Random hysteresis loops

www.imstat.org/aihp 2013, Vol. 49, No. 2, 307–339

DOI:10.1214/11-AIHP461

© Association des Publications de l’Institut Henri Poincaré, 2013

Random hysteresis loops

Gioia Carinci

Dipartimento di Matematica Pura e Applicata, Università dell’Aquila, via Vetoio 1, 67100 L’Aquila, Italy. E-mail:[email protected] Received 3 September 2010; revised 27 October 2011; accepted 4 November 2011

Abstract. Dynamical hysteresis is a phenomenon which arises in ferromagnetic systems below the critical temperature as a re- sponse to adiabatic variations of the external magnetic field. We study the problem in the context of the mean-field Ising model with Glauber dynamics, proving that for frequencies of the magnetic field oscillations of orderN^−2/3,Nthe size of the system, the “critical” hysteresis loop becomes random.

Résumé. L’hystérésis dynamique est un phénomène qu’on observe dans les systèmes ferromagnétiques au-dessous de la tempera- ture critique, en réponse à des variations adiabatiques du champ magnétique extérieur. Nous étudions le problème dans le contexte du modéle d’Ising de champ moyen avec la dynamique de Glauber, en montrant que, pour des fréquences d’oscillations du champ magnétique d’ordre deN^−2/3, avecNla taille du système, la boucle d’hystérésis«critique»devient aléatoire.

MSC:82C20

Keywords:Hysteresis; Ising; Mean field Glauber dynamics; Macroscopic fluctuations

1. Introduction

Hysteresis appears when a time dependent magnetic fieldh=h(t)is applied to a ferromagnet whose temperature is kept fixed below the critical value. The origin of the phenomenon lies in the fact that, at the equilibrium, at each value of the external magnetic fieldhmay not correspond a unique value of the magnetizationmof the system. The value ofm(t)is, thus, not determined byh(t)alone but also by the previous history of the input.

The phenomenon has been widely studied and modelled. Most classical theories (see for example [5,6,22]) consider hysteresis from a static point of view, by modelling it through integral operators not depending on the velocity of variation of the external input.

A dynamical approach to the study of the phenomenon has been proposed for the first time by Rao et al. [20] in the early nineties. The new theory aroused great interest and a number of experimental, numerical and theoretical works appeared on the argument in the last twenty years, investigating the response of the system to adiabatic oscillations of the magnetic field. They analyse, in particular, the dependence of shapes and areas of the hysteresis loops on amplitude and frequency of the input oscillations. Most of these results are essentially numerical. Monte Carlo simulations have widely been used to study the hysteretic response of a nearest-neighbor ferromagnetic Ising model (see for instance [1,10,11,15–17,20,23]). On the other hand, several theoretical and numerical results are concerned with those known as mean-field models (see [1,9,20,21]). In these models the dynamics is reduced to a single differential equation of the order parameter (the uniform magnetizationm(t)). These equations govern the dynamics of the magnetization in stochastic spin models in the limit of infinite system volume. Therefore they neglect both thermal fluctuations and finite system size effects. A first rigorous analysis of the effects of the stochastic fluctuations on the properties of the hysteresis cycles has been carried out by B. Genz and N. Berglund in a series of papers of about ten years ago

Fig. 1. The picture shows the dependence of the two equilibrium branchesX_±(h)(black lines) on the external magnetic fieldh. Adiabatic oscilla- tions of the magnetic field of amplitudeA > hcyield the typical hysteresis loop (blue line).

[2–4]. They model the thermal fluctuations by adding a stochastic noise to a mean-field type equation. They consider a Langevin equation with a Ginzburg–Landau potential:

dx=

F (x)+h

dt+N^−1/2dw(t), F (x)=x−x³, (1.1)

wherew(t)is the standard Brownian motion. We give toN >0 the physical interpretation of the total number of spin sites in a ferromagnetic system. Then, in the largeN regime, equation (1.1) can be thought of as a continuous counterpart of our Ising spin dynamics (see Section2).

In the present paper we shall study the problem for the Glauber process in the Curie–Weiss model, from which (1.1) is inspired.

Lethc>0 be the “coercive magnetic field” value, then for|h| ≤hc the magnetization density of the ferromagnet may have two equilibrium values,X₊(h)andX₋(h)(see Fig.1). The upper branchX₊(h)continues pasthcwhile it disappears forh <−h_c; the opposite holds for the lower branchX₋(h). Let us apply, now, a slowly oscillating magnetic fieldh(t). We denote, respectively, by Aandωthe amplitude and the frequency of the oscillations (we choose, for instance, h(t)= −Acos(ωt)). Let m(t) be the magnetization observed at time t and choose initially m(0)=X₊(h(0)). In the adiabatic (quasi-static) regime, whereωis very small, the following is observed. IfA≤h_c thenm(t)≈X₊(h(t))for anyt≥0. IfA > h_c,m(t)traces out the so called hysteresis loop, in the sense thatm(t)∈ {X₊(h(t)), X₋(h(t))}(approximately), jumping from the upper to the lower branch whenh(t)crosses−h_cand the opposite whenh(t)crossesh_c. A sharp statement (which avoids the above approximated statements) can be obtained in “the adiabatic limit” whereω→0.

The pediod of the magnetic field oscillations is of orderω⁻¹, thus, in the adiabatic regime the natural time-scale of the dynamics is very long. In long time intervals other phenomena may appear which in short time intervals are negligible and which may invalidate the picture. In the context of (1.1)X_±(h)are identified with the locally stable solutions of the stationary equationF (x)= −h. Ifhis constant, sayh∈(0, hc), thenX₋(h)is metastable and, on a time interval which diverges exponentially withN (asN → ∞), there is tunneling fromX₋(h)toX₊(h). Thus, ifωis exponentially small withN, the oscillations period is exponentially long withN, and then stochastic jumps between the two branches occur, essentially perturbing the hysteresis loop. We intend to consider a different regime for the frequencyω, i.e. we takeω=N^−κ,κ >0. We shall concentrate here on the critical amplitude caseA=h_c. In such a case the deterministic equation (i.e. (1.1) without the Brownian term) predicts that the magnetizationm(t) tracks always the upper branchX₊(h(t)), where it was initially. [3] proves that, with the addition of the stochastic effects, there exists a critical value forκ,κ=²₃. Ifκ <²₃the dynamics is still governed by the deterministic equation,

i.e. the magnetization tracks the upper branch, in the adiabatic limit. Whereas, ifκ > ²₃ there is hysteresis, thus the magnetization jumps to the lower branch as soon ash= −hcand then back to the upper one whenh=hcand so forth.

In the present work we will proveκ=²₃ to be the critical value even in our Ising spin context. We shall concentrate here on the critical caseκ=²₃ which is not covered by the analysis in [3,4]. We will see that forκ=²₃ the hysteresis loop becomes truly random. There is a positive and not one probability to leave the upper-lower branch at±hc. Our future aim is to extend our analysis to the Kac potential case by taking into account spatial effects.

2. Definitions and results The mean field Ising model

The configuration space is{−1,1}^N,N∈N; its elements are denoted byσ= {σ (i), i=1, . . . , N},σ (i)the spin at sitei. By

m_N=m_N(σ ):= 1 N

N i=1

σ (i) (2.1)

we denote the magnetization density of the configurationσ, so thatm_N∈MN, MN:= 1

N{−N,−N+2, . . . , N−2, N}. The mean field Hamiltonian is

H_h,N(σ ):=N

−m_N(σ )²

2 −hm_N(σ )

and the mean field Gibbs measure at the inverse temperatureβ >0 is the probabilityG_β,h,N on{−1,1}^N given by G_β,h,N(σ ):=e^{−βHh,N(σ )}

Z_β,h,N ,

where the partition functionZ_β,h,N is the normalization factor.

For an introduction to the mean field Ising model see Section 4.1 of [19].

The Glauber dynamics

A Glauber dynamics for the Ising system is the Markov process on{−1,1}^N with generator Lf (σ ):=

N i=1

c(i, σ;h)

f (σ_i)−f (σ )

, (2.2)

whereσi(j )=σ (j )fori=j andσi(i)= −σ (i);c(i, σ;h) >0, the spin flip intensity ati, is given by the formula c(i, σ;h)= e^{−β[Hh,N(σ(i))−Hh,N(σ )]}

e^{−βHh,N(σ(i))}+e^{−βHh,N(σ )}

withσ⁽ⁱ⁾the configuration obtained fromσ by flipping the spin ati. For more details on the Glauber dynamics for mean field Ising systems see Section 5.1 of [19].

h=h(t )is a smooth function of time, henceσ (t )is a time non-homogeneous Markov process. Since the Hamil- tonian depends onσ viam_N(σ ), the process{m_N(σ_t), t≥0}is itself Markov with state spaceMNand generatorL given by

Lhf (x):=c⁺(x, h)

f (x+2/N )−f (x)

+c⁻(x, h)

f (x−2/N )−f (x)

(2.3)

with

c^±(x, h)=N

2(1∓x)cˆ^±(x, h), cˆ^±(x, h)= e^{±β[h+(x±1/N )]}

e^{−β[h+(x±1/N )]}+e^{β[h+(x±1/N )]},

forx∈MN. Whenhis time independent there is a unique invariant measure (see Section 5.1.2 of [19]) which is the marginalμ_β,h,N ofG_β,h,N on the magnetization densitym_Ndefined in (2.1).μ_β,h,N is then the probability onMN

given by

μ_β,h,N(x):=e^{−βN φβ,h,N(x)} Zβ,h,N

, x∈MN,

where

φ_β,h,N(x):= −x²

2 −hx−SN(x) β and

e^NSN(x):=card

σ∈ {−1,1}^N: m_N(σ )=x .

Ifx_N∈MN,x_N→x∈ [−1,1]asN→ ∞thenφ_β,h,N(x_N)→φ_β,h(x)where φ_β,h(x)= −x²

2 −hx−S(x) β and

S(x)= −1−x

2 log1−x

2 −1+x

2 log1+x 2 . The mean field phase transitions

For anyβ≤1 and anyh∈Rthe mean field free energy density (see Section 4.1.2 of [19])φ_β,h(x)is a convex function ofx(absence of phase transitions). If insteadβ >1 (see Fig.2) there ish_c>0 such that, for any|h|< h_c,φ_β,h(x)is a double well function ofx with local minima atX₊(h) > X₋(h)and local maximum atX₀(h)∈(X₋(h), X₊(h));

X_±(h)andX₀(h)are solutions of the mean field equation:

x=tanh β(x+h) ,

X₊(h)is the absolute minimum forh≥0 andX₋(h)forh≤0, then only ath=0 there are two absolute minima and thus a phase transition; forh∈(0, hc),X₊(h)is the only pure phase whileX₋(h)is a metastable state, the opposite holds for negative fields. Whenh→ −h_c,X₊(h)−X₀(h)→0 and the limitx_c:=X₊(−h_c)ofX₊(h)is an inflection point for the functionφ_β,−hc(x). By symmetry the analogous picture describesX₋(h)whenh→h_c.

The macroscopic mean field dynamics

The infinite volume dynamics is governed by the ODE dx

dt =F (x, h), F (x, h):= −x+tanh β(x+h)

, (2.4)

in the following sense. LetmN(t)be the process of generatorLh(t ) (see (2.3)),h(t )a smooth function oft, which starts fromm⁰_N∈MN. We suppose thatm⁰_N→x⁰∈ [−1,1]asN→ ∞and denote byPNthe law ofmN(t), t≥0.

We have the following result.

Fig. 2. The picture shows some profiles ofφ_β,h(x)according to different values ofh, forβ >1.

Theorem 2.1. With the above notation,for anyδ >0and anyT >0,

Nlim→∞PN

sup

t≤T

m_N(t)−x(t)≥δ

=0, (2.5)

wherex(t)is the unique solution of dx

dt =F x, h(t)

, x(0)=x⁰. (2.6)

The proof of Theorem2.1is omitted. The proof in the case of constanthcan be found, for instance, in Section 5.1.5 of [19], the proof easily extends to the present case.

The adiabatic limit

Let

h(t):= −hccost (2.7)

we denote byxω(t)the solution of (2.6) withh=h(ωt )and initial conditionxω(0)=X₊(−hc). We omit the proof that

Theorem 2.2. For anyτ >0

ωlim→0 sup

t≤ω⁻¹τ

x_ω(t)−X₊h(ωt)=0. (2.8)

Theorem2.2proves that, for oscillations of critical amplitudeh_c, in the adiabatic limitω→0 there is not hysteresis (see Fig.3). The relevant time scale ist=ω⁻¹τ and the limit evolution is

ωlim→0x_ω ω⁻¹τ

=X₊ h(τ )

. (2.9)

Fig. 3. The functionxω(t)(blue line) for small values ofωtracks the positive branchX₊(h(t))(black line).

The main theorem

Theorem2.1asserts that the dynamics in the macroscopic limitN → ∞on finite time intervals is described by the deterministic mean field evolution equation (2.6). Whenωis small withN, the period of the magnetic field oscillations is large withN. Therefore the behavior exhibited by (2.9) in the adiabatic limit may not correspond to what the Glauber process does for large but finiteN. As it will turn out, it all depends on the wayω→0 asN→ ∞. As stated in the Introduction, the critical case isω=N^−2/3to which we restrict hereafter (the origin of the factor 2/3 will become clear from the proofs but it will also be explained in Section3in a heuristic way).

There are criticalities for values of the magnetic field in a neighborhood of±hc. Sincehis a periodic function of time and the process is invariant under change of sign we shall restrict ourselves to study the behavior in a semi-period.

We considert∈N^2/3[−^π₂,^π₂]and supposeh=hN(t), with h_N(t):=h

N^−2/3t

= −h_ccos N^−2/3t

(2.10) so that the critical time is set att =0. We shall denote by PN the law of the processm_N(t), t ∈N^2/3[−^π₂,^π₂] of generatorLh_N(t), withm_N(−N^2/3π₂)=m⁰_N. We choose such initial value in a neighborhood of sizeN^−1/2+γ,γ >0, of the positive branch, i.e.|m⁰_N−X₊(0)| ≤N^−1/2+γ (sinceh_N(−N^2/3π₂)=0). The main result is given by the following theorem. It provides the probability, for largeN, to find the magnetization in a neighborhood of one of the two equilibrium branchesX_±(h_N(t)), respectively, before and after the critical timet=0.

Theorem 2.3 (Main theorem). Consider the events H^±γ(I ):=

sup

t∈I

mN(t)−X_±

hN(t)≤N^−1/2+γ

, I ⊆R, γ >0. (2.11)

There isp₋∈(0,1)so that for anyγ , η >0andγ> γ,if|m⁰_N−X₊(0)| ≤N^−1/2+γ then

Nlim→∞PN

H⁺_γ

N^2/3

−π 2,−η

=1, (2.12)

Nlim→∞PN

H^±γ

N^2/3

η,π

2

=p_±, (2.13)

wherep₊=1−p₋.

The critical interval isN^2/3(−η, η),η >0 arbitrarily small. Equation (2.12) shows that, in the limit asN → ∞, the magnetization remains, almost surely, in a neighborhood of sizeN^−1/2+γ,γ> γ, of the positive branch before the criticality (i.e. fort <−ηN^2/3). Equation (2.13) provides the behavior after the criticality (fort > ηN^2/3), it states that there exists a non-trivial probability to find the magnetization either in the positive or in the negative equilibrium branches.

The result can be iterated, as the same arguments can be repeated every time the process runs into a criticality. The macroscopic dynamics is no more deterministic since, at every step there is a positive probability for the magnetization to jump or not, and the hysteresis loops observed become, in this sense, random.

3. Outline of proof

The proof of (2.12) is simple. Indeed, if we fixh >−hc+ε, for someε >0, and the magnetization is initially in a neighborhood of X₊(h), then m_N(t)has a drift towards X₊(h). Therefore, with large probability, it stays in a neighborhood of size N^−1/2+γ (as N^−1/2is the strength of the noise) of the positive branch. Only after a longer (exponential) time, tunneling to the negative branch will be observed. In our case his not fixed but it is so slowly varying that the above argument remains valid as long ash(t) >−h_c+ε, for someε >0 (see Section7). Whenh approaches−h_cthe above picture is wrong because at−h_cthe valuex_cis stationary but not stable. Lack of stability and slow changes of the frequency make the noise competitive with the drift (for the special choiceω=N^−2/3) as we are going to see.

Scalings

In order to understand the scalings let us go back to the stochastic ODE (1.1). Let the magnetic field oscillate as h(ωt )= −h_ccos(ωt), by expanding to leading ordersF (x)+h(F (x)given in (2.4)) aroundx_c,−h_c(i.e. forx−x_c andωtboth small) we get approximately

dx=

h_c(ωt)²

2 +F(x_c)

2 (x−x_c)²

dt+N^−1/2dw(t). (3.1)

We scaley=ω^a(x−x_c)andτ=ω^bt, thus ω^−ady=

ω^−2bh_cω²τ²

2 +ω^−2aF(x_c)y² 2

ω^−bdτ +N^−1/2ω^−b/2dw(τ ) (3.2)

which becomes independent ofωandN if

ω^a−b/2N^−1/2=1, 2+a−3b=0, a+b=0, (3.3)

which yieldsω=N^−2/3.

The same scalings apply to our case as we shall prove using extensively martingales techniques. In order to get rid of constants in the final equation, it is convenient to introduce suitable coefficients in the scaling transformation (3.2), we define, thus, the process

YN(t)=νN^1/3 mN

μN^1/3t

−xc

(3.4)

with μ=

2 βh_cx_c

1/4

and ν=(βx_c)^3/4 2

h_c 1/4

. (3.5)

We shall study the processY_N(t)in a time interval which starts from time−T, lettingT → +∞afterN → ∞.

The proof of (2.12) can be extended (see Section7) till time−μT N^1/3(which is the microscopic time corresponding to time−T forYN(·)) in the following sense:

Theorem 3.1. There isc >0so that,for anyT large enough,ε >0small enough, lim sup

N→∞ PN Y_N(−T )−T≤ε

≥1−e^−cε2T. (3.6)

One of the main points in the proof of (2.13) will be to show (see Sections4and8) that the law ofY_N(t)converges, asN→ ∞, to the law of the stochastic ODE

dY (t )=

t²−Y²(t)

dt+ξdwt, ξ=2

βμν², (3.7)

which is (modulo multiplicative coefficients) the same as (3.1) with parameters as in (3.3). Due to the quadratic dependence onY the solution can blow up in a finite time, therefore the process is defined with values onR∪ {−∞}, with the convention that, ifY (t )= −∞, thenY (t)= −∞for allt≥t. The drift in (3.7) vanishes on the two straight linesY = ±t. It is negative forY <−|t|and it points towards|t|forY >−|t|. A more careful analysis shows that there is a critical trajectoryy^∗(t) <0 solution of the deterministic version (i.e. withξ =0) of (3.7) such that any deterministic solution which starts above the critical curve is exponentially asymptotic to(t, t)ast→ ∞.

We denote byP₋T ,ythe law onR∪ {−∞}of the solutionY (t ), t >−T, of (3.7) starting fromY (−T )=y,T >0.

In Section5we prove the following theorem.

Theorem 3.2. LetP be the probability law with support on solutionsY (t )of(3.7)such that

t→−∞lim Y (t )+t=0 P-a.s. (3.8)

then there existp_±∈(0,1),p₊=1−p₋,such that P there ist: Y (t )= −∞

=p₋ and P

tlim→∞Y (t )−t=0

=p₊. (3.9)

For anyε >0small enough,for any bounded continuous functiong(y)with compact support and anyt∈R,

Tlim→∞1_|y−T|≤εE_{P−T ,y} g

Y (t )

=E_P g

Y (t )

. (3.10)

Moreover there existsc >0such that,for anyT large enough,εsmall enough, P Y (−T )−T≤ε

>1−e^−cε2T. (3.11)

Thus withP probability one eitherY (t )blows up in a finite time or it is asymptotic tot ast→ ∞, both events having non-zero probability. The next goal is to extend the above result to the finiteN processY_N(t). ForT >0 we define the rectangle:

RT = (t, y)∈R²: t∈ [−T , T],|y| ≤2T

(3.12) and, forε∈(0,1),

∂R⁺_T := {T} × [T −ε, T+ε], ∂R⁻_T := [−T , T] × {−2T}, (3.13)

∂R^±_T ⊆∂RT. For the processesY (t )such that(−T , Y (−T ))∈RT, we denote byτT the first exit time fromRT

τ_T :=inf t≥ −T: Y (t ) /∈RT

(3.14)

and define the sets E_T^±= Y:

τ_T, Y (τ_T)

∈∂R^±_T

. (3.15)

We shall prove in Section5

Proposition 3.3. LetY (t ), t≥ −T,be a solution of(3.7)starting at−T fromy: |y−T| ≤ε,ε >0small enough, then

Tlim→∞P−T ,y Y∈E_T⁺∪E_T⁻

=1 (3.16)

moreover

Tlim→∞P₋T ,y

tlim→∞Y (t )−t=0|Y ∈E_T⁺

=1 (3.17)

and

Tlim→∞P₋T ,y there ist: Y (t )= −∞|Y∈E_T⁻

=1. (3.18)

The following corollary is a direct consequence of Theorem3.2and Proposition3.3:

Corollary 3.4. ForY (t )as in the previous proposition we have

Tlim→∞P₋T ,y Y ∈E_T^±

−p_±=0. (3.19)

LetPN,−T ,y be the law ofY_N(t)givenY_N(−T )=y. Using martingale convergence theorems, in Section8we prove the following result.

Proposition 3.5. ForY (t )solution of(3.7)starting at−T fromy: |y−T| ≤ε,εsmall enough,we have

Nlim→∞PN,−T ,y Y_N∈E_T^±

=P−T ,y Y ∈E_T^±

(3.20) and

Nlim→∞PN,−T ,y Y_N∈E_T⁺∪E_T⁻

=P₋T ,y Y∈E_T⁺∪E_T⁻

. (3.21)

Proposition3.5allows us to extend the results obtained forY (t )to the finiteN processY_N(t). Finally in Section8 we prove the following proposition that is the last ingredient to conclude the proof of Theorem2.3.

Proposition 3.6. For anyη, γ >0,

Tlim→∞ lim

N→∞PN

H^±_γ

N^2/3

η,π

2

Y_N∈E_T^±

=1. (3.22)

4. Limit dynamics in the critical region

The study of the limit behavior asN→ ∞of the spin-flip evolution defined in Section2is based on some martingale theorems. In our dynamics we have two natural martingales:

M_N,T(t)=m_N(t)−m_N

−μT N^1/3

− _t

−μT N^1/3FN

m_N(s), h_N(s)

ds, (4.1)

whereFN(x, h):=Lhx,T >1, and M_N,T² (t)−

t

−μT N^1/3

GN

mN(s), hN(s)

ds (4.2)

withGN(x, h):=Lhx²−2xLhx.

In the following lemma we prove that, for large N, the function FN(x, h)is well approximated by the infinite volume driftF (x, h)= −x+tanh{β(x+h)}(see the infinite volume equation (2.4)).

Lemma 4.1. There existsc >0such that,for anyx∈ [−1,1],|h| ≤h_c,N large enough, FN(x, h)−F (x, h)≤ c

N (4.3)

and,forΛ(x, h)=1−xtanh{β(x+h)}, NGN(x, h)−2Λ(x, h)≤ c

N. (4.4)

Proof. We have FN(x, h)= 2

N

c⁺(x, h)−c⁻(x, h)

= ˆ

c⁺(x, h)− ˆc⁻(x, h)

−x ˆ

c⁺(x, h)+ ˆc⁻(x, h) then there existsc >0 such that

cˆ⁺(x, h)+ ˆc⁻(x, h)≤ c

N and cˆ⁺(x, h)− ˆc⁻(x, h)

−tanh β(x+h)≤ c

N (4.5)

for anyN large enough, that yields (4.3). Now GN(x, h)= 4

N²

c⁺(x, h)+c⁻(x, h)

thus

2−NGN(x, h)=2 1−

ˆ

c⁺(x, h)+ ˆc⁻(x, h) +2x

ˆ

c⁺(x, h)− ˆc⁻(x, h)

then (4.4) follows from (4.5).

LetY (t ), t≥ −T, be the solution of (3.7) starting fromY (−T )=y, andτ_T be the first exit time from the rectangle RT (see (3.12) and (3.14)). We denote by P₋^∗_{T ,y} the law of the stopped processY (t∧τ_T)onD[−T , T]. We call τN,T the corresponding stopping time for the finiteN-processY_N(t)(see (3.4)) and denote byP_N,^∗ _{−T ,y}the law of the corresponding stopped process. We are going to prove (see Proposition4.3) the convergence ofP_N,^∗ _{−T ,y}toP₋^∗_{T ,y} for suitableT , y. LetD[−T , T]be the space of functions on[−T , T]that are right-continuous and have left-hand limits.

The convergence results in this section are meant in the sense of the Skorohod metric onD[−T , T]. For more details on the spaceD[−T , T]and the weak convergence onD[−T , T]see Chapter 3 of [7].

For the martingaleMˆ_N,T(t):=νN^1/3M_N,T(μN^1/3(t∧τ_N,T)), Mˆ_N,T(t)=Y_N(t∧τ_N,T)−Y_N(−T )−νμN^2/3

_t∧τN,T

−T FN

m

μsN^1/3 , h_N

μsN^1/3 ds we have the following result

Proposition 4.2. Letw(t)be the standard Brownian motion andξ:=_β²μν²,then

MˆN,T(t)−→^D ξ w(T +t∧τT) asN→ ∞. (4.6)

Proof. By (4.2), the quadratic variation ofMˆN,T(t)is given by Vˆ_N,T(t):=ν²μN

t∧τ_N,T

−T

GN

m_N

μsN^1/3 , h_N

μsN^1/3 ds thus, forΛ(m, h)as in Lemma4.1, by(4.4), there existsc >0 such that

sup

t≥−T

Vˆ_N,T(t)−2ν²μ _t∧τN,T

−T

Λ m_N

μsN^1/3 , h_N

μsN^1/3 ds

≤cN⁻¹

for anyNlarge enough. In a neighborhood of(x_c,−h_c), Λ(x, h)= 1

β +O(h+h_c)+O(x−x_c)

moreover, fort < N^2/3there existsc >0 such that|h_N(t)+h_c| ≤c(tN^−2/3)²for anyN large enough. We have, thus

−T≤sups≤τN,T

Λ m_N

μsN^1/3 , h_N

μsN^1/3

− 1 β

≤cN^−1/3

for a suitablec >0 then sup

t≥−S

Vˆ_N,T(t)−ξ(T +t∧τ_N,T)≤cN^−1/3. (4.7)

We haveτ_N,T →^P τ_T, forN→ ∞, hence, by(4.7), Vˆ_N,T(t)→^P ξ(T +t∧τ_T) asN→ ∞

thus (4.6) follows sinceMˆ_N,T(−T )=0 andMˆ_N,T(t)has at most discontinuities of orderN^−2/3(see [7] and [18]).

Proposition 4.3. For anyT , y >0such thaty <2T,P_N,^∗ _{−T ,y} converges toP₋^∗_{T ,y} asN→ ∞.

Proof. As usual with martingale problems, we first need to prove tightness and then to identify the limiting points by proving that they satisfy a martingale equation which has unique solution. By Proposition4.2follows the tightness of Mˆ_N,T(t). It remains to prove the tightness of

Γ_N,T(t)=νμN^2/3 t∧τ_N,T

−T

FN

m_N

μsN^1/3 , h_N

μsN^1/3 ds.

We use the Chensov moment condition, indeed there existscsuch that, for allt > s≥ −T, E_PN,−T ,yΓ_N,T(t)−ΓN,T(s)²

≤c|t−s|², (4.8)

where (4.8) holds after using the Cauchy–Schwartz inequality, being the integrated function inL². It follows that the stopped processY_N(t∧τ_N,T)is tight and, consequently, its lawPN,−T ,y converges by subsequences. Moreover, any limiting point has support onC([−T , T],R), this follows from the fact that the jumps ofY_N are±N^−2/3.

By(4.3), we can approximate the termFN(x, h)in(4.1)withF (x, h)unless errors of orderN⁻¹. We perform the Taylor expansion ofF (x, h)in a neighborhood of(x_c,−h_c). BeingF (x_c,−h_c)=∂F /∂x(x_c,−h_c)=0, the leading terms are the first order in(h+h_c)and the second order in(x−x_c), we have

F (x, h)=(h+h_c)−βx_c(x−x_c)²+O

(h+h_c)(x−x_c) +O

(h+h_c)² +O

(x−x_c)³ .

On the other hand, for t N^−2/3vanishingly small asN → ∞,h_N(t)= −h_c+h_ct²N^−4/3/2+O((tN^−2/3)⁴), thus there existscsuch that

sup

t∈μN^1/3[−T ,τN,T]

FN

m_N(t), h_N(t)

− hc

2t²N^−4/3−βx_c

m_N(t)−x_c2

≤cN⁻¹ (4.9)

forNlarge enough, then, by (4.9), sup

t≥−T

MˆN,T(t)−YN(t∧τN,T)+YN(−T )+ _t∧τN,T

−T

h_c

2νμ³s²−βxcμν⁻¹Y_N²(s)

ds

≤cN^−1/3. (4.10) For our choice ofμandν (see (3.4)), the integrand in (4.10) becomess²−Y_N²(s). From (4.10) and Proposition4.2 we deduce that any limiting point satisfies a martingale relation that uniquely defines a process which is the law of the

solution of (3.7).

5. Behavior of the limit process

In this section we are going to investigate the behavior of a generic solutionY (t )of the SDE dY (t )=

t²−Y²(t)

dt+ξdwt, ξ >0. (5.1)

For any fixedt₀∈R∪ {−∞},y₀∈R, we denote byPt₀,y₀ the probability law of the processY (t ), t≥t₀, solution of (5.1) starting fromy₀at timet₀. Moreover we denote byP the law ofY (t ), t ∈R, solution of (5.1) conditioned to

|Y (t )+t| →0 ast→ −∞.

Deterministic analysis

One of the preliminary steps for the study of (3.7) is the analysis of the related deterministic equation

y(t)=t²−y²(t). (5.2)

Proposition5.1is proved in Section 2.3 of [8], it concerns the asymptotic behavior fort→ ∞of a generic solution y(t)of (5.2).

Proposition 5.1. There exists a decreasing solutiony^∗(t)of(5.2)such that−t > y^∗(t) >−√

t²+1,for anyt≥0.

Lety(t)be the solution of(5.2)starting at timet₀≥0fromy₀∈R,

• ify₀> y^∗(t₀),then,for anyδ∈(0,1)there existst_δ≥t₀such that|y(t)−t| ≤₂₍₁¹_−δ)t for anyt≥t_δ;

• ify0< y^∗(t0),theny(t)is decreasing fort≥0and it explodes to−∞in a finite time.

Asymptotic behavior ofY (t )fort→ ∞

In this first part of the section we prove the following theorem.

Theorem 5.2. Consider the sets E⁺:=

Y: lim

t→∞Y (t )−t=0

and E⁻:= Y: there ist: Y (t )= −∞

(5.3) thenPt0,y0{Y ∈E⁺∪E⁻} =1for anyt0∈R∪ {−∞},y0∈R.

The proof of Theorem5.2consists of three parts. We define the stopping time Π:=inf t: Y (t )= −∞

thenΠ ∈R∪ {+∞}. We fix T >0 large enough, supposeΠ > T and study the behavior of Y (t )for t≥T. In Proposition5.3we prove that ifY (t )is in a neighborhood ofy^∗(t)at time T thenY (t ) escapes from itP-a.s. In Propositions5.5and5.7we prove that the probability for the eventsY ∈E^∓to occur is close to the probability that Y (t )leaves such a critical neighborhood, respectively, from below or from above. Unless further indications, in this section we mean, byc, a positive constant not depending onT.

We will denote byy^∗(t)the solution of the ODE (5.2) defined in Proposition5.1, and define the processz^∗(t):=

Y (t )−y^∗(t).z^∗(t)verifies the equation dz^∗(t)= −z^∗(t)

z^∗(t)t+2y^∗(t)

dt+ξdw(t). (5.4)

For any fixedδ >0 small enough, we define the stopping timeτ_{T ,δ}^∗ :=inf{t≥T: |z^∗(t)| ≥δ}.

Proposition 5.3. For anyT >0,δ >0small enough, 1Π >TPT ,Y (T ) τ_{T ,δ}^∗ <∞

=1. (5.5)

Proof. Let us assumeΠ > T. We need to prove the assertion for the paths such that|z^∗(T )|< δ. Suitably applying the Ito’s formula to (5.4), we get

dz^∗2(t)=

−2z^∗2(t)

z^∗2(t)+2y^∗(t) +ξ²

dt+2ξ z^∗(t)dw(t), (5.6)

thus, forT ≤t≤τ_{T ,δ}^∗

z^∗2(t)≥z^∗2(T )+ξ²t+2ξ _t

T

z^∗_sdws, (5.7)

the inequality descending since, forδsmall enough,−2z^∗_t∧²_τ∗ T ,δ

(z^∗_t∧τ∗

T ,δ+2y_t^∗_∧τ∗ T ,δ

)≥0.

The process 2ξ^t∧τ_{T ,δ}^∗

T z^∗_sdws is a continuous martingale, thus its expected value is constantly zero and E

2ξ

_t∧τ∗ T ,δ

T

z^∗_sdws

2

=4ξ² _t

T E

z^∗_s²1_s≤τ∗ T ,δ

ds≤4ξ²δ²(t−T )

hence, by the Doob’s inequality, for anyn∈N, PT ,Y (T )

2ξ

_(T+n⁴_)∧τ∗ T ,δ

T

z^∗_sdws

≥n³

≤4ξ²δ² n²

thus, from the Borel–Cantelli Lemma and (5.7),PT ,Y (T )-a.s., there existsn˜such that, forn≥ ˜n, δ²≥z^∗2

T +n⁴

∧τ_{T ,δ}^∗

>−n³+ξ² T+n⁴

∧τ_{T ,δ}^∗

thenτ_{T ,δ}^∗ ≤(T +n⁴)∨(δ²+n³)/ξ², thus, for anyT >0 PT ,Y (T ) τ_{T ,δ}^∗ <∞

≥PT ,Y (T )

lim inf

n→+∞ τ_{T ,δ}^∗ ≤

T +n⁴

=1

and (5.5) is proved.

We omit the proof of the following lemma.

Lemma 5.4. Lett > s,for anyγ >0,we have e^{γ t2}

2γ t

1−e^{−γ (t2−s2)/2}

≤ _t

s

e^{γ u2}du≤e^{γ t2} 2γ t

2γ s² 2γ s²−1

(5.8) for anys >√¹

2γ,and e^{−γ s2}

2γ s

1−t⁻¹e^{−γ (t2−s2)/2} 2γ s² 2γ s²+1

≤ _t

s

e^{−γ u2}du≤e^{−γ s2}

2γ s (5.9)

for anys >0.

Proposition 5.5. There existsc >0such that,for anyT large enough,δ >0, 1Π >T1_τ∗

T ,δ<∞,z^∗(τ_{T ,δ}^∗ )<−δPτ_{T ,δ}^∗ ,Y (τ_{T ,δ}^∗ ) Y /∈E⁻

≤e^−cT. (5.10)