, however, the employer-landowners seldom complied with the rules. Several riots took place, against planters and, above all against their overlookers. 25 The Indian engagés resisted not only by reducing the amount of work they did and by rioting, but also by taking their cases to court. 26 However, faced with the unfavourable attitude of the magistrates and the administration, Indian engagés formed a trade union in which the members with the best mastery of the French language played a highly active role in formulating appeals, intervening with the authorities, and so on. 27 In 1837, the trade union was prohibited. 28 At that point, the engagés discovered a different instrument: they became "fugitives" and "deserters." 29 Most went into town while others hid up on the hills, and still others simply moved from small to large estates, where the working and living conditions were imagined to be better. And this was partially true: large estate owners lured labouring people from neighbouring estates, particularly small ones; and then, with the help of the local police, they did not return them to their original owner.

Faced with workers' resistance and the unfair competition between planters, local authorities decided to increase the number of migrant arrivals of migrants, starting with the Chinese engagés. A new decree was adopted in 1843 to regulate these engagés: their contracts were supposed to last at least five years, and the minimum age of the engagé was set at sixteen; the landowners had to agree to pay wages and the return trip to China; ill treatment or a two-month delay in wage payments was sufficient grounds for the administration to nullify a contract. However, once again, estate owners seemed unwilling to comply with the rules. 30 As a result, the few dozen Chinese who arrived soon adopted the same strategy as the Indians. At first, they protested against their living conditions and overdue wages, and then they started legal proceedings or left their employers.

Again, French planters and authorities invoked rules on rural labour existing in France and which, according to them, kept labourers under the strict authority of the master. In their eyes, indentured immigrants working on plantations were a kind of rural labourers who, as already mentioned, even in France, were under a strong dependence on their master. Moreover, because of their skin colour and provenance, the subordination of migrants became even more extreme than in France. In other words, the persistent harsh living and working conditions of migrants in Réunion island had two main roots: race and the still ongoing marginality of rural workers in France itself. This is intriguing: if the French revolutions of 1789 and 1848 had an ambiguous impact on slaves, and if they provided better conditions for urban workers, why did they marginalise rural labour?

Rural labour institutions in France across the 1789 Revolution

During the last decades, scholarship has stressed how France appears to be the first country to have abolished lifelong domestic service as well as criminal penalties in labour disputes. 31 As late as the eighteenth century, France's leading legal experts considered labour to be a service provision. Available studies show that prud'hommes protected the workers in the textile industry and certain urban milieus. 32 But what about the other sectors, especially agriculture?

Although the French Revolution eliminated lifelong domestic service, until 1901, rural labourers and workers could not apply to prud'hommes, but only to the justice of the peace. Both before and after the Revolution, legal texts classified people working in agriculture as labourers or "task workers" (tâcherons) or as servants in husbandry. 33 In the eighteenth century, servants in husbandry were by far the largest group of wage earners in French agriculture, just like in Great Britain. Prior to the Revolution, penalties were imposed on all labourers, pieceworkers or servants in husbandry who quit their jobs before the end of their contract or without the employer's authorisation. A variety of contractual arrangements to limit mobility existed at the time (bonuses for hardworking labourers, payment by task) along with general provisions. 34 Thus, from the sixteenth to the eighteenth century, agricultural labourers and servants were free to move about and change employers only at certain times of year -that is, according to the critical periods in the agricultural calendar. 35 It was not the same for day labourers; the seasonal nature of agricultural labour gave rise to a significant amount of regional mobility, which was already considerable in the seventeenth century and remained high until around the end of the nineteenth century. This mobility, together with the notion of labour as service in the legal and economic culture of the time, is precisely what helps to explain the harsh penalties imposed on labourers and servants. They were not allowed to leave their masters until the end of their contract, and, if they left prematurely, they were subject to heavy penalties as well as the loss of their earnings. The master, on the other hand, could discharge them at any time.

Disputes over the wages (gage) of servants in husbandry and labourers was one of the areas reserved for justices of the peace, 36 especially because masters were taken at their word (until 1868), unlike their dependants, regarding any issue concerning gages (different from wages), wages or advances.

A labourer could leave his employer at any time or be discharged without prior noticeand without providing or claiming any compensation. In practice, however, the need to ensure workers for urgent labour had an obvious corrective effect on this rule. For example, a labourer paid by the day might be kept on for one or two weeks, or even a month or two, in summer and autumn. In some regions, incidentally, there were forms of servants' contracts for six months or a year.

Labourers remained free to offer their services to multiple farmers, if their schedules permitted. Both the wage earner and the master paid for this freedom: the day labourer's employment status was precarious and he ran the risk of seasonal unemployment, while the employer faced a possible shortage of hands during peak seasons. 37 Lastly, domestic servants were most closely tied to their masters. What defined domestic servants and differentiated them from other agricultural wage earners was the content of the commitment, which was almost always tacit and which could not be broken "except for the most serious reasons." Domestic servants were subject to their master's will, which meant they "owed all [their] time to the master for any labour demanded." This subordination to the master's will resulted in receiving the promised gages in a lump sum. Without notice or compensation, the master could discharge the domestic servant for "dishonesty," "disobedience," "forgetting duties," cursing, or acts of violence. The domestic servants, for their part, complained of poor or inadequate food. 38 34 Philip Hoffman, Growth in a Traditional Society: The French Countryside, 1450-1815 (Princeton: Princeton University Press, 1996), 45-46. 35 Yvonne Crebouw, Salaires et salariés agricoles en France, des débuts de la révolution aux approches du Xxe siècle (Lille: ANRT, 1986).

36 Jean-Claude Farcy, Guide des archives judiciaires et pénitentiaires, 1800-1958 (Paris: CNRS éditions, 1992). Problems arose most often with regard to the gage. Most practices allowed the master to withhold wages equivalent to the amount of work due from the wage earner. This led to the proposal, renewed in 1848, to extend the worker's booklet (livret ouvrier)39 to agricultural labourers; however, the measure failed to pass. On the other hand, if the domestic servant demanded compensation, he had no other recourse but the justice of the peace. The situation changed during the second half of the century, when the rate of disputes went up and the demand for agricultural wage earners and domestic servants increased due to emigration to the cities. Employers accused the justices of the peace of being "on the side of labourers and domestic servants"40 -just like manufacturers during the same period accused the magistrates on industrial tribunals of being biased against them.

In short, before the Revolution, the status of French labourers and domestic servants resembled bondage; labour was assimilated to service provision. In the nineteenth century, although domestic servants and labourers were held far more accountable than their employers for breach of contract, they were no longer governed by criminal constraints, but merely by civil law. This marked a fundamental difference from the pre-revolutionary period. 41 In this context, rural labourers received far less protection than urban workers.

Rural labour in France

Conventional history claims that France stagnated in growth, productivity and in regard to main economic indicators throughout the eighteenth century. [START_REF] Patrick | Les voies de passage vers la société industrielle en Grande-Bretagne et en France[END_REF] Over the last decades, new studies have challenged this view and have shown that there was considerable growth in eighteenth-century French agriculture. [START_REF] Hoffman | Divisions of Labour: Agricultural Productivity and Occupational Specialization in Pre-Industrial France[END_REF] We now know [START_REF] Postel-Vinay | La terre et l'argent[END_REF] that credit markets were fully developed in pre-revolutionary France, and the same was true for property rights. Pluriactivity and seasonal labour markets were omnipresent.

George Grantham in particular evaluated the input of labour for the main agricultural operations in several French areas over a long period of time -roughly the eighteenth and nineteenth centuries -and concluded that labour productivity increased throughout the country during the period, particularly in intensive husbandry, and more so in tillage than in harvesting. In some areas, notably the Paris Basin, labour productivity was almost equal to the best agricultural areas in Britain at the time. The rural population decreased more slowly in France than in Britain, but more rapidly than in any other European country in the nineteenth century. This slow decrease was the result not just of the presumed protection of "peasant property" after the revolution, as conventional historiography has argued, but also of the strong coercion still exerted on rural workers. From this standpoint, the Revolution of 1789 brought much more "freedom" to urban than to rural workers.

In short, the Revolution of 1789 and the revolution in Saint-Domingue did not seem to bring much freedom to colonial workers or to rural labourers in mainland France. Both these groups were marginalised, in the colonies because from Napoleon onwards, the counterrevolutionary emancipation had been firmly established; and in France itself, because the advances of the revolution benefitted some finance, merchants and entrepreneurs, as well as limited groups of workers. Small units and rural labourers were left out. The reasons for this were both political and economic; in politics, the major divide in the world of labour was between the rural and the urban on the one hand, and between the mainland and the colonial on the other. From an economic standpoint, these attitudes were strengthened in the labour-intensive production in agriculture and in the colonies. The question is whether the revolution of 1848, the abolition of slavery and the Second Industrial Revolution changed this state of affairs.

Even as late as 1851, only 27.6 per cent of the active population was working in the industrial sector (16 per cent in 1815; 21 per cent at the end of the 1830s). In 1855-1864, production in cottage industry was still 1.6 times higher than that of industry. Market segmentation remained the rule in France until around the end of the nineteenth century and, as a result, there was no uniformity in prices, wages or skills. With a few exceptions, pluriactivity was the norm. [START_REF] Gayot | De la pluralité des mondes industriels: La manufacture royale des draps de Sedan[END_REF] Labour intensification in agriculture rose in France as well as in Britain, to such an extent that Gregory Clark came to the conclusion that, before 1850, labour intensification without any technical progress was the norm in Europe. [START_REF] Clark | Productivity Growth without Technical Change in European Agriculture before 1850[END_REF] At the same time, the output and yields were far higher in Britain than in France, despite an equal increase in the number and length of working days. 47 The organisation of labour and capital explain these differences. The contribution of labour dominated French growth until at least the 1870s, if not later.

The seasonal nature of agricultural labour gave rise to a significant amount of regional mobility, which was already considerable in the seventeenth century and remained high until around the end of the nineteenth century. In 1860, about 25 per cent of the industrial labour force was still moving from one sector to another during the summer. [START_REF] Magnac | Wage Competition between Agriculture and Industry in Mid-Nineteenth Century France[END_REF] Where possible, employers could try to compensate for the loss of men by employing women, adolescents and children. Although farm wages hovered below industrial wages for most of the year, the situation was reversed during the peak season, particularly for women. Their wages were systematically inferior to those of men, but they recovered the difference during the summer in both agriculture and industry. It was as if agriculture offered a high wage for a brief period every year in order to attract workers who were usually engaged in industrial production. [START_REF] Bompard | Migrations saisonnières de maind'oeuvre: Le cas de la France en 1860[END_REF] Seasonal labour was more intensive during the periods workers spent in towns and even more for peasant-workers in cottage industry. At the same time, wages, above all real wages, stagnated.

Day labourers were commonly found in the southern Mediterranean, Alsace-Lorraine, the Île-de-France and Picardy. It is estimated that in 1862 about half of the 4 million agricultural wage earners were day labourers; thirty years later, that figure had dropped to 1.2 million. This trend was linked, in large part, to a sudden reduction in the number of small landowners between 1862 and 1892; by contrast, servants in husbandry made up an increasingly high percentage of agricultural labourers. 50 These trends in different rural areas corresponded to those in various industrial branches; in industry, seasonal shutdowns were less frequent in capital-intensive firms. Industrial employers could either shut down production in the summer or increase wages. The decision they took depended on the branch of industry and the region. Indeed, a national market was still lacking in nineteenth-century France (at least until after the 1880s) and peasant-workers reasoned in terms of a comparison between local wages in agriculture and industry. This explains why, in departments where industrial wages were high, agricultural wages followed suit, and vice versa: workers compared and finally offset the imbalance between the two wages. For the same reason, summer shutdowns were more widespread among companies that paid their workers less than the summer wage they earned for farm labour.

This season shut-down was particularly well-suited to high labour-intensive crop areas. When labour was required at different intervals, as was the case for flax, hemp and vegetable and oil seeds, workers did not look to distant markets for compensation. In these cases, their wages would not be offset by going to distant markets.

In wine producing regions, on the other hand, workers did respond to distant higher wages, and so -to an even greater degree -did workers located in pastoral counties. In particular, those in pastoral counties -in northern France, the Vosges and parts of Normandy and Brittany -left industry in the summer to work in neighbouring grain-growing regions with excess demand -in Picardy, Champagne, Lorraine and the Paris Basin.

This situation disappeared almost entirely after 1875, when seasonal migration sharply declined. Between 1860 and 1890, the earlier practice of combining agricultural and industrial employment largely vanished. In 1860, at least five hundred thousand and probably as many as eight hundred thousand workers quit their jobs during summer. By 1890, only one hundred thousand continued to do so.

In short, after 1848, official rules on the rural labour market did not change and rural workers still were penalised compared to urban workers. However, over the years, their legal strength increased and so did urbanisation. This double movement produced slowly improving conditions for rural workers, who could now better negotiate their seasonal activities in town and in the countryside. However, during the second half of the nineteenth century, this slow progress in the condition of rural workers on the mainland seemed to be in sharp contrast with the persistent bondage of former slaves and new immigrants in the colonies.

Indentured migrants after 1848

Immediately after the decrees of May 1848, the elites of Réunion Island gathered to prepare a petition and a series of documents intended to demonstrate the need to postpone abolition in order to preserve the economy and stability of the island. Otherwise, they argued, the whole population, including the slaves, would suffer from the sudden collapse of sugar production.

In 1848, the General-Commissioner of the Republic, Sarda-Garriga, was sent to the island to enforce the abolition decree. He immediately ordered the plantation owners to free their slaves and ordered the latter to respect public order and not turn into vagrants. Freedom, he declared, makes work a duty. In principle, on 20 December 1848, sixty-two thousand slaves on the island were manumitted. 51 Indeed, when the French abolished slavery in 1848, unlike the British they did not impose an intermediate period of "apprenticeship" but, instead, practised disguised forms of enslavement. Recruitment in India, Madagascar, Mozambique and the East Coast of Africa relied on networks that had been in place since eighteenth century. It employed the same practices as the slave trade, often involving violence, sometimes with the help of local tribal chiefs. [START_REF] François | Libération d'esclaves et nouvelle servitude: Les rachats de captifs africains pour le compte des colonies françaises après l'abolition de l'esclavage[END_REF] In 1847, there were a total of 6,508 engagés -Indians, Chinese, Africans and Creoles. The lack of available labour encouraged several landowners to call for the importation of additional engagés, this time from Africa, especially since France was moving towards the abolition of slavery. Indeed, like in the British Empire in the 1830s and 1840s, the abolition of slavery in the French colonies in 1848 was followed by a revival of the engagement system. While only 153 African engagés entered into service in 1853, thereafter, on average, about 4,000 Africans arrived each year between 1851 and 1854; 10,008 were imported in 1858 and 5,027 the following year. [START_REF] Anom | tableau de l'immigration africaine à la réunion de 1848 à 1869[END_REF] The routes supplying engagés and immigrants partly reproduced those of the slave trade and commercial networks already in place. Indian labourers were brought to the island with the help of Arab and Indian middlemen, who were often in competition with each other. Taking advantage of the British peace and their permission to sail under the British flag, Indian merchants (often from Bombay and the Malabar Coast) gained a considerable competitive advantage in transporting coolies and goods between the Indian Ocean and India. 54 On the western route, between Réunion Island and Africa, the annexation of Mayotte opened up new supply sources in the Comoros archipelago itself, in Madagascar and on the West Coast of Africa. Between 1856 and 1866, about 8,000 engagés, nearly all of them from Mozambique, were transported via Mayotte to Réunion Island. [START_REF] Anom | [END_REF] According to the 1866--1867 census, of the population of 11,731 inhabitants, 3,716 were Africans (31.7 per cent); the vast majority (2,245) of them were engagés from Mozambique. [START_REF] Alpers | A complex relationship: Mozambique and the Comoro islands in the Nineteenth and Twentieth Centuries[END_REF] The demand for manpower was so high that Réunionese estate owners even tried to annex Madagascar; in this, they had the support of the Arab and Sakalava communities in the Mozambique Channel islands. Trading houses and transport offices in Nantes, Bordeaux and Marseille also played an active part in these operations. 57 Relying on already existing slavery and the support of the local sultans, these merchants established stable, large-scale commercial contacts in Réunion Island for engagés in Madagascar, Gabon, the Congo and even West Africa. [START_REF] Mohamed | Les différentes formes de l'esclavage et leurs abolitions successives à Mayotte[END_REF] In 1853, France built new centres in Gabon and Senegal to expand the zone for "redeeming" engagés; similar operations were carried out in Zanzibar and Mozambique. [START_REF] Reu | [END_REF] The "redeemed" (actually newly purchased) Madagascans and Africans finally boarded ships for Réunion Island, often passing through the Seychelles. In all, between 1851 and 1860, 20,000 engagés were exported from Portuguese Africa and the same number was shipped from the Swahili coast; 10,000 landed in Réunion Island and Comoros. [START_REF] Campbell | The Structure of Slavery in Indian Ocean Africa and Asia[END_REF] The British ships sequestered several cargoes of engagés bought by the French or the Portuguese. [START_REF] Anom | [END_REF] Finally, in 1861, a Franco-British agreement provided for the cessation of French prior redemption in Africa in exchange for the possibility of resorting to Indian coolies.

Initially, those put to work under contract of engagement were either new immigrants or local freed slaves, in almost equal numbers. Thus, in 1854, 34,650 men and 4,709 women immigrants, were added to 22,650 "local" male indentured labourers and 9,022 females. Local workers under other agreements than engagement were a tiny minority: 3,763 males and 3,116 females. The total number of workers came to 86,028. 62 Seasonal variations were nevertheless significant: in the first quarter of 1855, there were 61,191 workers in all, including 37,062 new male immigrants, 5,049 female immigrants, 9,545 local males and 4,785 local females. At the moment, contracts for one year (or longer) were the rule; only 1,683 workers were under shorter terms. These numbers did not reflect reality: the law required long-term contracts; the planters therefore provided this kind of contract, but they actually employed short-term workers as the seasonal variations demonstrate. In the second half of 1855, during the peak agricultural season, the number of working people jumped to 91,276, including 41,155 new immigrants, 33,228 members of the "local population" and 9,214 without a contract of engagement. 63 The Second Empire imposed tighter restrictions not only on rural labourers, as we have seen, but also, and above all, on emancipated slaves and engagés. A contract of engagement was imposed on all workers in the colonies; the legal rules governing the livret ouvrier were widely implemented and enforced. Anyone without fixed employment (defined as a job lasting more than one year) was considered a vagrant and punished as such. The penalties were considerable, but the law was also frequently circumvented through fictitious contracts of engagement that some -especially women -signed with landowners who were interested in having occasional labourers.

In principle, engagés had the right to go to court and denounce cases of mistreatment and abuse. We have seen that, under slavery, those rights had been largely ignored. Abolition did little to change those attitudes; in practice, it was still extremely difficult to make use of the rules, mainly because colonial law courts were in the hands of local elites. Thus, when immigrants went to court to denounce abuses, they were often sent back to their employer, who, at best, punished them and docked their wages for insubordination; at worst, the employer would sue them for breach of contract and slander. In the face of these difficulties, workers sometimes joined together to denounce illegal practices, but they risked being sentenced by the judge and the police to two months of forced labour in a workhouse for illicit association and breach of the peace. 64 Following protests by Indian immigrants and the British consul, in the late 1850s, permission was given to form a union for the protection of immigrants. The union was granted the authority to inspect estates and was supposed to safeguard immigrants' legal rights. However, the union performed its mission poorly, at least until the late 1860s; inspections were seldom held, and legal assistance was offered only to those immigrants who had completed less than five years of a renewed contract. This approach provoked a counteraction on the part of immigrants and the British consul, but the initial decisions of the courts validated a conservative interpretation and rejected claims denouncing unequal treatment under the law. 65 Contract renewals, wage payments and corporal punishment were the most common issues in the lawsuits filed by engagés. Unlike slaves, engagés had the right to return home; terms were negotiated in the contract, which was supposed to comply with the general provisions of the law. In practice, however, repatriation was difficult. During the 1850s and 1860s, a third of the indentured immigrants returned home (mostly Indians). This percentage was close to the one in Mauritius, the Caribbean, Surinam and Jamaica at the time, but it was far from the repatriation figure of 70 per cent recorded in Thailand, Malaya and Melanesia. Distance and the cost of transport were just two of the variables affecting repatriation; politics and concrete forms of integration were also important factors. [START_REF] Northrup | Indentured Labour in the Age of Imperialism, 1834-1922[END_REF] On Réunion Island, in particular, urban traders and certain colonial officers encouraged engagés to return home. The former group argued that once the immigrants had completed their commitment, they settled in towns and engaged in illegal trade and unfair competition. Colonial administrators were inclined to support this view: the defence of public order required the repatriation of immigrants. [START_REF] Reu | [END_REF] By contrast, several employers and estate owners, especially small ones, were hostile to the resettlement of immigrants in town or their repatriation, and they pushed for the renewal of contracts. Their attitude can be explained by the fact that unlike large estate owners, they faced increasing problems in finding the financial resources, networks and diplomatic support for new recruits. For that reason, they made use of every legal and illegal means to retain workers at the end of their contracts. In particular, they seized immigrants' wages and livrets and added severe penalties whenever possible ("laziness" and the failure to accomplish assigned tasks in due time were the most common arguments for applying penalties). Hence, the worker's "debt" was never repaid, and the contract was protracted. Day labour standards and objectives were gradually raised so that few workers could meet them; they were thus subject to stiff penalties while working eighteen to twenty hours a day instead of the ten mentioned in contracts and official rules. 68 And as if all this were not enough, employers had no qualms about using physical force to make workers renew their commitments.

These practices had been informally denounced from the 1850s, but it was not until the 1860s that they were brought before the courts, under pressure from British diplomats and French central government authorities. 69 Even then, lawsuits often dragged on for years and involved only a very small percentage of workers. In very few cases, employers were forced to pay their workers due wages, with no damages or interest, though many immigrants were also granted permission to terminate (illegal) contracts and abuses without paying penalties. 70 In first-level courts throughout the 1870s, no more than seven employers were sentenced each year for inflicting injuries and other violence. At the appeals court level, the figure dropped to one per year, the sole exception being four individuals convicted in 1875, but this was a single lawsuit and the three people receiving sentences were themselves immigrants working as foremen.

Conversely, every year employers sued several hundred workers for breach of contract. Sentences were usually favourable to the plaintiffs, and the workers had to face severe monetary penalties, which often translated into forced labour. Immigrants were also dragged into court for robbery, for which the sentences were very harsh -such as five years of forced labour for a stolen chicken. 72 As time went by, the use of coercion became increasingly essential in public governmental activities such as building and repairing roads, bridges and ports. Forced labour as a disciplinary measure and workhouses were already in existence under slavery, for the punishment of slaves who were under the jurisdiction of the public authorities; accordingly, slaves were assigned to workhouses or put to work on public works projects, rather than sent to their masters to be punished. In February 1831, colonial workhouses had a total workforce of 888 (612 men, 180 women and six children) in their so-called "mobile" department, that is, forced labour performed in various parts of the island; the "sedentary" department had a total of 202 labourers (148 men, twenty-eight women and twenty-six children). The mortality rate in these activities was very high: by 1831, an estimated 303 deaths (out of 12,347 prisoners) had taken place in the workhouse since it was set up in 1826. 73 Such coercion continued after the abolition of slavery. In Saint-Leu, in 1859, there were 321 people in the public workhouses, with between seventy and eighty prison sentences per quarter. By contrast, only sixteen people were engaged in disciplinary forced labour by the end of 1858. 74 In part, this outcome can be attributed to the violence and the poor living and working conditions of French planters, who received much less fiscal and financial support than their neighbours in Mauritius. Contingency and structural dynamics converged in a world of violence, exploitation and inequality. Violent colonialism responded not only to economic pressures and local conditions on the island, but also to the way the French managed labour and their empire.

The French Revolution suppressed lifelong domestic bondage, whereas the nineteenth century progressively abolished slavery, first in the British colonies, and then in the French colonies. Still, this process did not accompany the rise of a free labour market between legally equal actors. In Britain, France and their colonies, workers and indentured immigrants were not disguised slaves (as much of the literature in the nineteenth century argued), 75 but they did have an inferior legal status and far fewer rights than their masters. From this perspective, colonies were territories not only of slavery but, above all, of forms of bondage inspired by status inequalities entrenched in France itself. Status inequalities in France served as the model for those in the colonies. However, unlike Britain, where the Masters and Servants Acts opposed any kind of worker -including both domestic and rural labourers -to their masters, in France this opposition was only partially replicated before the Revolution of 1789. The difference was that, even at that time, legal differences between aristocrats, workers, peasants, merchants and the clergy overlapped with the emerging capitalist social differentiation. The Revolution consolidated this trend and provided special legal, political and social rights to estate owners, old aristocracies and new capitalists while proclaiming the equality of the whole population before the law. In practice, workers had no right to unionise or even to vote until universal suffrage in the twentieth century. Within this context, informal unions, industrial law courts and social movements were able to win some legal and social protection in some urban and industrial sectors while rural labour was marginalised. Corporatism replicated from pre-revolutionary to capitalist France. This became a source of inspiration for regulating labour in the colonies, in particular in the aftermath of the abolition of slavery. The rights of indentured immigrants and former slaves were constantly ignored in post-1848 French colonies and this was so not only under the Second Empire, that is, since 1851, but even in the aftermath of the revolution, as was shown at the very beginning of this chapter. In fact, the fragmentation of rights, interests and action of labouring people (weak solidarities between different branches of workers, deep suspicion between urban and rural workers, their joint hostility towards immigrants from the colonies) responded to the alliance between aristocrats and capitalists. The latter divided the former. As a consequence, under the Second Empire, two trends were at work: on the one hand, in commercial, trade and financial relationships, Napoleon III gradually reduced regulations and moved towards free trade and deregulated markets. On the other hand, in labour relationships, the Second Empire put severe limitations on workers' rights, sought to preserve the worker's booklet and denied the right to strike or even to unionise. In the colonies, this attitude took a distinctly repressive turn and inequalities existed both in the law (fewer statutory rights for immigrants and non-French residents) and in the way it was implemented (legal procedures unfavourable to labouring people) as well as outside the law (unpunished abuses and violence, economic exploitation). The notion of freedom was for a world to come and it appealed to immediate exclusion as a foundation for an eventual future inclusion. 75 For more on this debate, see Alessandro Stanziani, "Free Labour-Forced Labour: An Uncertain Boundary? The Circulation of Economic Ideas between Russia and Europe from the Eighteenth to the Mid-Nineteenth Century," Kritika: Explorations in Russian and Eurasian History 9, no. 1 (2008): 1-27.

  37AN (National Archvies, Paris) F 10 452 "Fixation des salaires agricoles," an II, an III; AN C 1157-61. 38 Antoine Pages, Usages et règlements locaux, servant de complément à la loi civile et topographie légale du département de l'Isère (Grenoble: Baratier Frères, 1855). See also the 1870 parliamentary enquiry in AN C 1157-61.

  ); Seymour Drescher, Capitalism and Antislavery: British Mobilization in Comparative Perspective (New York and Oxford: Oxford University Press, 1987); David Brion Davis, The Problem of Slavery in the Age of Revolution, 1770-1823 (New York: Oxford University Press, 1999); Marcel Dorigny, ed, Les abolitions de l'esclavage (Saint-Denis: Presses Universitaires de Vincennes, 1995); Yves Benot, La révolution française et la fin des colonies (Paris: La Découverte, 2004); Lawrence Jennings, French Antislavery: The Movement for the Abolition of Slavery in France, 1802-1848 (Cambridge: Cambridge University Press, 2000); Nelly Schmidt,

Abolitionnistes de l'esclavage et réformateurs des colonies, 1820-1851: Analyses et documents (Paris: Karthala, 2000). 10 Richard Allen, European Slave Trade in the Indian Ocean (Athens, OH: Ohio University Press, 2015); Edward Alpers, The Indian Ocean in World History (New York and Oxford: Oxford University Press, 2013); Abdul Sheriff, The Indian Ocean: Oceanic Connections and the Creation of New Societies (London: Hurst, 2014). 11 Gabriel Debien, Les engagés pour les Antilles 1634-1715 (Paris: Société de l'histoire des colonies françaises, 1952), 45.

Général Ambert, Abolition de l'esclavage (Paris: Imprimerie de Guirodet et Jouaust, 1848).

ANOM (French Colonial Archives), Généralités 162 d. 1326, "Comptes rendus des séances de la Commission d'abolition de l'esclavage."

Jean-Claude Farcy, "Les archives méconnues de la justice civile," in Histoire et archives: L'histoire contemporaine et les usages des archives judiciaires, ed. Frédéric Chauvaud and Jacques-Guy Petit (Paris: Champion, 1998), 397-408.

Jean-Marie Fillot, La traite des esclaves vers les Mascareignes au XVIIIe siècle (Paris: Orstom, 1974); Sudel Fuma, L'esclavagisme à laRéunion, 1794-1848 (Paris: Harmattan, 1992).

Frédéric Régent, La France et ses esclaves (Paris: Grasset 2007), 25.

Fillot, La traite.

Marina Carter, "Indian Slaves in Mauritius (1729-1834)," Indian Historical Review 15, nos.1-2 (1988- 1989), 233-47.

[START_REF]Fixation des salaires agricoles," an II, an III; N C 1157-61[END_REF] Fillot, La

traite.17 Régent, La

France, 61. 18 Claude Wanquet, Histoire d'une Révolution: La Réunion, 1789-1803 (Paris:

Editions Laffitte, 1980-1984).

Fréderic Régent, Esclavage, métissage, liberté (Paris: Grasset,

2004).20 Monica Schuler, "The Recruitment of African Indentured Labourers for European Colonies in the Nineteenth Century," in Colonialism and Migration: Indentured Labour before and afterSlavery, ed. Pieter Emmer (Dordrecht, Boston and Lancaster: Martinus Nijhoff, 1986), 125-61. 21 Alessandro Stanziani, Labour in the Fringes of Empire: Voice, Exit, and the Law (New York: Palgrave, 2018).22 TNA (The National Archives, Kew), CO 415/9/A.221, 1827.23 Archives départementales de la Réunion (henceforth ADR) in 57 M1, for example: "Exposé de la situation intérieure de la colonie en 1832 par le directeur de l'intérieur" and "Rapport sur les différents services de la colonie," 1828.25 ADR 168 M 3, Lettre du Directeur de l'intérieur concernant les Indiens mutins de l'établissement Adam, 2 August 1831; Lettre du régisseur de l'établissement de Champ-Borne, 24 July 1831.25 ADR 168 M 3, Lettre du Directeur de l'intérieur concernant les Indiens mutins de l'établissement Adam, 2 August 1831; Lettre du régisseur de l'établissement de Champ-Borne, 24 July 1831.26 Megan Vaughan, Creating the Creole Island Slavery in Eighteenth Century Mauritius (Durham and London: Duke University Press, 2007).27 ANOM FM SG/Reu c 406, c 432 d 4603 à 4606 (immigration chinoise). ("FM" stands for Fonds ministériels; "SG" for Série Géographiques (geographical series), and "Reu" stands for Réunion Island; "c" is the box and "d" the file in the box.)28 Sudel Fuma, Esclaves et citoyens, le destin de 62000 Réunionnais, histoire de l'insertion des affranchis de 1848 dans la société réunionnaise (Saint-Denis, La Réunion: Fondation pour la Recherche et le Développement dans l'Océan Indien, 1979), 116.

The livret ouvrier was a discharge certificate; it had to certify being hired for a specific job and its completion (quittance) or acknowledge that the worker had not yet paid off advances received as wages and that his debt remained to be deducted from future wages by the new employer.

Cottereau, "Droit et bon droit."

Jean-Luc Mayaud, "Salariés agricoles et petite propriété dans la France du XIXe siècle," In La moisson des autres, ed. Jean-Claude Farcy and Ronald Hubscher (Nice: Créaphys édition, 1996): 29-56.