J Ignacchiti

A Reigue

F Amrani

B Debord

F Gérôme

F Benabid
email: f.benabid@xlim.fr

Strong phase-locking in quantum seeded of Raman comb

We demonstrate strong and stable phase-locking between the pump and the quantum generated Stokes/anti-Stokes spectral-pair in a comb generated in the regime of a single spatiotemporal coherent mode transient stimulated Raman scattering in H2-filled HCPCF. © 2022 The Author(s)

In conclusion, we demonstrated strong and stable phase-locking between the pump, 1 st Stokes and 1 st anti-Stokes of a Raman comb generated using H2-filled IC-HCPCF.

Transient stimulated Raman scattering (TSRS) in inhibited-coupling guiding hollow-core photonic crystal fiber (IC-HCPCF) proved to be a promising means to generate several octave bandwidth coherent optical spectral comb [1][2]. This in turn could represent a powerful means for optical waveform synthesis [3] and attosecond optical pulse generation. Because the TSRS spectrum is initiated from the vacuum noise, the phase locking between the different spectral components of the comb generated by TSRS has been a raised question as it strongly depends on the quantum initial conditions of the spontaneous Stokes field. One approach to ensure robust phase locked spectral components is a scheme that could "single out" and amplify only one spatio-temporal mode (STM) from the spontaneously emitted Stokes field Es. STMs are coherent modes that can form a basis for Es. More specifically, the latter can be expressed as a linear combination of individual coherent STM using Karhunen-Loève modal decomposition, i.e. ))) [4]. If only one STM is amplified, the resulting Stokes pulse is a transformlimited pulse with a well-defined phase, and in turn the generation of a broadband coherent comb via Raman cascade and four wave mixing processes. This remarkable dynamics was experimentally demonstrated using hydrogen molecule filled IC-HCPCF excited with nanosecond optical pulses [1][2]. Here, the fiber close-to single mode guidance allows to act as a "spatial filter" for the Stokes spatial modes ,𝜙 𝑘 (𝑟), so only the spatial mode with the highest Raman gain survives the propagation. Furthermore in [5], we extended this filtering approach to the time-domain by pumping a H2-filled IC-HCPCF with optical pulses with duration of less than 10 ps. This allowed the generation of a train of sub-20 fs pulses spaced by 57 fs. However, to date no direct experimental evidence was reported on the phase relation between the pump and the first order Stokes and anti-Stokes, which is a key indicator to the coherence of the generated comb and for the potential pulse to pulse locking. Here, we demonstrate strong phase locking between the pump, the Stokes and the anti-Stokes pulse of TSRS based spectral comb using a novel spectrally resolved interferometry. The results show a very stable phase with a Gaussian statistical distribution with a standard deviation of ~2°. Fig. 1(a) shows the contour plot of the calculated energy extinction ratio between the highest gain STM, and the 80 first higher order STM, (𝑅 𝑒 = 10 * |log(𝑏 1

-1 ∑ 𝑏 𝑘 ) 𝑘 2
| ) in function of pump-pulse energy and duration. The dashed region of the contour plot shows the pulse energy-duration range where 𝑅 𝑒 >30dB and the TSRS is void of parasitic phenomena such as Kerr effect. The result shows that for pulse energy range ~1.5-7 J, the pulse duration range for single STM excitation and amplification is ~1-9ps. Fig. 1(bc) shows the typical frequency-resolved optical gating (FROG) and auto-correlator traces of the first 3 Raman orders from the generated comb [5]. Fig. 1(d) presents the experimental set-up used in this work. The TSRS spectral comb generated in the H2-filled IC-HCPCF is split into two arms. The first beam (arm 1) is spectrally filtered to have only the pump and Stokes lines and acts as the reference signal with a variable delay line achieved using stable motorized plate with 0.25 m resolution. The second beam (arm 2) can be spectrally filtered to have a chosen number of lines and acts as the spectrum under test. The two beams are then focused on a BBO crystal and the generated sum frequency spectrum (SFS) is recorded for varying delays on a photo-spectrometer. Fig. 1(e-g) shows examples of spectra for both arms and the resulted sum frequency spectrum generated in the BBO for a given position of the movable mirror. Under the assumption of a Raman comb field of the form 𝐸 = ∑ 𝐴 𝑛 (𝑡)𝑒 -𝑖(𝜔 𝑛 𝑡+𝜙 𝑛 ) 𝑛 with 𝑛, 𝐴 𝑛 , 𝜔 𝑛 , and 𝜙 𝑛 the Raman line order, amplitude, angular frequency and phase respectively, the intensity of the line 𝑝 of the SFS (i.e. 𝜔 ̃𝑝 = 2𝜔 0 + 𝑝Ω) can be express as function of the delay time:

𝐼 (𝑝) (𝜏) = 𝐽 1 0,𝑝 (𝜏) + 𝐽 1 -1,𝑝+1 (𝜏) + 2𝐽 2 0,𝑝,-1,𝑝+1 (𝜏) cos(Ωτ + Δϕ p )
(1) with Δϕ p = (𝜙 0 + 𝜙 𝑝 ) -(𝜙 -1 + 𝜙 𝑝 +1 ), and where 𝐽 1 𝑛,𝑚 (𝜏)and 𝐽 2 𝑛,𝑚,𝑝,𝑞 can be seen as correlation functions of the different Raman lines combined to produce the line 𝑝 in the BBO. Eq. 1 predicts an oscillation of the intensity with the delay as experimentally observed in Fig. 2(a-b) for the line 𝑝 = 0, (𝜆=515 nm), and a laser pulse of 3 ps and 5 ps respectively. This oscillation is due to the beating between the pump, Stokes and anti-Stokes lines. The large value of the interference contrast clearly demonstrates the strong phase locking between these thre lines of the Raman comb. This is further supported in Fig. 2 which shows the oscillations realized over a large number (200 repetitions) (Fig. 2ab) and the histogram of Δ𝜙 0 demonstrating the stability of the phase for the Raman lines involved (Fig. 2c-d).

Additionally, for a spectrum under test comprising only the pump and 1 st Stokes, and assuming a Gaussian pulse profile, the temporal profile of the SFS lines of order p=0 (i.e.𝜆=515 nm) , p=-1(530 nm) and p=-2 (550 nm) are

{ 𝐼 (0) (𝜏)𝛼𝑒 -( 𝜏 𝜏0 ) 2 𝐼 (-1) (𝜏)𝛼𝑒 -( 𝜏+Δ𝑡 𝜏0 ) 2 + 𝑒 -( 𝜏-Δ𝑡 𝜏0 ) 2 + 2𝐶𝑒 -( Δ𝑡 𝜏0 ) 2 𝑒 -( 𝜏 𝜏0 ) 2 cos(Ωτ) 𝐼 (-2) (𝜏)𝛼𝑒 -( 𝜏 𝜏0 ) 2 (2)
where Δ𝑡is the time delay between the pump and the Stokes emission and 𝐶 is a parameter due to experimental imperfection. Eq. 2 predicts reconstruction of the pulse temporal profile of the pump and the Stokes (i.e. 𝐼 (0) and 𝐼 (-2) ) and oscillation of the intensity in the presence of phase coherence between the 1 st Stokes and the pump (Fig. 2ef).

Fig. 1 .

 1 Fig. 1. (a) Extinction ratio between the highest gain STM and the 80 first higher order STM. (bc) FROG and auto-correlator traces of the first three Raman orders from the generated comb. (d) Experimental scheme. (efg) Optical spectra recorded on arm 2, arm 1 and after the BBO respectively.

Fig. 2 .

 2 Fig. 2. (ab) Measured 515 nm oscillations repeated over 200 times for an unfiltered spectrum under test (cd ) Statistical distribution of the fitted phase of 515 nm oscillations. (ef) Measured SFS intensity temporal profile for orders 0 (515nm), -1 (530nm), and -2 (550nm) for a spectrum under test comporting only the pump and the 1 st Stokes. The insets show a plot of Eq. 2 for Δ𝑡 = 200𝑓𝑠(e) and Δ𝑡 = 700𝑓𝑠(f).

We acknowledge financial supports from BPI (Grant


Thomas Billotte

Jonas Osório

Foued Amrani

Frédéric Gérôme

Benoît Debord

Fetah Benabid
email: f.benabid@xlim.fr

Saturated absorption spectroscopy using azimuthally structured mode in hollow-core fiber

Sub-Doppler spectroscopy such as electromagnetically induced transparency (EIT) or saturated absorption spectroscopy (SAS) is usually performed with a Gaussian mode profile both in free-space and guided optics configurations. Recently, in free space, works have been reported where Laguerre-Gaussian (LG) modes have been set as pump or probe beams in EIT [1][2][3][4][5] and SAS [6] schemes in Rb vapors. Radial variation of LG mode induces narrowing of Doppler-free transparencies profiles. Chanu et al. [1] demonstrated decrease of resonances linewidths of Rb of almost factor 6 switching from Gaussian to LG mode. For [5], LG modes contribute to increase the Rb Zeeman level lifetime and so decrease the FWHM. Furthermore, influence of LG modes order has also been studied in [2] demonstrating that larger order induces narrower signal due to its azimuthal component and authors of [4] present the influence of transverse mode overlap (pump and probe) on the signal. Gas-filled HCPCF technology already demonstrated its potential and impact on the enhancement of the light-gas interactions phenomena and especially in spectroscopy. The various works deriving from the seminal demonstration [7] have been limited to pump (or pump-probe) Gaussian beam configurations due to the fact that most of the developments on HCPCFs have been focused on the design and fabrication of fibers supporting the lowest loss LP01 like mode with the main objective to increase transmission performances. Moreover, the intrinsic cladding modal properties of microstructured hollow-core fibers render difficult the control on the loss mode hierarchy and to guide light in a high order single mode fashion. This restriction has been lifted by the emergence of the tubular cladding HCPCF providing a large flexibility on the tubes cladding position. Thanks to alteration of the azimuthal position of the cladding tubes, Osorio et al. [8] demonstrated, theoretically and experimentally, for the first time in a HCPCF, an inversion of the classical loss hierarchy in fibers developing fibers whose lowest loss modes are the LP11 and LP21, instead of the LP01. In this study, we specifically design and fabricate a fiber based on this geometry allowing to guide preferentially high order mode (HOM) around 1550nm without any specific light coupling precaution, we report for the first time to the best of our knowledge on SA spectroscopy generated in HCPCF with a LP11 high-order mode control beam. Comparison with Gaussian control beam configuration demonstrates an increase of the contrast and peak position shift.

Theoretical approach on saturable absorption process during beam propagation in a gas-filled fiber shows that absorption could be separated in two distinguish expressions. The first one corresponds to the "linear" absorption as observed in usual Doppler limited absorption scheme. The second one, noted Snm, relates to the saturable part corresponding to the effect of a pump beam and could be expressed as a function of frequency  :

𝑆 𝑛𝑚 = 4(𝛼 𝑓 + 𝛼 0 )𝐸 0 2 𝜋𝑣 0 2 ∫ ∫ 𝑒 (𝑣 𝜃 2 +𝑣 𝑧 2 ) 𝑣 0 2 Γ 2 𝜓 𝑛𝑚 (𝑟, 𝜃) 4 (Δ + 𝑘𝑣 𝑧 + 𝑚𝑣 𝜃 𝑟 ) 2 + Γ 2 𝑟𝑑𝑟𝑑𝜃𝑑𝑣 𝜃 𝑑𝑣 𝑧 𝑣 ⃗ 𝑟 (1)
where  is the broadening, f the optical loss of the fiber, 0 the absorption at resonance =0, vz the speed on longitudinal axis, v speed along transversal axis, and m the mode azimuthal number. By comparing the expression of the saturable part of the absorption for LP01 and LP11 mode, one can notice an extra azimuthal term 𝑚𝑣 𝜃 𝑟 that should induce a different Doppler shift. This effect has been investigated experimentally by comparing SA signals generated with LP11 and LP01 control modes inside acetylene filled HCPCFs. For this purpose, we have designed and fabricated three HCPCF (Fig. 1(a)): two tubular lattice fibers with relatively strong 580dB/km LP01-loss level for fiber #1, and lower 50dB/km@1530nm loss for fiber #2. Fiber #3 exhibits an asymmetric tubular lattice to guide preferentially LP11 mode with 420dB/km loss at 1530nm and a core diameter of 52µm. Care has been taken to develop fibers with almost similar core size (~50µm) and an identical 2.2m length is taken for experiments in order to normalize SA measurements and only consider the influence of HOM. The two first fibers are used as reference and compare to deduce the impact of optical loss on SA signal. The experimental set-up (shown in Fig. 1(a)) is similar to a counter-propagative probe-pump scheme in gasfilled fiber. The different fibers have been filled with acetylene at pressure of 0.15mbar. The contrast of Doppler limited absorption profile is measured to ensure that the same gas pressure is set during the three distinct measurement campaigns (with an estimated deviation of less than 0.005mbar). SA measured profiles are presented in Fig. 1(b) for an input intensity of 6µW/cm². One can notice that the peak position is shifted by 10MHz in case of LP11 mode as pump beam and LP01 as probe beam in accordance with theory expectation. Finally, Fig. 1(c) shows the contrast evolution with coupled intensity for different probe-pump mode configurations. For same input intensity, and same LP01 pump and probe modes, fiber #1 with higher loss demonstrates a lower contrast than fiber #2. Fiber #3 with 420dB/km LP11 loss exhibits higher contrast (for both LP11 pump-LP01 probe and LP11 pump-LP11 probe configurations) than the two other fibers despite its higher loss level with contrast enhancement (compared to LP01 mode with comparable loss level) reaching up to a factor 2. Furthermore, the contrast is also higher than the theoretical calculation for Gaussian mode with 420dB/km loss (dotted curve). These results demonstrate that implementation of LP11 guided mode profile for SAS allow us to obtain higher contrast with less intensity. Authors acknowledge support from Région Nouvelle Aquitaine.

Fig. 1 .

 1 Fig. 1. (a) Setup scheme and description of the used hollow core fibers. (b) SAS measurement at 6µW/cm² of input intensity (c) evolution of contrast with coupled intensity for different probe-pump mode configurations. Measured SA spectrum at 14µW/cm² of input intensity for LP11 probe and pump (d) and LP01 probe and pump configuration (e).


S Herrmann

J Tugaut

Non uniqueness of stationary measures for self-stabilizing processes

Keywords: self-interacting diffusion, stationary measures, double well potential, perturbed dynamical system, Laplace's method, fixed point theorem. 2000 AMS subject classifications: primary 60H10, secondary: 60J60, 60G10, 41A60

We investigate the existence of invariant measures for self-stabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a double-well landscape and attracted by its own law. This specific self-interaction leads to nonlinear stochastic differential equations and permits to point out singular phenomenons like non uniqueness of associated stationary measures. The existence of several invariant measures is essentially based on the non convex environment and requires generalized Laplace's method approximations.

Introduction

The aim of this paper is to present some new and surprising results concerning the existence of invariant probability measures for one-dimensional selfstabilizing diffusions. The specificity of such diffusion is the attraction of its paths by the own law of the stochastic process. The dynamical system solved by self-stabilizing diffusions can be characterized by three essential elements: first the system is governed by a double-well potential V which represents roughly the environment of the process, secondly some interaction potential F describes how strong the attraction between the process and its own law is, and finally the system is perturbed by some Brownian motion with small amplitude ( √ ǫB t , t ≥ 0).

Let us denote by u ǫ t (dx) the law of the self-stabilizing diffusion (X ǫ t , t ≥ 0), then the SDE satisfied by (X ǫ t ) is given by:

X ǫ t = X 0 + √ ǫB t - t 0 V ′ (X ǫ s )ds- t 0 R
F ′ (X ǫ s -x)du ǫ s (x)ds, ǫ > 0.

(E ǫ,X0 )

Introducing the notation of the convolution product, (E ǫ,X0 ) can be written as follows:

X ǫ t = X 0 + √ ǫB t - t 0 (V ′ + F ′ * u ǫ s ) (X ǫ s ) ds. (1.1)
Let us just note that the interaction part of the drift term is related to the diffusion in some simple way:

F ′ * u ǫ t (x) = E[F ′ (x -X ǫ t )
]. This way of characterizing the drift term essentially points out the structure of the attraction between the paths of the diffusion and its law. Self-interaction corresponds obviously to mean fields stabilization.

Self-stabilizing diffusion paths can usually be approximated by the movement of some specific Brownian particle belonging to a huge ensemble of identical ones. In this global system each particle is submitted to the same forces. First it moves in the potential landscape characterized by the double-well function V and accordingly it is attracted by positions which minimize the potential. The second force which acts on the system is the interaction between all the particles. More precisely each one is attracted by all the others. This attraction can for instance be thought of as being generated by electromagnetic effects. In this case, the solution of the global system doesn't represent some spatial position but some electromagnetic charge.

The huge particle system containing N elements is governed by the following stochastic differential equation

dX i,N t = √ ε dW i t , -V ′ (X i,N t ) dt - 1 N N j=1 F ′ (X i,N t -X j,N t ) dt, X i,N 0 = x 0 ∈ R, 1 ≤ i ≤ N. (1.2)
Here the W i are independent Brownian motions. In the limit, as N becomes large, the interaction part of the drift term is approximatively the average with respect to the law of one characteristic particle of the system (law of large numbers framework). More precisely the empirical measures 1 N N j=1 δ X j,N t converges to some law u ε t for each fixed time and noise intensity, and each individual particle's motion converges in probability to the solution of the diffusion equation

dX i t = √ εdW i t -V ′ (X i t ) dt - R F ′ (X i t -x) du ε t (x) dt. (1.3)
Interacting particle systems such as (1.2) have been studied from various points of view. A survey about the general setting for interaction (under global Lipschitz and boundedness assumptions) may be found in [START_REF] Sznitman | Topics in propagation of chaos[END_REF].

The aim of this paper is to consider both the existence and the uniqueness of stationary measures for the self-stabilizing diffusion (E ǫ,X0 ). In [START_REF] Herrmann | Large deviations and a Kramers' type law for self-stabilizing diffusions[END_REF] Herrmann, Imkeller and Peithmann proved the existence of some unique strong solution to equation (1.1) generalizing previous results obtained by Benachour, Roynette, Talay and Vallois [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF] in the context of constant environment potential V (V ′ (x) = 0 for all x ∈ R). We choose their work as basis for developing our study. Nevertheless there exist several different papers dealing with the existence problem for self-stabilizing diffusion, each of them concerning other families of interaction functions. Let us cite McKean who studied in some earlier work a class of Markov processes that contains the solution of the limiting equation under restrictive global Lipschitz assumptions for the interaction [START_REF] Mckean | A class of Markov processes associated with nonlinear parabolic equations[END_REF], Stroock and Varadhan who considered some local form of interaction [START_REF] Stroock | Multidimensional diffusion processes, volume 233 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF], Oelschläger who studied the particular case where interaction is represented by the derivative of the Dirac measure at zero [START_REF] Oelschläger | A law of large numbers for moderately interacting diffusion processes[END_REF] and finally Funaki who addressed existence and uniqueness for the martingale problem associated with self-stabilizing diffusions [START_REF] Funaki | A certain class of diffusion processes associated with nonlinear parabolic equations[END_REF].

Let us now focus our attention to the stationary measures. In [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF] the authors emphasize that the invariant measure, corresponding to some given average, is unique in this particular constant potential V situation. This feature is essential for further developments. The natural convergence question between the law of the process and the invariant measure, as time elapses, can then be analyzed, see [START_REF] Benachour | Nonlinear self-stabilizing processes. II. Convergence to invariant probability[END_REF]. This kind of convergence was also considered by Tamura under different assumptions on the structure of the interaction, see [START_REF] Tamura | Free energy and the convergence of distributions of diffusion processes of McKean type[END_REF] and [START_REF] Tamura | On asymptotic behaviors of the solution of a nonlinear diffusion equation[END_REF].

The presence of some potential gradient which describes the environment of the self-stabilizing diffusion is essential for the question of existence and uniqueness of invariant measures. In particular, if the landscape is represented by some symmetric double-well potential then surprising effects appear due to the lack of convexity: we shall prove that, under suitable conditions, there exist at least three invariant measures of which one is symmetric (Theorem 4.5) and two are asymmetric or so-called outlying (Theorem 4.6). In the particular linear interaction case (F ′ (x) = αx with α > 0), these three measures constitute the whole set of invariant measures (Theorem 3.2) provided that V ′′ is a convex function.

The material of this paper is organized as follows: first we list several assumptions concerning both the interaction function F and the environment potential V which permit in particular to assure the existence of the self-stabilizing diffusion (E ǫ,X0 ). In Section 2 preliminary results concerning the structure of the invariant measure (if it exists !) are developed. These results are essential for the construction of such measures. The question of existence starts to be addressed in Section 3 in the particular linear interaction context. After pointing out some symmetric and asymmetric invariant measures, we point out some nice context for which the whole set of stationary measures can be described. This study is finally extended to the general interaction case in the last section. We postpone different tools concerning asymptotic analysis based on Laplace's method to the Appendix.

Main assumptions

In order to study invariant measures for self-stabilizing diffusions, we especially need that (1.1) admits some unique strong solution. For this reason, we assume that both the potential landscape V and the interaction function F satisfy some growth conditions and some regularity properties. Moreover we add some technical assumptions which permit to simplify the statements. We assume the following properties for the function V :

(V-1) Regularity: V ∈ C ∞ (R, R). C ∞ denotes
the Banach space of infinitely bounded continuously differentiable function.

(V-2) Symmetry: V is an even function.

(V-3) V is a double-well potential. The equation V ′ (x) = 0 admits exactly three solutions : a, -a and 0 with a > 0; V ′′ (a) > 0 and V ′′ (0) < 0. The bottoms of the wells are reached for x = a and x = -a.

(V-4) There exist two constants

C 4 , C 2 > 0 such that ∀x ∈ R, V (x) ≥ C 4 x 4 -C 2 x 2 . V -a a Figure 1: Potential V (V-5) lim x→±∞ V ′′ (x) = +∞ and ∀x ≥ a, V ′′ (x) > 0.
(V-6) The growth of the potential V is at most polynomial: there exist q ∈ N * and C q > 0 such that

|V ′ (x)| ≤ C q 1 + x 2q . (V-7) Initialization: V (0) = 0.
Typically, V is a double-well polynomial function. But our results can be applied to more general functions: regular functions with polynomial growth as |x| becomes large. We introduce the parameter ϑ which plays some important role in the following:

ϑ = sup x∈R -V ′′ (x). (1.4)
Let us note that the simplest example (most famous in the literature) is

V (x) = x 4 4 -x 2
2 which bottoms are localized in -1 and 1 and with parameter ϑ = 1. Let us now present the assumptions concerning the attraction function F .

(F-1) F is an even polynomial function. Indeed we consider some classical situation: the attraction between two points x and y only depends on the distance F (x -y) = F (y -x).

(F-2) F is a convex function.

(F-3) F ′ is a convex function on R + therefore for any x ≥ 0 and y ≥ 0 such that x ≥ y we get

F ′ (x) -F ′ (y) ≥ F ′′ (0)(x -y).
(F-4) The polynomial growth of the attraction function F is related to the growth condition (V-6):

|F ′ (x) -F ′ (y)| ≤ C q |x -y|(1 + |x| 2q-2 + |y| 2q-2 ).
Let us define the parameter α ≥ 0 which shall play some essential role in following:

F ′ (x) = αx + F ′ 0 (x) with α = F ′′ (0) ≥ 0. (1.5)
In [START_REF] Herrmann | Large deviations and a Kramers' type law for self-stabilizing diffusions[END_REF], Herrmann, Imkeller and Peithmann present sufficient conditions for the SDE (1.1) to admit a unique strong solution. In particular, if E[X 8q 2 0 ] < +∞, with q defined in (V-6), and if all the main assumptions just defined are satisfied, the existence and uniqueness of the solutions are proved. In the following we will always assume that the (8q 2 )-th moment of the initial value X 0 is finite. This permits to study further the self-stabilizing diffusions and exhibit invariant measures.

General structure of the invariant measures

This section deals with different preliminary results describing the main structure of the invariant measures of E ǫ,X0 . First of all, there is some classical link between the stochastic differential equation and the associated parabolic partial differential equation which permits to characterize stationary measures.

Lemma 2.1. Let u ǫ t (x) denote the density of (X ǫ t ; t ≥ 0) with respect to the Lebesgue measure. Then u ǫ is solution of the following PDE:

∂ ∂t u ǫ t (x) = ǫ 2 ∂ 2 ∂x 2 u ǫ t (x) + ∂ ∂x u ǫ t (x) V ′ (x) + (F ′ * u ǫ ) (t, x) (2.1)
for all t > 0, x ∈ R and u ǫ 0 (dx) = P (X 0 ∈ dx). We recall that R x 8q 2 u ǫ 0 (dx) < ∞.

Proof. Let f ∈ C 2 (R, R) such that lim x-→±∞ f (x) = lim x-→±∞ f ′ (x) = 0. (2.2)
By Itô's formula, we obtain

E [f (X ǫ t )] = E [f (X ǫ 0 )] + E t 0 f ′ (X ǫ s ) √ ǫdB s -E t 0 f ′ (X ǫ s ) V ′ (X ǫ s ) + F ′ * u ǫ s (X ǫ s ) ds + ǫ 2 t 0 f ′′ (X ǫ s )ds .
Taking the time derivative, we get

d dt E [f (X ǫ t )] = -E f ′ (X ǫ t ) V ′ (X ǫ t ) + F ′ * u ǫ t (X ǫ t ) + ǫ 2 f ′′ (X ǫ t ) = - R f ′ (x) V ′ (x) + F ′ * u ǫ t (x) u ǫ t (x)dx - R ǫ 2 f ′′ (x)u ǫ t (x)dx.
Since f is a C 2 function, integration by parts leads to

d dt E [f (X ǫ t )] = R f (x) ∂ ∂x (V ′ (x) + F ′ * u ǫ t (x)) u ǫ t (x) + ǫ 2 ∂ 2 ∂x 2 u ǫ t (x) dx.
Using the equality:

d dt R f (x)u ǫ t (x)dx = R f (x) ∂ ∂t u ǫ t (x)dx,
we deduce for all C 2 -functions satisfying (2.2):

R f (x) ∂u ǫ t (x) ∂t dx = R f (x) ∂ ∂x (V ′ (x) + F ′ * u ǫ t (x)) u ǫ t (x) + ǫ 2 ∂ 2 u ǫ t (x) ∂x 2 dx
We obtain (2.1) by identification.

The density of (X ǫ t , t ≥ 0) with respect to the Lebesgue measure is solution to the parabolic PDE (2.1) (non-linear Kolmogorov equation): this implies in particular that any stationary measure (if it exists !) satisfies some elliptic differential equation. This link between non-linear differential equations and self-stabilizing diffusions permits to express the invariant measure in some exponential form.

Lemma 2.2. If there exists an invariant measure u ǫ to (E ǫ,X0 ) whose (8q 2 )moment is finite, then:

u ǫ (x) = 1 λ(u ǫ ) exp - 2 ǫ x 0 F ′ * u ǫ (y)dy + V (x) (2.3)
= 1 λ(u ǫ ) exp - 2 ǫ F * u ǫ (x) -F * u ǫ (0) + V (x) ,
where λ(u ǫ ) denotes the normalization factor: R u ǫ (x)dx = 1.

Proof. By (2.1), any stationary measure u ǫ satisfies

ǫ 2 u ′′ ǫ (x) + u ǫ (x) (V ′ (x) + F ′ * u ǫ (x)) ′ = 0, for all x ∈ R.
By integrating the previous equality, we obtain the existence of some constant

C ǫ ∈ R such that ǫ 2 u ′ ǫ (x) + u ǫ (x)(V ′ (x) + F ′ * u ǫ (x)) = C ǫ , for all x ∈ R.
Using the method of variation of parameters, the solution u ǫ takes the following form

u ǫ (x) = Λ ǫ (x) exp - 2 ǫ x 0 F ′ * u ǫ (y)dy + V (x) , with Λ ′ ǫ (x) = 2 ǫ C ǫ exp 2 ǫ x 0 F ′ * u ǫ (y)dy + V (x) .
Hence

u ǫ (x) = Λ ǫ (0) exp - 2 ǫ x 0 F ′ * u ǫ (y)dy + V (x) + 2 ǫ C ǫ x 0 exp 2 ǫ y 0 F ′ * u ǫ (z)dz + V (y) dy × exp - 2 ǫ x 0 F ′ * u ǫ (y)dy + V (x) .
Let us assume that C ǫ = 0. Applying Lemma A.1 to the function U (x) =

x 0 F ′ * u ǫ (y)dy + V (x), whose second derivative is positive for |x| large enough (using hypotheses (V-5) and (F-2)), permits to exhibit the equivalent of Λ ǫ (x):

Λ ǫ (x) ≈ 2 ǫ C ǫ exp 2 ǫ x 0 F ′ * u ǫ (y)dy + V (x) 2 ǫ V ′ (x) + F ′ * u ǫ (x) as x → ±∞. Hence u ǫ (x) ≈ C ǫ V ′ (x) + F ′ * u ǫ (x)
.

Due to the conditions (V-6) and (F-4), there exists some constant K > 0 such that

|V ′ (x) + F ′ * u ǫ (x)| ≤ K(1 + |x| 4q-1
), for all x ∈ R.

We deduce that x → x 8q 2 can't be integrated with respect to u ǫ : that contradicts the essential assumption of the statement. We deduce that C ǫ = 0 and obtain (2.3) after normalization.

Lemma 2.2 presents the essential structure of any invariant measure. The global exponential form will play a crucial role in next sections: to prove the existence of some stationary measure, it is necessary and sufficient to solve equation (2.3).

The linear interaction case

First we shall analyze the existence problem for stationary measures in the simple linear case. In this case F ′ (x) = αx with α > 0, the interaction gradient function is quadratic: F (x) = α 2 x 2 and the stochastic differential equation takes an interesting simple form. The non-linearity of the drift term is limited to the average of the density u ǫ t (x):

X ǫ t = X 0 + √ ǫB t - t 0 V ′ (X ǫ s )ds -α t 0 X ǫ s - R xdu ǫ s (x) ds, ǫ > 0.
The study of this particular case emphases the existence of several invariant measures. The interesting problem is then to determine in which situations the number of such measures is perfectly known.

Existence of invariant measures

The existence question is really simplified in the linear interaction case, it is just reduced in fine to the following parametrization problem. Let us denote the first moment of an invariant measure u ǫ by

m 1 (ǫ) = R xu ǫ (x)dx, (3.1)
then (2.3) becomes

u ǫ (x) = exp -2 ǫ V (x) + α x 2 2 -αm 1 (ǫ)x R exp -2 ǫ V (y) + α y 2 2 -αm 1 (ǫ)y dy . ( 3.2)
We now come to the essential equivalence: u ǫ is an invariant measure if and only if (3.1) and (3.2) are satisfied. It suffices then to point out the convenient parameters m 1 (ǫ) since there is a one to one correspondence between these parameters and the invariant measures. In other words, we shall find the solution of the equation

m = Ψ ǫ (m) with Ψ ǫ (m) = R x exp -2 ǫ V (x) + α x 2 2 -αmx dx R exp -2 ǫ V (x) + α x 2 2 -αmx dx . (3.3)
Obviously, m 0 1 (ǫ) = 0 is a candidate. The corresponding measure u 0 ǫ is invariant and symmetric:

u 0 ǫ (x) = exp - 2 ǫ V (x) + α x 2 2 R exp - 2 ǫ V (y) + α y 2 2 dy -1
.

In fact u 0 ǫ is the unique symmetric stationary measure. Of course the natural question concerns the existence of others reals m 1 (ǫ) solutions of (3.2). In fact the basic dynamical system associated to self-stabilizing diffusions is symmetric since F and V are assumed to be even functions. The consequence is immediate: if the initial law of the diffusion (X ǫ t , t ≥ 0) is symmetric so will be the law of X ǫ t for all t > 0. In [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF], the authors consider self-stabilizing diffusions without the environment potential V . They proved the existence of some unique symmetric invariant measure and describe the behavior of the diffusion: for any initial law satisfying the moment condition of order 8q 2 the law of X t -E[X 0 ] converges to the invariant symmetric law as time elapses.

Adding some double-well potential V in the main structure of the stochastic differential equation changes drastically the situation. In particular we prove the existence of several invariant measures, one of them being symmetric. Proposition 3.1. Let a be the unique positive real which minimizes V (see (V-3)). For all δ ∈]0, 1[, there exists ǫ 0 > 0 such that for all ǫ ≤ ǫ 0 , the equation

(3.
3) admits a solution satisfying the estimates:

a - (1 + δ)V (3) (a) 4V ′′ (a) (α + V ′′ (a)) ǫ ≤ m 1 (ǫ) ≤ a - (1 -δ)V (3) (a) 4V ′′ (a) (α + V ′′ (a))
ǫ.

(3.4)

Moreover -m 1 (ǫ) satisfies (3.3) too.
Let us note that, for ǫ small enough, the preceding proposition implies the existence of at least three invariant measures corresponding to the averages: 0, m 1 (ǫ) and -m 1 (ǫ).

Proof. Set τ > 0. Let's proceed to the first order asymptotic development of the expression Ψ ǫ (a -τ ǫ).

Ψ ǫ (a -τ ǫ) = R x exp -2 ǫ V (x) + α x 2 2 -α(a -τ ǫ)x dx R exp -2 ǫ V (x) + α x 2 2 -α(a -τ ǫ)x dx = R xe -2ατ x exp -2 ǫ V (x) + α x 2 2 -αax dx R e -2ατ x exp -2 ǫ V (x) + α x 2 2 -αax dx
.

By Lemma A.5 applied to the context:

f (x) = -2ατ x, n = 1, U (x) = V (x) + α 2
x 2 -αx and µ = 0, we get:

Ψ ǫ (a -τ ǫ) = a - 1 4a (α + V ′′ (a)) 2 aV (3) (a) + 4aατ (α + V ′′ (a)) ǫ + o(ǫ) = a -τ ǫ + V ′′ (a) α + V ′′ (a) τ - V (3) (a) 4V ′′ (a) (α + V ′′ (a)) ǫ + o(ǫ). Set τ 0 = V (3) (a)
4V ′′ (a)(α+V ′′ (a)) . Then a -τ 0 ǫ is the first order approximation of the fixed point. Indeed for δ ∈]0; 1[ we can define

d ± := Ψ ǫ a -τ 0 (1 ± δ)ǫ -a -τ 0 (1 ± δ)ǫ = ±δ V ′′ (a) α + V ′′ (a) τ 0 ǫ + o(ǫ).
For ǫ small enough,

d + > 0 and d -< 0. Since the function Ψ ǫ is C 0 continuous, there exists m 1 (ǫ) ∈ [a -τ 0 (1 + δ)ǫ; a -τ 0 (1 -δ)ǫ] which satisfies Ψ ǫ (m 1 (ǫ)) = m 1 (ǫ).
Finally, by the change of variable x := -x in the integral expression

(3.3), we obtain Ψ ǫ (-m 1 (ǫ)) = -Ψ ǫ (m 1 (ǫ)) = -m 1 (ǫ).

Description of the set of invariant measures

According to Proposition 3.1, we know there are at least three invariants measures. One of them is symmetric corresponding to the average 0 and two others will be called outlying measures, one wrapped around a and the other one around -a. The aim of this section is to study if there are exactly three invariants measures or more.

For this purpose, we study the asymptotic behavior of the function Ψ ǫ defined by (3.3) in the small noise limit.

Theorem 3.2. If V ′′ is a convex function then, in the small noise limit, there exist exactly three stationary measures.

Proof. Let m > 0. Let us recall that the interaction function is linear:

F ′ (x) = αx with α > 0.
In order to study the invariant measures, we have to consider the fixed points of the application Ψ ǫ (m) defined by (3.3). We introduce the following potential function:

W m (x) = V (x) + α 2 x 2 -αmx. Since V ′ (0) = 0, we have W ′ m (0) < 0. Moreover lim x→+∞ W ′ m (x) = +∞.
So we denote by x m the positive real for which the potential W m admits its global minimum. It is uniquely determined since V ′′ is a convex function. In particular,

x m satisfies V ′ (x m ) + α(x m -m) = 0 and V ′′ (x m ) + α ≥ 0. Fur- thermore V ′′ (x m ) + α > 0. Indeed, since x m is a global minimum, the equality V ′′ (x m ) + α = 0 implies that V (3) (x m ) = 0 that is x m = 0 which contradicts the assumption concerning the positivity of x m . We define χ ǫ (m) = Ψ ǫ (m) -m and χ 0 (m) = x m -m.
We obtain the expression:

χ ǫ (m) = x m -m + R (x -x m ) exp -2 ǫ V (x) + αx 2 2 -αmx dx R exp -2 ǫ V (x) + αx 2 2 -αmx dx . (3.5)
It suffices to prove that χ ǫ has just one zero in R * + .

Step 1: For all ǫ > 0 and m > 0, we observe that χ ǫ (m) ≤ χ 0 (m) = x m -m. We apply the change of variable x := y + x m to the integrals in (3.5) and obtain

χ ǫ (m) = χ 0 (m) + R y exp -2 ǫ V (y + x m ) + αy 2 2 + α (x m -m) y dy R exp -2 ǫ V (y + x m ) + αy 2 2 + α (x m -m) y dy = χ 0 (m) + ∞ 0 y exp -α ǫ y 2 Ω ǫ,m (y)dy R exp -2 ǫ V (y + x m ) + αy 2 2 + α (x m -m) y dy , with Ω ǫ,m (y) = exp - 2 ǫ V (y + x m ) + α (x m -m) y -exp - 2 ǫ V (y -x m ) -α (x m -m) y .
We introduce the function

Λ m (y) = V (y + x m ) -V (y -x m ) + 2α (x m -m) y Since V is an even function, Λ m (0) = 0 and Λ ′′ m (0) = 0. According to the definition of x m , Λ ′ m (0) = 0. V ′′ is a convex function therefore V (3) is increasing. So Λ (3) m (y) = V (3) (y + x m ) -V (3) (y -x m ) ≥ 0 for all y. We deduce that Λ ′′ m is increasing. Hence Λ ′′
m is nonnegative on R * + so does Λ m (y) for y > 0. Finally we get Ω ǫ,m (y) ≤ 0 for all y > 0. We obtain the announced result: χ ǫ (m) ≤ χ 0 (m) for m > 0.

Step 2. χ 0 has a unique zero on R * + . Let us compute χ 0 (a) with a defined in (V-3). We know that a is solution of

V ′ (x) + α (x -a) = 0 with V ′′ (x) + α > 0. Hence χ 0 (a) = 0.
Let us focus our attention to the variations of the function χ 0 on the interval ]0, a]. Since V ′ (x m ) + αx m = αm, and α + V ′′ (x m ) > 0 we deduce that m → x m is derivable; we obtain

χ ′ 0 (m) = d dm x m -1. and d dm x m = α α + V ′′ (x m ) > 0 which implies χ ′ 0 (m) = - V ′′ (x m ) α + V ′′ (x m ) . (3.6)
The denominator is positive due to the definition of x m . According to (V -5), V ′′ (x) > 0 for all x > a. Hence χ ′ 0 (m) < 0 for all m > a. Since χ 0 (a) = 0 we deduce that, for all m > a, χ 0 (m) is strictly negative and therefore the function χ 0 has no zero on ]a; +∞[. It remains to study χ 0 on the interval ]0, a]. Since V ′′ is a convex function, we deduce that the derivative of χ 0 is non positive for x m ≥ c with c > 0 satisfying V ′′ (c) = 0. We know that c > 0 is unique since V ′′ (0) < 0 and V ′′ is a convex function. Moreover c < a. Since the function m → x m is increasing for m > 0, we deduce that χ ′ 0 is negative for x ∈] max(0, m c ), a] where m c = c + V ′ (c) α . By construction, if m c > 0 then the equality x mc = c holds. We observe then two different cases:

• If m c ≤ 0 i.e. α < |V ′ (c)| c : χ 0 is decreasing on R * + with χ 0 (a) = 0. The unique zero of χ 0 on R * + is a. • If m c > 0 then χ 0 , which is a continuous function on R *
+ , is increasing on ]0, m c [ and decreasing on ]m c , +∞[ with χ 0 (a) = 0. It suffices to prove that lim m→0+ χ 0 (m) ≥ 0 in order to conclude that a is the unique zero of χ 0 on R * + . Due to the definition of x m we get:

lim m→0+ χ 0 (m) = lim m→0+ x m ≥ 0. Indeed m → x m is continuous from ]0, +∞[ to ]0, +∞[ so the extension to m = 0 is non negative.
In these two cases, there is a unique zero of χ 0 on R * + .

Step 3. The family of functions (χ ǫ ) ǫ (respectively

(χ ′ ǫ ) ǫ ) converges uniformly towards χ 0 (resp. χ ′ 0 ) on each compact subset of R * + . First we prove the convergence of χ ǫ (m) for m > 0. Recall that χ ǫ (m) = R x exp -2 ǫ V (x) + αx 2 2 -αmx dx R exp -2 ǫ V (x) + αx 2 2 -αmx dx .
By Lemma A.5 with U (x) = V (x) + αx 2 2 , n = 1, µ = m and G = -αx we obtain the announced convergence result:

χ ǫ (m) -χ 0 (m) = χ ǫ (m) -x m + m = - V (3) (x m ) 4 (α + V ′′ (x m )) 2 ǫ + o(ǫ).
Moreover this convergence is uniform with respect to the variable m on compact subsets of R * + . We estimate now the asymptotics of χ ′ ǫ (m) as ǫ becomes small. Taking the derivative of Ψ ǫ , we obtain

Ψ ′ ǫ (m) = 2α ǫ    R x 2 exp -2 ǫ W m (x) dx R exp -2 ǫ W m (x) dx - R x exp -2 ǫ W m (x) dx R exp -2 ǫ W m (x) dx 2    .
We recognize the variance of the measure u (m) ǫ which is the measure associated to the average m by (3.2). Hence

χ ′ ǫ (m) = 2α ǫ Var(u (m) ǫ ) -1. (3.7)
Applying again Lemma A.5 with

U = V (x) + αx 2 2 , G = -αx, µ = m and n = 2, we obtain R x 2 exp -2 ǫ W m (x) dx R exp -2 ǫ W m (x) dx = x 2 m - x m V (3) (x m ) -(α + V ′′ (x m )) 2 (α + V ′′ (x m )) 2 ǫ + o(ǫ).
Applying the same lemma with n = 1 permits to compute the first moment:

R x exp -2 ǫ W m (x) dx R exp -2 ǫ W m (x) dx = x m - x m V (3) (x m ) 4x m (α + V ′′ (x m )) 2 ǫ + o(ǫ).
By (3.7) and the computations of the two first moments, we get

χ ′ ǫ (m) = -V ′′ (x m ) α + V ′′ (x m ) + o(1) = χ ′ 0 (m) + o(1). (3.8)
Furthermore this convergence is uniform with respect to the variable m on compact subsets of R * + .

Step 4. For any δ > 0 small enough, there exists ǫ 0 > 0 such that χ ǫ has a unique zero on [δ, ∞[ for all ǫ ≤ ǫ 0 . Since there is no zero of χ ǫ on the interval ]a, +∞[ (Step 1 and 2), we focus our attention to the interval ]0, a]. On each compact subset of this interval, χ ǫ converges uniformly towards the limit function χ 0 (Step 3). Hence the zeros of χ ǫ are in a small neighborhood of the unique zero of χ 0 namely a (Step 2). Let us study the derivative of χ ǫ in a neighborhood of a. Since χ ′ ǫ converges uniformly towards χ ′ 0 (Step 3) and χ ′ 0 (m) < 0 in a neighborhood of a (Step 2), we obtain that χ ′ ǫ (m) < 0 in a neighborhood of a for ǫ small enough. Finally we proved that, as soon as ǫ is small enough, the function χ ǫ can't admit two zeros or more on R * + .

Step 5. There exists δ > 0 and ǫ 0 > 0 such that χ ǫ doesn't vanish on ]0, δ] for all ǫ ≤ ǫ 0 . In this last step, we have to distinguish three different cases depending on the values ϑ and α defined by (1.4) and (1.5).

Step 5.1. We assume α < ϑ. In this particular case W 0 (x) = V (x) + αx 2 /2 reaches a unique global minimum on R + for x = x 0 > 0. Let us fix some small δ > 0 (depending on x 0 : we shall precise it in the following). We prove that, for ǫ small enough,

χ ǫ (m) = Ψ ǫ (m) -m > 0 on ]0, δ]. By the definition of Ψ ǫ , see (3.3), it suffices to prove that N ǫ (m) > 0 for m ∈]0, δ] where N ǫ (m) = R x exp - 2 ǫ W m (x) dx -m R exp - 2 ǫ W m (x) dx. (3.9)
Obviously N ǫ (0) = 0. Let us prove that N ǫ is non decreasing. Taking the derivative, we get

N ′ ǫ (m) = 2α ǫ R x 2 -mx - ǫ 2α exp - 2 ǫ W m (x) dx.
This expression is in fact non negative. Indeed, using the symmetry property of W 0 (x) and the upper bound m ≤ δ, we obtain

N ′ ǫ (m) = 2α ǫ ∞ 0 x 2 - ǫ 2α cosh 2αmx ǫ -mx sinh 2αmx ǫ e -2 ǫ W0(x) dx ≥ α ǫ ∞ 0 P δ (x)e 2αmx ǫ e -2 ǫ W0(x) dx with P δ (x) = x 2 -δx - ǫ α .
We split the preceding integral into two parts: the first integral I 0 concerns the support [0, 2δ] and the second integral I 2δ the complementary support [2δ, ∞[. We get N ′ ǫ (m) ≥ α ǫ (I 0 + I 2δ ). Since the roots of the polynomial function P δ satisfy

x ± = 1 2 δ ± δ 2 + 4ǫ α < 2δ,
the polynomial is positive on the interval [2δ, ∞[ and can be lower bounded by P δ (2δ) = 2δ 2 -ǫ/α. Lemma A.3 implies the existence of some constant C > 0 leading to the following estimate as ǫ → 0:

I 2δ ≥ (2δ 2 -ǫ/α) x0+1 2δ e -2 ǫ W0(x) dx ≥ Cδ 2 √ ǫ e -2 ǫ (V (x0)+αx 2 0 /2) (3.10)
provided that x 0 > 2δ (it suffices then to chose δ small enough).

Let us finally focus our attention to the lower bound of the integral term I 0 . Since the minimum value of P δ is -(δ 2 /4 + ǫ/α) and since W ′′ (0) < 0, we have

I 0 ≥ - δ 2 4 + ǫ α 2δ 0 e 2αmx ǫ e -2 ǫ W0(x) dx ≥ -2δ δ 2 4 + ǫ α e -V (2δ) ǫ . (3.11)
For δ > 0 small enough, V (2δ) > V (x 0 ) + αx 2 0 /2 (since the minimum of V (x) + αx 2 /2 is only reached for x = x 0 ). Consequently the negative lower bound of I 0 (3.11) is negligible with respect to the positive lower bound of I 2δ as ǫ becomes small. We deduce that there exists ǫ 0 such that N ′ ǫ (m) > 0 for all m ∈ [0, δ] and ǫ ≤ ǫ 0 . Since N ǫ (0) = 0 we conclude that N ǫ (m) > 0 on ]0, δ] and so is χ ǫ .

Step 5.2. We assume α > ϑ. In this case W 0 (x) admits a unique minimum reached for x = 0 and x m converges continuously to 0 as m → 0. Using similar arguments as those presented in Step 3, we claim that χ ǫ (resp. χ ′ ǫ ) converges towards χ 0 (resp. χ ′ 0 ) uniformly on [0, a] as ǫ → 0. Due to the regularity of χ 0 and by the inequality χ ′ 0 (0) = -V ′′ (0) α+V ′′ (0) > 0 we obtain the existence of δ > 0 and ǫ 0 > 0 such that χ ′ ǫ (m) > 0 for m ∈ [0, δ] and ǫ ≤ ǫ 0 . χ ǫ starts in 0 and is strictly increasing on [0, δ] which implies the announced result.

Step 5.3. We assume that α = ϑ. It suffices then to note that χ ǫ depends continuously on the parameter α. The following results can be directly deduced from the preceding case (Step 5.2) by continuity: χ ′ ǫ (0) > 0 and χ ′ ǫ (m) ≥ 0 for m ∈ [0, δ] and ǫ ≤ ǫ 0 . In fact χ ǫ vanishes for x = 0 and is increasing on [0, δ]. The inequality χ ǫ (m) > 0 for all m ∈]0, δ] and ǫ ≤ ǫ 0 is an obvious consequence.

Conclusion:

Step 4 and 5 lead to the existence of ǫ 0 > 0 such that for all ǫ < ǫ 0 , χ ǫ has exactly three zeros: 0 and two other reals, one in the neighborhood of a, the other one near -a. To each of these averages corresponds a unique invariant measure obtained by (3.2).

Example: In Theorem 3.2, for all α > 0, as soon as ǫ is small enough, there exist exactly three invariant measures. There is a one to one correspondence between these measures and their average through (3.2). It suffices to determine the averages which are in fact solutions to the equation

χ α ǫ (m) := Ψ ǫ (m) -m = 0.
These solutions are really close to the solutions of χ α 0 (m) = x α m -m = 0 in the small noise limit. We recall that x α m is the global minimum of

W α m (x) := V (x) + α 2 x 2 -αmx on R * + .
Let us observe these averages in the particular case:

V (x) = x 4 4 -x 2 2 and F (x) = α 2 x 2 .
In this case, we compute the parameter c > 0 which vanishes V ′′ and the corresponding parameter

m c = c + V ′ (c)
α . We obtain:

c = 1 √ 3 and m c = 3α -2 3 √ 3α . (3.12)
We shall for this example present graphs of the functions χ α 0 (dotted line) and χ α ǫ for different values of α. We choose ǫ = 1/4. Even if it seems to be not very small, this value suffices in this example to observe three invariant measures for each interaction parameter value considered.

First of all we have to determine the value of x α m which is solution of the system (E α,m ):

X 3 + (α -1)X -αm = 0 and 3X 2 + (α -1) ≥ 0.
Its discriminant is equal to

∆ α (m) = α 2 m 2 4 + (α -1) 3 27 .
We distinguish different cases:

• α = 0: the solution is evident, we get x α m = 1 and χ 0 0 (m) = 1 -m for m > 0 and by symmetry χ 0 0 (m) = -1 -m for all m < 0. Moreover χ 0 0 (0) = 0.

• α > 1 (Figure 2): for all m ∈ R, we get ∆ α (m) > 0. Hence

χ α 0 (m) = 3 αm 2 + ∆ α (m) + 3 αm 2 -∆ α (m) -m.
The function χ α 0 is C ∞ -continuous and odd. We observe also that χ α 0 (0) = χ α 0 (1) = 0 and χ α 0 ′ (m c ) = 0 with m c defined by (3.12). Hence

χ α 0 is increasing on ]0, m c [ and decreasing on ]m c , ∞[. • α = 1 (Figure 3) then χ α 0 (m) = m 1 3 -m. The limit function is odd, continuous on R and C ∞ on R * . Moreover the path is increasing for m ∈]0, m c [, decreasing for m ∈]m c , ∞[ with m c = 1 3 √ 3 .
• 2 3 < α < 1 (Figure 4): the discriminant can be negative. Therefore let us define m 0 (α) such that ∆ α (m 0 (α)) = 0. Then for all m between 0 and m 0 (α), the discriminant is negative and for all m larger than m 0 (α) it is positive. We get m 0 (α) = 2(1 -α) [START_REF] Funaki | A certain class of diffusion processes associated with nonlinear parabolic equations[END_REF]. We obtain the following function: χ α 0 (0) = 0 and

3 2 /(3α √
χ α 0 (m) = ϕ (α) 1 (m) ∀m ∈ [-m 0 (α); 0[ ]0; m 0 (α)] ϕ (α) 2 (m) ∀m ∈] -∞; -m 0 (α)] [m 0 (α); +∞[ with ϕ (α) 1 (m) = 2 1 -α 3 cos 1 3 arccos αm 2 27 (1 -α) 3 -m ϕ (α) 2 (m) = 3 αm 2 + ∆ α (m) + 3 αm 2 -∆ α (m) -m.
Let us note that χ α 0 (0

+ ) = √ 1 -α = 0 and χ α 0 (0 -) = - √ 1 -α = 0. The function is C ∞ -continuous on ]0; m 0 (α)[∪]m 0 (α); +∞[ and continuous in m 0 (α).
Moreover the function is increasing on the interval ]0, m c [ and decreasing for m > m c . The maximum is therefore reached for m = m c . We observe that m c ≤ m α 0 for α ∈ [2/3, 3/4] and m c ≥ m α 0 for α ∈ [3/4, 1]. We remark also that the increasing part is smaller and the decreasing part is longer for smaller values of α. Furthermore, the part where χ α is equal to ϕ (α) 1 is longer.

• α ≤ 2 3 (Figure 5): the function χ α 0 is defined in the same way as in the preceding case. The important difference is that the function is decreasing on R * + since m c defined by (3.12) is non positive.

The general interaction case

We assumed for this study that the self-attraction phenomenon is represented by a polynomial function F ′ , see (F-1). In previous section, we analyzed the particular linear situation: F ′ (x) = αx and proved under suitable conditions that there exist exactly three invariant measures in the small noise limit. In this section we shall focus our attention to the general case: the polynomial function F is of degree n ≥ 2. First we shall present results concerning the symmetric invariant measure and secondly we discuss the presence of asymmetric measures.

Symmetric invariant measures

In the linear case we proved the existence of a unique symmetric invariant measure. The result is obvious since it suffices to solve the equation (3.3) with m 1 (ǫ) = 0. In the general case in order to find the symmetric measure we have to solve some equation like (3.2) but depending on much more parameters than just the mean m 1 (ǫ). The total number of parameters depends in fact on the degree of F . Instead of trying to solve such system, we choose some other kind of proof based on a fixed point theorem which permits to prove the existence of symmetric invariant measures in even more general cases: the interaction function does not need to be polynomial. In [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF], Benachour, Roynette, Talay and Vallois introduced this method of proof for a self-stabilizing diffusion in the constant environment case (V ′ (x) = 0). This proof can be adapted to our situation and is based on the following Schauder's theorem (see for instance [ In order to use this proposition we introduce some definitions and notations:

1. Let us choose p > 4q where q is defined in (V-6).

D

= v : R -→ R + | v is symmetric and sup x∈R + (1 + |x| p ) v(x) < ∞ . 3. B = {f : R -→ R ; sup x∈R (1 + |x| p ) |f (x)| < ∞}. Let us note that D ⊂ B. B is equipped with the norm | • | ∞ where |f | ∞ = sup x∈R (1 + |x| p ) |f (x)|.
4. For all M > 0 we define the function space C M as the subset of all non negative and even function belonging to B which satisfy: 5. For any function f ∈ D we define the operator:

A ǫ (f )(x) = exp -2 ǫ V (x) + x 0 (F ′ * f ) (y)dy z∈R exp -2 ǫ V (z) + z 0 (F ′ * f ) (y)dy (4.1) = 1 λ ǫ (f ) exp - 2 ǫ V (x) + x 0 (F ′ * f ) (y)dy ,
where λ ǫ (f ) is the normalization factor.

6. For any function u ∈ D, we define the moments

γ k (u) = R |x| k u(x)dx with 0 ≤ k ≤ p -2.
Let us just point out that C M is a closed and convex subset of B. Moreover we have C M ⊂ D ⊂ B. The aim of this section will consist in proving that the application A ǫ is C 0 (C M , C M )-continuous and that A ǫ (C M ) is compact. Therefore Schauder's theorem implies the existence of a fixed point and as the matter of fact the existence of an invariant measure in the function space D.

Lemma 4.2. For all u ∈ C M , we have:

1. γ k (u) ≤ M C 1 where C 1 = 1 + max 0≤r≤p-2 R |x| r 1 + |x| p dx for all 0 ≤ k ≤ p -2.

there exists a constant

C 2 > 0 independent of M such that α 2 x 2 ≤ x 0 (F ′ * u)(y)dy ≤ C 2 M x 2 (1 + x 2q ) for all x. (4.2) Proof. 1. Let u ∈ C M then the function x → |x| k 1+|x| p is integrable on R since k ≤ p -2. Moreover the definition of C M implies that (1 + |x| p ) u(x) ≤ M for all x ∈ R. Therefore γ k (u) = R |x| k 1 + |x| p (1 + |x| p )u(x)dx ≤ M R |x| k 1 + |x| p dx ≤ M C 1 .
2. Let x ≥ 0. Since u ∈ C M , u is an even function. By (1.5) we have F ′ (x) = αx + F ′ 0 (x) and (F-3) implies that F ′ and F ′ 0 are non negative odd functions so is F ′ 0 * u. Using the inequality developed in the statement of Lemma 4.3 in [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF] and the assumption (F-3), we have

F ′ 0 (x) ≤ 1 2 F ′ 0 (x -y) + F ′ 0 (x + y) for y ∈ R, x ≥ 0.
Therefore, for x ≥ 0:

x 0 (F ′ 0 * u) (y)dy = x 0 ∞ 0 (F ′ 0 (y -z) + F ′ 0 (y + z)) u(z)dzdy ≥ x 0 ∞ 0 2F ′ 0 (y)u(z)dzdy ≥ 0
From the preceding inequality we deduce

x 0 (F ′ * u) (y)dy = x 0 (F ′ 0 * u) (y)dy + α 2 x 2 ≥ α 2
x 2 for all x ≥ 0.

Since

x 0 (F ′ * u)(y)dy is an even function, we get the inequality for all x ∈ R. 3. Due to the symmetry of F ′ * u we restrict our study to x ≥ 0.

x 0 (F ′ * u) (y)dy = 1 2 x 0 ∞ 0 F ′ (y -z) + F ′ (y + z) u(z)dzdy.
According to the assumptions (F-1) and (F-4), F is an even polynomial function of degree smaller than 2q with q ≥ 1. We can therefore write F ′ as follows

F ′ (x) = q-1 k=0 α k x 2k+1 .
Therefore defining F (y, z) = F ′ (y -z) + F ′ (y + z) we get

F (y, z) = y q-1 k=0 α k k j=0 C 2j+1 2k+1 y 2j z 2k-2j ≤ y max 0≤k≤q-1 |α k |2 2q max 0≤j≤q k j=0 y 2j z 2k-2j ≤ Cy 1 + y 2q 1 + z 2q .
Finally since p > 4q, there exists some constant C ′ > 0 such that:

∞ 0 F (y, z)u(z)dz ≤ Cy 1 + y 2q ∞ 0 1 + z 2q u(z)dz ≤ Cy 1 + y 2q ∞ 0 1 + z 2q 1 + z p (1 + z p )u(z) dz ≤ C ′ yM 1 + y 2q .
By integration we obtain

x 0 (F ′ * u)(y)dy ≤ C 2 M x 2 (1 + x 2q ) for all x ∈ R + . Lemma 4.3. There exists M 0 > 0 such that for any M ≥ M 0 , A ǫ (C M ) ⊂ C M .
Proof. By construction A ǫ u is a non negative even function which satisfies R A ǫ u(x)dx = 1. It suffices then to prove that:

sup x∈R (1 + |x| p ) A ǫ u(x) ≤ M.
By (4.1) and according to Lemma 4.2 we obtain some lower bound for the normalization factor:

λ ǫ (u) = +∞ -∞ exp - 2 ǫ V (x) + x 0 (F ′ * u)(y)dy dx ≥ +∞ -∞ exp - 2 ǫ V (x) + C 2 M x 2 1 + x 2q dx.
According to both (V-3) and (V-7), we know that V (x) ≤ 0 for all x ∈ [-a; a]. Hence

λ ǫ (u) ≥ +a -a exp - 2 ǫ C 2 M x 2 (1 + a 2q ) dx. Let us define ξ(M ) = ǫ 1/2 (2C 2 M (1 + a 2q )) -1/2 then lim M→∞ ξ(M ) = 0.
By the change of variable x := ξ(M )y and Lemma A.1, the following development holds

+a -a exp - 2 ǫ C 2 M x 2 (1 + a 2q ) dx = 2ξ(M ) a/ξ(M) 0 e -x 2 dx = ξ(M ) √ π 2 - ξ(M ) a exp - a 2 ξ(M ) 2 + o ξ(M ) a exp - a 2 ξ(M ) 2 .
As soon as M is large enough, we have

λ ǫ (u) ≥ √ πξ(M ) 1 4 = πǫ 32C2(1+a 2q ) 1 √ M . Therefore 1 λǫ(u) ≤ C(ǫ) √ M
where C(ǫ) is a positive constant determined by parameters of the global system and ǫ. By (4.1) and the preceding upper bound, we prove that

(1 + |x| p )A ǫ u(x) ≤ C(ǫ) √ M (1 + |x| p )e -2 ǫ V (x) ≤ C ′ (ǫ) √ M ,
where C ′ (ǫ) is a positive constant similar to C(ǫ). In order to conclude, it is sufficient to choose M ≥ C ′ (ǫ) 2 : we get immediately A ǫ u ∈ C M .

Lemma 4.4. A ǫ is a continuous operator on C M with respect to the uniform norm.

Proof. We shall find some upper bound for the following expression |A ǫ u -A ǫ v|.

Step 1. Let u, v ∈ C M . We define:

Λ ǫ (x) = e -2 ǫ V (x) exp - 2 ǫ x 0 (F ′ * u)(y)dy -exp - 2 ǫ x 0 (F ′ * v)(y)dy = e -2 ǫ V (x)-α ǫ x 2 exp - 2 ǫ x 0 (F ′ 0 * u)(y)dy -exp - 2 ǫ x 0 (F ′ 0 * v)(y)dy .
It is well known that |e -a -e -b | ≤ |a -b| for a, b ≥ 0. In order to apply this inequality we have to prove that

(F ′ * f ) (y)dy is lower bounded by α 2 x 2 . So x 0 (F ′ 0 * f ) (y)dy is non negative due to the relation: F ′ (y) = F ′ 0 (y) + αy. Hence |Λ ǫ (x)| ≤ 2 ǫ e -2 ǫ V (x)-α ǫ x 2 Λ ǫ 0 (x), (4.3)
with Λ ǫ 0 defined by

x 0 (F ′ 0 * u)(y)dy - x 0 (F ′ 0 * v)(y)dy = x 0 R F ′ 0 (y -z)(u(z) -v(z))dzdy .
Since u and v are elements of C M , they are even functions and the integral with respect to the variable z becomes

Λ ǫ 0 = x 0 ∞ 0 (F ′ 0 (z + y) -F ′ 0 (z -y)) (u(z) -v(z)) dzdy ≤ x 0 ∞ 0 |F ′ 0 (z + y) -F ′ 0 (z -y)| |u(z) -v(z)| dzdy. (4.4)
The assumption (F-4) gives informations about the increments of the interaction function: there exist two positive constants C q and C such that

|F ′ 0 (z + y) -F ′ 0 (z -y)| ≤ 2|y|C q 1 + |z + y| 2q-2 + |z -y| 2q-2 ≤ 2|y|C q 1 + 2 2q-1 |z| 2q-2 + 2 2q-1 |y| 2q-2 ≤ C|y| 1 + |y| 2q-1 + |z| 2q-1 ≤ C|y| 1 + |y| 2q-1 1 + |z| 2q-1 . (4.5)
We shall now find some upper bound for |u(z

) -v(z)| in (4.4). Since u, v ∈ C M then u(z)(1 + |z| p ) ≤ M and v(z)(1 + |z| p ) ≤ M , ∀z ∈ R. The obvious upper bound |u(z)-v(z)|(1+|z| p ) ≤ 2M permits to obtain |u(z) -v(z)| ≤ 2M 1+|z| p . Consequently, for all z of R, |u(z) -v(z)| ≤ ||u -v|| ∞ 2M
1+|z| p where • ∞ denotes the uniform norm. Using this inequality, (4.5) and (4.4) in order to estimate Λ ǫ 0 , we get

|Λ ǫ 0 (x)| ≤ C ||u -v|| ∞ x 0 |y| 1 + |y| 2q-1 dy ∞ 0 2M 1 + |z| p 1 + z 2q-1 dz.
Since p > 4q the integral with respect to the variable z is finite and can be considered like a constant term. By (4.3) and using the positivity of αx 2 , we obtain directly the existence of some positive constant C > 0 such that

|Λ ǫ (x)| ≤ C M ǫ ||u -v|| ∞ x 2 1 + |x| 2q-1 e -2 ǫ V (x) .
According to (V-2), the expression x 2 1 + |x| 2q-1 e -2 ǫ V (x) can be bounded by some constant independent of ǫ. Therefore

||Λ ǫ || ∞ ≤ C(M, ǫ) ||u -v|| ∞ .
Two results can be deduced: firstly ||Λ ǫ || ∞ is finite and secondly ||Λ ǫ || ∞ becomes small as ||u -v|| ∞ decreases towards 0.

Step 2. For any x ∈ R, we introduce:

Ω ǫ (x) = 1 λ ǫ (u)λ ǫ (v) exp - 2 ǫ x 0 (F ′ * v)(y)dy + V (x) . (4.6)
Then the difference A ǫ u(x) -A ǫ v(x) can be decomposed as follows:

A ǫ u(x) -A ǫ v(x) = 1 λ ǫ (u) Λ ǫ (x) + (λ ǫ (v) -λ ǫ (u))Ω ǫ (x). (4.7)
Taking the uniform norm, we get

||A ǫ u -A ǫ v|| ∞ ≤ 1 λ ǫ (u) ||Λ ǫ || ∞ + |λ ǫ (v) -λ ǫ (u)| ||Ω ǫ || ∞ . (4.8)
We have shown in the proof of Lemma 4.3 that 1 λǫ(u) ≤ C(ǫ) √ M and moreover

||Λ ǫ || ∞ ≤ C(M, ǫ) ||u -v|| ∞ . We deduce that 1 λ ǫ (u) ||Λ ǫ || ∞ ≤ C ′ (M, ǫ) ||u -v|| ∞ .
It is then sufficient to find a similar inequality for the term

|λ ǫ (v) -λ ǫ (u)| ||Ω ǫ || ∞ in order to conclude the proof. |λ ǫ (v) -λ ǫ (u)| = R Λ ǫ (x)dx ≤ C M ǫ ||u -v|| ∞ +∞ -∞ x 2 1 + |x| 2q-1 e -2 ǫ V (x) dx.
According to (V-4), the integral with respect to the variable x is finite and does not depend on M . We have immediately

|λ ǫ (v) -λ ǫ (u)| ≤ C(M, ǫ) ||u -v|| ∞ .
It remains to estimate Ω ǫ (x). By (V-4) and (4.2), we have

x 0 (F ′ * v) (y)dy + V (x) ≥ C 4 x 4 + α 2 -C 2 x 2
for all x positive. Furthermore the symmetry property of V and F permits to extend the bound to all x ∈ R. The function exp -2 ǫ x 0 (F ′ * v)(y)dy + V (x) is then bounded by a constant depending on ǫ. Moreover we have already proved that 1 λǫ(f ) ≤ C(ǫ) √ M for all elements f of the function space C M . This bound can therefore be applied to u and v. Finally we obtain the existence of some constant C(ǫ) > 0 such that, for all real value x, |Ω ǫ (x)| ≤ C(ǫ)M . By (4.7), we have

||A ǫ u -A ǫ v|| ∞ ≤ C ′ (M, ǫ) ||u -v|| ∞ + C(M, ǫ) ||u -v|| ∞ C(ǫ)M.
In other words,

||A ǫ u -A ǫ v|| ∞ ≤ C ′′ (M, ǫ) ||u -v|| ∞
what finishes the proof. Here C, C ′ and C ′′ are positive constants.

We have now all the keys for proving the existence of some symmetric invariant measure. Indeed we have just presented some continuous mapping which stabilizes a convex subset of the Banach space B. Theorem 4.5. There exists a symmetric invariant measure for (1.1).

Proof. Let M 0 defined by Lemma 4.3. Taking M ≥ M 0 , let us prove that A ǫ (C M ) is a compact set. For this reason we shall estimate the following derivative:

(A ǫ u) ′ (x) = - 2 ǫ (F ′ * u)(x) + V ′ (x) λ ǫ (u) exp - 2 ǫ x 0 (F ′ * u)(y)dy + V (x) .
Let us analyze the different elements of this derivative. We have already seen in the proof of Lemma 4.3 that for any u ∈ C M the normalization factor λ ǫ (u)

satisfies 1 λ ǫ (u) ≤ C(ǫ) √ M . (4.9)
By (4.2), we obtain the bound: 0

≤ x 0 (F ′ * u) (y)dy ≤ C 2 M x 2 1 + x 2q
. Furthermore by (V-4) and (V-7), we get some estimation of V and its derivative:

V (x) ≥ C 4 x 4 -C 2 x 2 and |V ′ (x)| ≤ C q 1 + |x| 2q
for all x ∈ R. (4.10)

It remains to find some upper bound for the convolution term: |(F ′ * u) (x)| with x ∈ R + . By (F-4) and since u is an even function,

|(F ′ * u) (x)| = R F ′ (x -z)u(z)dz ≤ ∞ 0 F ′ (x + z) + F ′ (x -z) u(z)dz ≤ C q ∞ 0 |x + z| 1 + |x + z| 2q-2 + |x -z| 1 + |x -z| 2q-2 u(z)dz.
Therefore:

|(F ′ * u) (x)| ≤ R + C q 2 2q-1 |x| 2q-1 + |z||x| 2q-2 +|x| 1 + |z| 2q-2 + |z| 1 + |z| 2q-2 u(z)dz.
By definition of C M , we have u(z) ≤ M 1+|z| p for p > 4q. Hence the moments of order 1, 2q -2 and 2q -1 are bounded: there exist some constants C and C ′ , independent of the different parameters appearing in the system, such that

|(F ′ * u) (x)| ≤ C 1 + |x| + |x| 2q-2 + |x| 2q-1 ≤ C ′ 1 + |x| 2q+1 . (4.11)
To sum up: using (4.9), (4.10) and (4.11) we obtain

(A ǫ u) ′ (x) ≤ 2 ǫ C(ǫ) √ M (1 + |x| 2q+1 ) exp - 2 ǫ C 4 x 4 -C 2 x 2 .
Finally we deduce that there exists some constant C ǫ such that (A ǫ u)

′ (x) ≤ C ǫ for all x ∈ R.
Let us prove now that A ǫ C M is compact. To this end, we take some sequence of functions (u n ) n∈N in C M and focus our attention to the sequence

(A ǫ u n ) n∈N .
According to the definition of A ǫ , for all x real the set {A ǫ u n (x), n ∈ N} is compact. Furthermore the bound of (A ǫ u) ′ (x) is independent of the variables x and u ∈ C M : the equicontinuity condition for the application of Ascoli's theorem is satisfied. Hence, we deduce that there exists some subsequence of A ǫ u n which converges to a limit function v belonging to A ǫ C M . By Lemma 4.3 and Lemma 4.4 we can apply Schauder's theorem (Proposition 4.1) for the operator A ǫ on the function space C M with M ≥ M 0 . We deduce the existence of some fixed point which is, by construction, a symmetric stationary measure for the diffusion (1.1).

Example:

F (x) = β 4 x 4 + α 2 x 2
We have just shown the existence of a symmetric invariant measure for general self-stabilizing diffusions using fixed point arguments. Now let us study some particular case by a completely different way: the procedure shall be close to that developed in section 3.2. Let V be a potential satisfying (V-1)-(V-7). Let u ǫ be a symmetric invariant measure (Theorem 4.5). We denote by m 2 (ǫ) its second moment. The couple (m 2 (ǫ), u ǫ ) is solution to some system like (3.1)-(3.2). Indeed

F * u ǫ (x) = R F (x -z)u ǫ (z)dz = α 2 x 2 + β 4 x 4 + 3βm 2 (ǫ) 2 x 2 + α 2 m 2 (ǫ) + β 4 R z 4 u ǫ (z)dz , with β ≥ 0 since F ′ is a convex function on R + .
The expression delimited by the brackets is just a constant so we obtain the following system of equations for m 2 (ǫ) and u ǫ :

m 2 (ǫ) = R x 2 ν(m 2 (ǫ), x)dx and u ǫ (x) = ν(m 2 (ǫ), x) where ν(m, x) = exp -2 ǫ V (x) + F (x) + 3βm 2 x 2 ∞ 0 exp -2 ǫ V (z) + F (z) + 3βm 2 z 2 dz .
Therefore we introduce the function χ ǫ (m) = ∞ 0 x 2 ν(m, x)dx-m. By Theorem 4.5, we know that χ ǫ admits at least one zero on R + . Computing the derivative of χ ǫ , we prove that the considered function is decreasing:

χ ′ ǫ (m) = - 3β 2 ∞ 0 x 4 ν(m, x)dx - ∞ 0 x 2 ν(m, x)dx 2 -1 < 0.
The conclusion is immediate: there is a unique symmetric invariant measure. Obviously this result and the kind of method used to prove it are particular to our simple example. If the degree of the interaction function is strictly larger than 4 then it isn't enough to know the second moment in order to define the invariant measure: we need more moments and the proof of the uniqueness becomes awkward.

Outlying invariant measures

This section is essentially motivated by the uniqueness question for invariant measures. The existence of some symmetric measure was just proved in Section 4.1. It suffices now to point out asymmetric stationary measures for selfstabilizing diffusions. In the general setting, the interaction function is polynomial: set

F (x) = n k=1 F (2k) (0) (2k)
! x 2k . Let u be the density of some probability measure with respect to the Lebesgue measure and µ 1 , • • • , µ 2n-1 denote its moments of orders 1 to 2n-1 respectively. We assume they are finite. Then the difference D(x)

:= F * u(x) -F * u(0) satisfies D(x) = F (x -a) -F (-a) + 2n-1 p=1 (-1) p p! (µ p -a p ) n j≥ 1+p 2 F (2j) (0) (2j -p)! x 2j-p = F (x -a) -F (a) + 2n-1 p=1 (-1) p p! (µ p -a p ) F (p) (x) -F (p) (0) . Hence D(x) = Z m (x) -Z m (0)
where

Z m (x) = F (x -a) + 2n-1 p=1 (-1) p p! (m p -a p )F (p) (x). (4.12)
Since the convolution product can be expressed as a polynomial function which coefficients just depend on the moments of u, then the exponential expression of invariant measure (2.3) can be specified. Indeed equation (2.3) can be transformed into some system of equations whose unknown factors are the moments of the measure. In order to introduce this system, let us define, for all k ∈ [1; 2n -1], the function

ϕ (ǫ) k (m 1 , • • • , m 2n-1 ) = R x k exp -2 ǫ (V (x) + Z m (x) -Z m (0)) dx R exp -2 ǫ (V (x) + Z m (x) -Z m (0)) dx = R x k exp -2 ǫ W m (x) dx R exp -2 ǫ W m (x) dx (4.13)
with the potential W m (x) = V (x) + Z m (x). We construct the mapping:

Φ (ǫ) = (ϕ (ǫ) 1 , . . . , ϕ (ǫ) k , . . . , ϕ (ǫ) 2n-1 ). (4.14)
The measure associated to the density function u is invariant if and only if its moments vector

(µ 1 , • • • , µ 2n-1 ) is a fixed point of the map Φ (ǫ) .
We are going to show the existence of an asymmetric invariant measure defined by 2n -1 parameters close to a, • • • , a 2n-1 respectively, in other words the outlying measure is close to the Dirac mass in the point a. More precisely, we shall prove that there exists a parallelepiped stable by Φ (ǫ) , which converges to the point (a, a 2 , • • • , a 2n-1 ) as ǫ tends to 0. As in the linear case, we shall proceed by applying the mean value theorem in order to obtain asymptotic developments in the small noise limit.

Theorem 4.6. Let (η ǫ ) ǫ some sequence satisfying lim ǫ→0 η ǫ = 0 and lim ǫ→0 ǫ/η ǫ = 0.

Under the condition

2n-2 p=0 F (p+2) (a) p! a p < α + V ′′ (a), (4.15)
for any ρ > 0, there are at least two outlying measures u + ǫ and u - ǫ satisfying, for ǫ small enough

R x k u ± ǫ (x)dx -(±a) k ≤ ρ η ǫ . (4.16)
Proof. Let λ > 0. Let us define the parallelepiped

C(ǫ) = 2n-1 p=1 [a p -pa p-1 λη ǫ , a p + pa p-1 λη ǫ ].
Let m be an element of C(ǫ) then there exist some coordinates (r p ) 1≤p≤2n-1 which determine m through the equations m p = a p +r p η ǫ . By (4.12) and (4.13), we get

ϕ (ǫ) k (m) = R x k e -2 ǫ (V (x)+F (x-a)) exp -2ηǫ ǫ 2n-1 p=1
(-1) p rp p! F (p) (x) dx R e -2 ǫ (V (x)+F (x-a)) exp -2ηǫ ǫ 2n-1 p=1
(-1) p rp p! F (p) (x) dx .
We apply Lemma A.7 and Remark A.8 to the functions U (x) = V (x)+F (x-a), f (x) = x k , µ p = r p and G p (x) = (-1) p p! F (p) (x). We obtain:

ϕ (ǫ) k (m) = a k -η ǫ ka k-1 α + V ′′ (a) 2n-1 p=1 (-1) p r p p! F (p+1) (a) + o(η ǫ ),
uniformly with respect to the coordinates (r p ) p . By definition of the parallelepiped C(ǫ) the coordinates satisfy |r p | ≤ pa p-1 λ. Therefore, under condition (4.15),

ϕ (ǫ) k (m) -a k ≤ η ǫ λ ka k-1 α + V ′′ (a) 2n-1 p=1 F (p+1) (a) p! pa p-1 + o(η ǫ ) < η ǫ ka k-1 λ + o(η ǫ ).
Since this estimate is uniform with respect to the coordinates, as soon as ǫ is small enough, we have |ϕ

(ǫ) k (m) -a k | < ka k-1 λη ǫ , that means that Φ (ǫ) (m) ∈ C(ǫ).
Let us note that C(ǫ) is a convex, closed and bounded subset of R 2n-1 . Since the space dimension is finite, the continuity of Φ (ǫ) implies that the closure of the parallelepiped's image is a compact set. We can apply Schauder's Theorem (Proposition 4.1) and obtain that there exists some fixed point in the compact. In other words there exists m ∈ C(ǫ) such that the measure associated to the density

u ǫ,m (x) = exp -2 ǫ W m (x) R exp -2 ǫ W m (z) dz (4.17)
is invariant. In a similar way, the measure defined by m -is also invariant; here m -(k) = (-1) k m k . To conclude: we have at least two outlying measures, one around a and the second one around -a.

We can not prove at this stage the uniqueness of the couple of outlying invariant measures (this question shall be explored in a subsequent work). We can effectively imagine that other outlying measures could exist around a, around -a or even around other areas. Nevertheless we can develop a sharper description of one particular outlying measure: the measure close to δ a where δ represents the Dirac measure. To do this it suffices to estimate its different moments, that requires the following preliminary result. Lemma 4.7. There exists a unique solution (τ 0 1 , • • • , τ 0 2n-1 ) to the following Cramer's system

2n-1 p=1 (-1) p p! F (p+1) (a)τ p + α + V ′′ (a) ka k-1 τ k = V (3) (a) 4(α + V ′′ (a)) - k -1 4a , (4.18
)

for 1 ≤ k ≤ 2n -1.
This solution is given by

τ 0 k = ka k-1 aV (3) (a) -(k -1)V ′′ (a) 4aV ′′ (a) (α + V ′′ (a)) , 1 ≤ k ≤ 2n -1. (4.19)
Proof. Let us denote by I 2n-1 the unit matrix of dimension 2n -1 and for A ∈ R 2n-1 , A T represents the transpose of the vector A. Moreover we adopt the following notation (x k ) 1≤k≤2n-1 = (x 1 , . . . , x 2n-1 ). The system (4.18) can be written in this way: we define T = (τ k ) T 1≤k≤2n-1 then

(α + V ′′ (a))I 2n-1 + C 1 C T 2 T = ka k-1 V (3) (a) 4α + 4V ′′ (a)) - k -1 4a T 1≤k≤2n-1 with the vectors C T 1 = (ka k-1 ) 1≤k≤2n-1 and C T 2 = (-1) k k! F (k+1) (a) 1≤k≤2n-1 .
We define therefore

A = (α + V ′′ (a))I 2n-1 + C 1 C T 2 . (4.20) Let us note that C 1 C T 2 C 1 C T 2 = (C T 2 C 1 )C 1 C T 2 and C T 2 C 1 = 2n-1 p=1 (-1) p p! F (p+1) (a)pa p-1 = - 2n-2 p=0
(-1) p p! F (p+2) (a)a p = -F ′′ (0).

Since F ′′ (0) = α, we obtain

A 2 = (α + V ′′ (a)) 2 I 2n-1 + 2(α + V ′′ (a)) + C T 2 C 1 C 1 C T 2 = (α + V ′′ (a)) 2 I 2n-1 + 2(α + V ′′ (a)) -F ′′ (0) C 1 C T 2 = (α + V ′′ (a)) 2 I 2n-1 + α + 2V ′′ (a) C 1 C T 2 = (α + 2V ′′ (a))A -V ′′ (a) (α + V ′′ (a)) I 2n-1 ,
We deduce that A is invertible, that is (4.18) is a Cramer's system, and using (4.20) we get explicitly the inverse:

A -1 = 1 V ′′ (a)(α + V ′′ (a)) (α + 2V ′′ (a))I 2n-1 -A = 1 V ′′ (a)(α + V ′′ (a)) V ′′ (a)I 2n-1 -C 1 C T 2 .
Therefore the Cramer's system (4.18) admits a unique solution given by

τ 0 k = 1 V ′′ (a)(α + V ′′ (a)) V ′′ (a)ka k-1 aV (3) (a) -(k -1)(α + V ′′ (a)) 4a (α + V ′′ (a)) -ka k-1 2n-1 p=1 (-1) p p! F (p+1) (a)pa p-1 aV (3) (a) -(p -1)(α + V ′′ (a)) 4a (α + V ′′ (a)) = ka k-1 4aV ′′ (a)(α + V ′′ (a)) 2 aV (3) (a) V ′′ (a) - 2n-1 p=1 (-1) p a p-1 (p -1)! F (p+1) (a) -(α + V ′′ (a)) (k -1)V ′′ (a) - 2n-1 p=2 (-1) p (p -2)! F (p+1) (a)a p-1 = ka k-1 aV (3) (a) -(k -1)V ′′ (a) 4aV ′′ (a)(α + V ′′ (a)
) .

Indeed, we use

2n-1 p=1 (-1) p (p -1)! F (p+1) (a)a p-1 = -F ′′ (0) = -α, and
2n-1 p=2 (-1) p (p -2)! F (p+1) (a)a p-1 = aF (3) (0) = 0.
Theorem 4.6 points out the existence of two outlying measures, one concentrated around a and an other around -a. According to Lemma 4.7 we get some sharper upper bound for the distance between δ a and some asymmetric invariant measure. Theorem 4.8. Under the condition (4.15), for any δ > 0, there exists ǫ 0 such that Φ (ǫ) admits two fixed points m ± with

m ± k (ǫ) -(±1) k a k -(±1) k τ 0 k ǫ ≤ δ τ 0 k ǫ, 1 ≤ k ≤ 2n -1, ǫ ≤ ǫ 0 . (4.21)
Proof. It is similar to the proof of Theorem 4.6. Let δ > 0 and C(ǫ

) = 2n-1 p=1 [a p -(τ 0 p + pa p-1 δ)ǫ, a p -(τ 0 p -pa p-1 δ)ǫ].
We choose an element m in the parallelepiped C(ǫ). For all 1 ≤ p ≤ 2n -1, there exists a coordinate δ p ∈ [-δ; δ] such that m p = a p -(τ 0 p + pa p-1 δ p )ǫ. By (4.12) and (4.13), we obtain

ϕ (ǫ) k (m) = R x k exp 2 2n-1 p=1 (-1) p p! (τ 0 p + pa p-1 δ p )F (p) (x) e -2 ǫ (V (x)+F (x-a)) dx R exp 2 2n-1 p=1 (-1) p p! (τ 0 p + pa p-1 δ p )F (p) (x) e -2 ǫ (V (x)+F (x-a)) dx
We apply Lemma A.5 and Remark A.8 with the following functions:

U (x) = V (x) + F (x -a), µ p = τ 0 p + pa p-1 δ p , G = 0 and f p (x) = 2 (-1) p p! F (p) (x). Hence ϕ (ǫ) k (m) = a k - ka k-2 4(α + V ′′ (a)) 2 aV (3) (a) -(α + V ′′ (a)) (k -1) + 4a 2n-1 p=1 (-1) p p! (τ 0 p + pa p-1 δ p )F (p+1) (a) ǫ + o(ǫ) = a k - 1 α + V ′′ (a) ka k-1 V (3) (a) 4(α + V ′′ (a)) -ka k-1 2n-1 p=1 (-1) p τ 0 p p! F (p+1) (a) - k(k -1)a k-2 4 -ka k-1 2n-1 p=1 (-1) p δ p a p-1 (p -1)! F (p+1) (a) ǫ + o(ǫ).
This estimate is uniform with respect to the variables (δ p ) p . We denote by d ǫ k the difference |ϕ

k (m)-a k +τ 0 k ǫ|. We compute this expression:

d ǫ k ≤ ka k-1 V (3) (a) 4(α + V ′′ (a)) 2 - ka k-1 α + V ′′ (a) 2n-1 p=1 (-1) p p! τ 0 p F (p+1) (a) - k(k -1)a k-2 4(α + V ′′ (a)) -τ 0 k - ka k-1 α + V ′′ (a) 2n-1 p=1 (-1) p (p -1)! δ p F (p+1) (a)a p-1 ǫ + o(ǫ).
According to the Lemma 4.7 and using the condition (4.15), we obtain, for ǫ small enough,

ϕ (ǫ) k (m) -a k + τ 0 k ǫ ≤ ka k-1 α + V ′′ (a) 2n-1 p=1 a p-1 (p -1)! |δ p ||F (p+1) (a)|ǫ + o(ǫ) ≤ δ ka k-1 α + V ′′ (a) 2n-2 p=0 1 p! |F (p+2) (a)|a p ǫ + o(ǫ) < ka k-1 δǫ.
In other words, Φ (ǫ) (m) ∈ C(ǫ) in the small noise limit. The application of Schauder's Theorem (Proposition 4.1) permits to prove the existence of some fixed point in the compact. Therefore there exists m ∈ C(ǫ) such that the associated measure u ǫ,m (x) defined by (4.17) is invariant. In the same way, the measure defined by m -is invariant with m -(k) = (-1) k m k . Finally the continuous map Φ (ǫ) admits two fixed points m ± (ǫ) satisfying (4.21).

Remark 4.9. 1. In the particular case: F (p) (a) ≥ 0 for all p ∈ N, the condition for the existence of outlying measures becomes V ′′ (a) > F ′′ 0 (2a) where F 0 is defined by F (x) = α 2 x 2 + F 0 (x). 2. In the linear interaction case: F (x) = α 2 x 2 , (4.15) is equivalent to the simple condition V ′′ (a) > 0 which is in fact always satisfied according to (V-3). In other words we obtain the existence result presented in the linear interaction case.

A Annexe

We shall present here some useful asymptotic results which are close to the classical Laplace's method.

Lemma A.1. Let M > 0. Let us assume that U is C 2 ([M, ∞[)-continuous, U (x) = 0 and U ′′ (x) > 0 for all x ∈ [M, ∞[ and lim x→∞ U ′′ (x) (U ′ (x)) 2 = 0. If x → e -U(x) is integrable on R then for any m ∈ R: +∞ x e -U(t) dt ≈ e -U(x) U ′ (x) and x m e U(t) dt ≈ e U(x) U ′ (x) as x → ∞. (A.1)
Proof. Since x → e -U(x) is integrable and since these properties are satisfied: U (x) = 0 and U ′′ (x) > 0 for x ≥ M , we know that lim x→∞ U (x) = +∞. Furthermore there exists some M 0 > M such that U ′ (x) > 0 for x ≥ M 0 . Hence for t ≥ M 0 we obtain t) .

e -U(t) = - e -U(t) U ′ (t) ′ - U ′′ (t) (U ′ (t)) 2 e -U(
Therefore

I := ∞ x e -U(t) dt = e -U(x) U ′ (x) - ∞ x U ′′ (t) (U ′ (t)) 2 e -U(t) dt, x ≥ M 0 .
Using the assumptions of the statement we have

∞ x U ′′ (t) (U ′ (t)) 2 e -U(t) dt ≥ 0. Hence I ≤ e -U(x) U ′ (x) -1 . Moreover lim x-→±∞ U ′′ (x) (U ′ (x)) 2 = 0.
As a consequence for any δ > 0, there exists M 1 (δ) > M 0 such that (1 + δ)I ≥ e -U (x) U ′ (x) . The estimation of I can be deduced easily. The second equivalence can be obtained by similar arguments.

Lemma A.2. Set ǫ > 0. Let U and G two C ∞ (R)-continuous functions. We define U µ = U + µG for µ belonging to some compact interval I of R. Let us introduce some interval [a, b] satisfying: U ′ µ (a) = 0, U ′ µ (b) = 0 and U µ (x) admits some unique global minimum on the interval [a, b] reached at x µ ∈]a, b[ for all µ ∈ I. We assume that there exists some exponent k 0 independent of µ ∈ I such that 2k 0 = min r∈N * U (r) µ (x µ ) = 0 . Then taking the limit ǫ → 0 we get

I 0 := b a e -Uµ (t) ǫ dt = 1 k 0 ǫ(2k 0 )! U 2k0 µ (x µ ) 1 2k 0 Γ 1 2k 0 e -Uµ (xµ) ǫ (1 + o I (1)), (A.2)
where Γ represents the Euler function and o I (1) converges towards 0 uniformly with respect to µ ∈ I.

Proof. We define η µ = U (2k 0 ) µ (xµ) (2k0)! . Let us note that η µ depends continuously on µ. Since U µ is regular and admits some unique global minimum for x = x µ , there exists τ 0 > 0 independent of the parameter µ such that τ 0 < min {x µ -a; b -x µ } for all µ ∈ I and such that the minimum on the interval [a;

x µ -τ ] [x µ + τ ; b] denoted by U µ (τ ) is reached on the boundary {x µ -τ ; x µ + τ } for all τ < τ 0 . Consequently xµ-τ a exp - U µ (t) ǫ dt + b xµ+τ exp - U µ (t) ǫ dt ≤ (b -a) exp - U µ (τ ) ǫ . Defining I τ = xµ+τ xµ-τ exp - Uµ(t) ǫ
dt, we obtain the following bound:

|I 0 -I τ | ≤ (b -a) exp - U µ (τ ) ǫ . (A.3)
Let us first estimate I τ . By the mean value theorem, there exists some constant C > 0 independent of µ ∈ I such that, in a neighborhood of x µ , the following bound is satisfied:

U µ (t) -U µ (x µ ) -η µ (t -x µ ) 2k0 ≤ C|t -x µ | 2k0+1 . Hence J 1 exp - Cτ 2k0+1 ǫ ≤ I τ 2 exp U µ (x µ ) ǫ ≤ J 1 exp Cτ 2k0+1 ǫ , (A.4)
where In order to conclude we choose a particular value for the variable τ namely τ = exp log(ǫ)

J τ = τ 0 exp 1 ǫ η µ t 2k0 dt = ǫ η µ 1 2k 0 1 2k 0 τ 2k 0 ηµ ǫ 0 t 1 2k 0 -1 e -t dt,
2k0+ 1 2 . Then we get: for C ∈ R, l > 0, lim ǫ→0 e C τ 2k 0 +1 ǫ = 1, lim ǫ→0 e -ηµ τ 2k 0 ǫ τ 1-2k0 ǫ 1 2k 0 -1 = lim ǫ→0 ǫ -l e Uµ(xµ )-U µ (τ ) ǫ = 0.
These convergences are uniform with respect to the parameter µ. Applying these asymptotic results to (A.3), (A.4) and (A.5) permits to prove the statement of the lemma.

Lemma A.3. Let U and G be two C ∞ ([a, b])-functions. We define U µ = U +µG for µ belonging to some compact interval I of R. We assume that U µ admits a unique global minimum on the interval ]a; b[ reached at x = x µ , with U ′′ µ (x µ ) > 0. Let f m be a C 3 -continuous function for any parameter value m belonging to some compact set M. We assume that there exists some constant λ such that |f with some arbitrary ρ > 0 which should be specified in the following.

Step 1. We shall prove that the second integral is negligible as Step 2. Let us focus our attention to the integral on the domain [x µ -ρ; x µ +ρ]. The function f m can be developed in the neighborhood of x µ : 3) m (w m,µ (x))(x-x µ ) 3

f m (x) = f m (x µ )+f ′ m (x µ )(x-x µ )+ 1 2 f ′′ m (x µ )(x-x µ ) 2 + 1 6 f (
with the value w m,µ (x) between x µ and x. Taking into account these different terms, the integral I 1 can be split into 4 different integrals respectively Ĩ0 ,..., Ĩ3 .

For each integral we shall analyze the asymptotic behavior.

Step (A.12)

Step 3. To sum up: in Step 1, we proved that it suffices to estimate the integral I 1 which can be split into 4 terms. Each of them has been estimated in equations (A.9), (A.10), (A.11) and (A.12). The whole integral has the asymptotic equivalence (A.6) as soon as ρ 3 /ǫ → 0, ρ 18 /ǫ 7 → 0 and ρ 5 /ǫ 2 → 0.

The particular choice ρ = ǫ 9 20 fulfills all these conditions.

We can extend the statement of the preceding lemma to integrals with unbounded supports.

Lemma A.4. Let U and G be two C ∞ (R)-continuous functions. We define U µ = U + µG for the parameter µ belonging to some compact interval I of R We assume that U µ (t) ≥ t 2 for |t| larger than some R independent of µ and that U µ admits a unique global minimum at x µ with U ′′ µ (x µ ) > 0. Let f m be a C 3 -continuous function depending on some parameter m which belongs to a compact set M. Furthermore we assume that there exists some constant λ > 0 such that |f m (t)| ≤ exp [λ|U µ (t)|] for all t ≥ R, µ ∈ I, m ∈ M and |f

Figure 4 : 2 / 3 <Figure 5 :

 4235 Figure 2: χ α 0 (dotted line) and χ α ǫ

Rf( 1 +

 1 (x) dx = 1 and sup x∈R |x| p ) f (x) ≤ M.

x0

  (F ′ 0 * v)(y)dy and x 0 (F ′ 0 * u)(y)dy are non negative. By Lemma 4.2, for each function f ∈ C M the convolution term x 0

m

  (x)| ≤ λ for all m ∈ M, x ∈ [a, b] and 0 ≤ i ≤ 3. Then the following asymptotic result holds:

µ

  x µ ) + γ 0 (µ)ǫ + o IM (ǫ) (A.6) with γ 0 (µ) = f m (x µ ) (x µ) and o IM (ǫ)/ǫ converges to 0 as ǫ becomes small uniformly with respect to the parameters m and µ.Proof. First we split the integral into two parts:I = xµ+ρ xµ-ρ f m (t)e -2Uµ (t) ǫ dt + [xµ-ρ;xµ+ρ] c T [a;b] f m (t)e -2Uµ(t) ǫ dt = I 1 + I 2

ρ 2 ǫ

 2 → ∞ that means that I 2 = o IM {ǫ 3/2 e -2U (xµ ) ǫ }. We get I 2 ≤ (b -a) sup z∈[a,b] |f m (z)| exp -2 inf z∈[xµ-ρ;xµ+ρ] c U µ (z) ǫ (A.8)Since the global minimum of U µ is unique and due to the regularity of U µ with respect to the parameter µ, we deduce that the minimum of the function on the interval [x µ -ρ; x µ + ρ] c [a; b] is reached on the boundary provided that ρ is small enough. The developmentU µ (x µ ± ρ) = U µ (x µ ) + 1 2 U ′′ µ (x µ )ρ 2 + o I (ρ 2) implies, as already claimed that I 2 = o IM ǫ 3/2 e -2U (xµ) ǫ as ρ 2 /ǫ → ∞.

2 . 1 .

 21 Asymptotic behavior of Ĩ3 . By definition w m,µ (t) ∈ [x µ -ρ; x µ + ρ] when t ∈ [x µ -ρ; x µ + ρ]. Moreover, by assumption |f

( 3 ).z 4 1 5 ρ -ρ e -U 2 ǫ z 2 - U 3 3ǫ z 3 - U 4 12ǫ z 4 + 1 60ǫ |U ( 5 ) 2 ǫ

 345234152 m (w m,µ (t))| is upper bounded by some constant λ > 0 independent of m and µ. By Lemma A.2 applied to 2U µ , for ρ < 1 and ǫ small, we obtain the existence of some constant C > 0, independent of the parameters m and µ, such that| We denote by D = e 2Uµ (xµ) ǫ Ĩ0 -f m (x µ )I ′ 0 this distance.Then there exists some constant C > 0 independent of µ and m such that|D| ≤ |f m (x µ )| -e -1 60ǫ U (5) (yµ(z+xµ)z 5 dz ≤ |f m (x µ )|Cλ 60ǫ ρ (yµ(z+xµ))z 5 | dz.If both conditions ρ 2 /ǫ → ∞ and ρ 3 /ǫ → 0 are satisfied then the integral term in the preceding inequality is obviously equivalent to πǫ U ′′ µ (xµ) . The following equivalence holds for the initial integral Ĩ0 : under the assumption that ρ 5 + o IM (ǫ) .

  locally bounded uniformly with respect to the parameter m ∈ M for 0 ≤ i ≤ 3. Then the following asymptotic result holds as ǫ tends to 0:R f m (t)e -2Uµ (t) ǫ dt = e -2Uµ (xµ ) ǫ πǫ U 2 f m (x µ ) + γ 0 (µ)ǫ + o IM (ǫ) , (A.13)where γ 0 (µ) is defined by (A.7) and o IM (ǫ)/ǫ converges to 0 as ǫ → 0 uniformly with respect to the parameters m and µ.Proof. Let R > 0 such that U µ (t) ≥ t 2 for t ≥ R.The initial integral can be split into two integrals: the first one denoted by I 1 concerns the compact support [-R, R] and the other one I 2 concerns the complementary support. For I 1 it suffices to apply Lemma A.3 in order to get the asymptotic development. It remains then to prove that I 2 is negligible with respect to I 1 that is I 2 = Remark A.8. The statements of Lemmas A.2-A.7 can be easely generalized, replacing the parametrized function U µ = U + µG by U µ = U + k i=1 µ i G i where µ = (µ 1 , . . . , µ k ) ∈ I 1 × . . . × I k . The convergence results are then uniform with respect to all parameters.

Hence, if ρ 3 = o(ǫ) then the following asymptotic result holds

(A.9)

Step 2.2. Asymptotic behavior of Ĩ2 . Using the C 3 -regularity of U µ that is

µ (y µ (t))(t-x µ ) 3 with y µ (t) belonging to [x µ -ρ; x µ + ρ], we get

µ (yµ (t)) 3ǫ

(t-xµ) 3 dt.

Since y µ (t) belongs to some compact set, the third derivative U

µ (y µ (t)) is bounded by some constant independent of µ. Applying the following change of variable u = (t -x µ ) 2 U ′′ µ (x µ )/ǫ yields

,

(1 + o I (1)).

(A.10)

Step 2.3. Asymptotic behavior of Ĩ1 . Let us develop the function U µ in the neighborhood of

µ (y µ (t))t 4 where y µ (t) ∈ [x µ -ρ, x µ + ρ]. The regularity of U µ (x) with respect to both x and µ implies the existence of some constant C > 0 independent of µ which bounds the forth derivative of U µ on the integral support. Therefore we have

ǫ z 2 dz. Some estimation of the integral J ρ points out directly:

Symmetry arguments permit easily to deduce that m ρ (1) = m ρ (7) = 0. Finally it suffices to compute m ρ (4) and n ρ (1). To this end we introduce the change of variable u := U2 ǫ z 2 and let ρ 2 /ǫ tend to infinity:

To sum up: if

Step 2.4. Asymptotic behavior of Ĩ0 . Let us first study the following integral

We recall the usual notations

Adopting the same notations as in Step 2.3 and using symmetry properties, the following bound (uniform with respect to the parameter µ) yields

By the usual change of variable u :=

z 2 we emphasize some asymptotic estimation of I ′ 0 as ρ 2 /ǫ → ∞ and ρ 3 /ǫ → 0:

We apply the mean value theorem to the function U µ :

From this equality we deduce an estimation of the distance between the integrals Ĩ0 and

2 s the following bound holds:

Lemma A.1 permits to prove as claimed that I 2 can be neglected.

Lemma A.4 can be applied to particular functions f m .

Lemma A.5. Let U and G be two C ∞ (R)-continuous functions. We define U µ = U + µG with µ belonging to some compact interval I of R. We assume that U µ (t) ≥ t 2 for |t| larger than some R independent of µ and that U µ admits a unique global minimum at x µ with U ′′ µ (x µ ) > 0. Let f m be a C 3continuous function depending on some parameter m which belongs to a compact set M. Furthermore we assume that there exists some constant λ > 0 such that

m | is locally bounded uniformly with respect to m ∈ M for 0 ≤ i ≤ 3. Then, for any n ≥ 1 and asymptotically as ǫ → 0 we obtain the estimate

where

µ (x µ ) and o IM (ǫ)/ǫ converges to 0 as ǫ → 0 uniformly with respect to the parameters m and µ.

Proof. We just apply two times Lemma A.4: the first time to the denominator D ǫ that is for the function t → e fm(t) and the second time to the numerator N ǫ for the function t → t n e fm(t) . The following asymptotic result holds

where

The numerator normalized by x n µ i.e. N ǫ /x n µ satisfies some similar identity as D ǫ , namely (A.14) with γd replaced by γn :

The estimation of the ratio is then a classical exercise of asymptotic analysis.

The next lemmas are generalizations of Lemma A.4 and Lemma A.5 to functions G depending on the small parameter ǫ.

Lemma A.6. Let U and G be two C ∞ (R)-continuous functions such that U (t) ≥ t 2 for |t| large enough and |G(t)| ≤ λ|U (t)| + C for some constants λ > 0 and C > 0. Moreover we assume that U admits some unique global minimum reached at x 0 with U ′′ (x 0 ) > 0. For any sequence (η ǫ ) ǫ satisfying lim ǫ→0 η ǫ = 0 and lim ǫ→0 ǫ/η ǫ = 0 we define U ǫ,µ = U + η ǫ µG depending on the parameter µ which belongs to some compact interval I of R. Let f a C 3 -continuous function such that |f (t)| ≤ e λ|U(t)| for all |t| large enough and such that |f

m | is locally bounded uniformly with respect to m ∈ M for 0 ≤ i ≤ 3. Then, there exists ǫ 0 > 0 such that the potential U ǫ,µ admits a unique global minimum reached at x ǫ,µ for all ǫ ≤ ǫ 0 . Furthermore the following asymptotic results hold

where

and o I (η ǫ )/η ǫ tends to 0 as ǫ → 0 uniformly with respect to the parameter µ.

Proof. Let us first prove that the potential U ǫ,µ (x) admits a unique minimum for x = x ǫ,µ with lim ǫ→0 x ǫ,µ = x 0 . By the definitions of (η ǫ ) ǫ and U ǫ,µ , the following convergence holds

Since x 0 is the unique global minimum of U , for any small R > 0 there exists

We deduce the existence of two small constants ρ ′ R and ǫ 0 such that

. By (A.17) and (A.18) we obtain: for any R > 0 the global minimum of the parametrized potential U ǫ,µ is reached in the interval x ∈ [x 0 -R, x 0 + R] provided that ǫ is small enough (uniformly with respect to µ). Moreover this global mimimum is unique. Indeed U ′′ (x 0 ) > 0 and the regularity of U implies that U ′′ (x) > 0 for all x in some small neighborhood of x 0 . Since U ′′ ǫ,µ converges towards U ′′ as ǫ → 0 uniformly on each compact subset of R, we obtain that U ′′ ǫ,µ (x) > 0 for all x ∈ [x 0 -R, x 0 + R] provided that R and ǫ are small enough. The minimum is actually unique, we denote its localization x ǫ,µ and point out that, for ǫ small, U ′′ ǫ,µ (x ǫ,µ ) > 0 uniformly with respect to µ.

Let us determine x ǫ,µ . By applying the mean value theorem to U ǫ,µ , we get

, where x is in between x 0 and x ǫ,µ . Since the second derivative is continuous, U ′′ ǫ,µ (x) is uniformly bounded. Moreover U ′ (x 0 ) = 0. Consequently x ǫ,µ -x 0 = O I (η ǫ ). Using the same argument for the second order asymptotic development of U ′ ǫ,µ (x ǫ,µ ), that is

we obtain the announced estimate (A.15). Finally let us prove the estimate (A.16). The statement of Lemma A.4 can be applied to U ǫ,µ since the asymptotic result (A.13) is uniform with respect to the parameter µ. So it suffices to consider the case when µ is replaced by µη ǫ . We immediately obtain

It remains to approximate f (x ǫ,µ ) and U ′′ ǫ,µ (x ǫ,µ ) using (A.15). Due to the regularity of both f and U , the following developments hold

The statement of Lemma A.6 is obtained just by combination of the two preceding asymptotics and (A.19).

We are now able to present a statement similar to Lemma A.5 for some potential U µ depending on the small parameter ǫ. It suffices to consider a ratio of two integral terms. Then an immediate application of Lemma A.6 leads to the following result.

Lemma A.7. Let U and G be two C ∞ (R)-continuous functions such that U (t) ≥ t 2 for |t| large enough and |G(t)| ≤ λ|U (t)| + C for some constants λ > 0 and C > 0. Moreover we assume that U admits some unique global minimum reached at x 0 with U ′′ (x 0 ) > 0. For any sequence (η ǫ ) ǫ satisfying lim ǫ→0 η ǫ = 0 and lim ǫ→0 ǫ/η ǫ = 0 we define U ǫ,µ = U + η ǫ µG depending on the parameter µ which belongs to some compact interval I of R. Let f a C 3 -continuous function such that |f (t)| ≤ e λ|U(t)| for all |t| large enough and such that |f where o I (η ǫ )/η ǫ tends to 0 as ǫ → 0 uniformly with respect to the parameter µ.