Anne De Bouard
email: debouard@cmapx.polytechnique.fr

Eric Gautier
email: gautier@ensae.fr

Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise

Keywords: Stochastic partial differential equations, Korteweg-de Vries equation, soliton, large deviations AMS 2000 Subject Classification: 35Q53, 60F10, 60H15, 76B25, 76B35

We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude ǫ. The initial datum gives rise to a soliton when ǫ = 0. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of ǫ -2 . We prove exponential upper and lower bounds for the small noise limit of the probability that the exit time from a neighborhood of this randomly modulated soliton is less than T , of the same order in ǫ and T . We obtain that the time scale is exactly the right one. We also study the similar probability for the exit from a neighborhood of the deterministic soliton solution. We are able to quantify the gain of eliminating the secular modes to better describe the persistence of the soliton.

Introduction

The Korteweg-de Vries (KdV) equation is a model for the evolution of weakly nonlinear, shallow water, unidirectional long waves. It is of the form

∂ t u + ∂ 3 x u + ∂ x (u 2 ) = 0 (1.1)
1 where the space variable x is in R. The results of this paper could easily be extended to generalized subcritical KdV equations for which the nonlinearity is ∂ x (u p ) for p < 5, but we consider throughout this article the p = 2 case for simplicity. The KdV equation is famous for its soliton solutions confirming the observation of the solitary wave propagating on a channel by Russell in 1844. These solitons are traveling waves of the form u c,x0 (t, x) = ϕ c (x -ct + x 0 ) where c is the constant velocity, x 0 ∈ R is the initial phase and

ϕ c (x) = 3c 2 cosh 2 ( √ cx/2) (1.2)
These waves are localized, i.e. they decay exponentially to zero as x goes to infinity. Their shape is stable against perturbations of the initial state. A first notion of stability, for initial data close to ϕ c , which takes into account the symmetries of the evolution equation, is that of orbital stability. This notion of stability was first considered, for the solution ϕ c (x -ct + x 0 ) of the KdV equation, by Benjamin [START_REF] Benjamin | The stability of solitary waves[END_REF]. The set {ϕ c (• -s), s ∈ R} is the orbit of ϕ c . The functional Q c (u) = H(u)+ cM(u) is used as a Lyapunov functional in the proof. It involves two important invariant quantities of the evolution equation (1.1): the Hamiltonian, defined for u in H 1 (R), the space of square integrable functions with square integrable first order derivatives, by

H (u) = 1 2 R (∂ x u(x)) 2 dx - 1 3 R u 3 (x)dx (1.3)
and the mass defined by

M (u) = 1 2 R u 2 (x)dx.
(1.4)

The space H 1 (R) is the energy space, and it is a natural space for the solutions of (1.1) : indeed, if u ∈ C([0, T ]; H 1 (R)) is a solution of (1.1), then H(u(t)) = H(u 0 ) and M(u(t)) = M(u 0 ) for any t ∈ [0, T ], where u 0 is the initial datum in H 1 . The shape of the soliton ϕ c is a solution of the constrained variational problem which consists in minimizing the Hamiltonian for a constant mass. Orbital stability means that when the initial datum is close to ϕ c in H 1 then the solution remains close to the orbit of ϕ c . The second stronger notion of stability is that of asymptotic stability. It states that, for initial datum close in H 1 to ϕ c , the solution converges in some sense as time goes to infinity to a soliton where the velocity and phase have been shifted. Convergence may correspond to weak convergence in H 1 , see [START_REF] Martel | Asymptotic stability of solitons for subcritical generalized KdV equation[END_REF] or [START_REF] Martel | Asymptotic stability of solitons for subcritical gKdV equations, revisited[END_REF], or strong convergence in some weighted Sobolev space, see [START_REF] Pego | Asymptotic stability of solitary waves[END_REF], for less general perturbations of the initial datum, when the solution is written in the soliton reference frame. Note that strong convergence in H 1 is not expected due to the possibility of a dispersive tail moving away from the soliton as time goes to infinity.

It is often physically relevant to consider random perturbations of equation (1.1), see [START_REF] Scalerandi | Korteweg-de Vries solitons under additive stochastic perturbations[END_REF]. It is also interesting from a theoretical perspective to study the stability of the soliton shape under these random perturbations. We consider as in [START_REF] Scalerandi | Korteweg-de Vries solitons under additive stochastic perturbations[END_REF] the case of an additive noise which could model a random pressure at the surface of the water. The corresponding stochastic partial differential equation (SPDE for short) written in Itô form is the following

du + ∂ 3 x u + ∂ x (u 2 ) dt = ǫdW (1.5)
where (W (t)) t≥0 is a Wiener process and ǫ is the small noise amplitude. As we work in infinite dimensions, and in the absence of global smoothing property of the group S(t) on H 1 associated to the unbounded operator -∂ 3

x , W needs to be a proper Wiener process on H 1 . Thus, the components of W (1) need to be correlated for the law of W (1) to be a bona-fide Gaussian measure. It can then always be seen as the direct image via a Hilbert-Schmidt self-adjoint mapping Φ of a cylindrical measure and we assume that Φ is a mapping from L 2 into H 1 . Recall that Φ is Hilbert-Schmidt from L 2 into H 1 if it is a bounded linear operator and for a complete orthonormal system (e i ) i∈N of L 2 , i∈N Φe i 2 H 1 is finite. The sum does not depend on the complete orthonormal system and, endowed with its square root as a norm, the space of Hilbert-Schmidt operators

L 2 (L 2 , H 1 ) = L 0,1 2
is a Hilbert space. As a consequence, the Wiener process could be written as

W (t, x) = i∈N β i (t)Φe i (x), t ≥ 0, x ∈ R (1.6)
where (β i ) i∈N is a collection of independent standard real valued Brownian motions and (e i ) i∈N a complete orthonormal system of L 2 . Existence of path-wise mild solutions, almost surely continuous in time for all t positive with values in H 1 , of the SPDE supplemented with the initial datum u(0) = u 0 ∈ H 1 and uniqueness among those having almost surely paths in some subspace X T ⊂ C [0, T ]; H 1 has been obtained in [START_REF] De Bouard | On the stochastic Korteweg-de Vries equation[END_REF]. In [START_REF] De Bouard | White noise driven Korteweg-de Vries equation[END_REF][START_REF] De Bouard | Periodic solutions of the Korteweg-de Vries equation driven by white noise[END_REF] global well posedness is obtained for rougher noises and less regular solutions. It should be noticed that in the physics literature the space-time white noise, corresponding to Φ = I is often considered, which we are not able to treat mathematically. For simplicity, we consider the sequence of operators

Φ n = I -∆ + 1 n (x 2 I -∆) k -1/2 , (1.7)
which are Hilbert-Schmidt from L 2 into H 1 for k large enough, in order to prove lower bounds on exit times. As n goes to infinity the Hilbert-Schmidt assumption tends to be relaxed and the noise mimics a spatially homogeneous noise with covariance (I -∆) -1 , which is a white noise in the Hilbert space H 1 . It should also be noted that these operators are uniformly bounded in the space L 0,1 of bounded operators from L 2 into H 1 ; indeed we have for every integer n, Φ n L 0,1 ≤ 1. It is possible to work with a more general approximating sequence, see for example the kind of assumptions made in [START_REF] Debussche | Small noise asymptotic of the timing jitter in soliton transmission[END_REF].

The linearized operator around the soliton is particularly interesting to study the stability. It has a general null-space spanned by the two secular modes ∂ x ϕ c and ∂ c ϕ c . These modes are associated with infinitesimal changes in the velocity and location of the solitary wave. In Remark 2.3 in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF], the following heuristic argument then implies that the random solution should at most remain close to the deterministic solution up to times of the order ǫ -2/3 . It is based on an analogy with the behavior of a linear system of SDEs such that 0 is a degenerate simple eigenvalue corresponding to a Jordan block

dX 1 = X 2 dt + ǫdW 1 (t) dX 2 = ǫdW 2 (t)
with Brownian motions W 1 and W 2 . In such a case,

X 1 (T ) = ǫ T 0 W 2 (s)ds + ǫW 1 (T )
has variance of the order of ǫ 2 T 3 for large T . Thus, for a first approximation of the solution u ǫ,ϕc 0 of (1.5) with initial datum ϕ c0 , of the form

u ǫ,ϕc 0 (t, x) = ϕ c0 (x -c 0 t) + ǫη ǫ (t, x -c 0 t), (1.8)
and with an exit time defined for a neighborhood of the soliton in H 1

B (ϕ c0 , α) = f ∈ H 1 : f -ϕ c0 H 1 < α by τ ǫ α = inf {t ∈ [0, ∞) : u ǫ,ϕc 0 (t, • + c 0 t) ∈ B (ϕ c0 , α) c } ,
exit is expected to occur on a time scale of the order of ǫ -2/3 . However, it is believed that the soliton shape is preserved over a longer time scale. A general approach which works for the deterministic equation, see [START_REF] Martel | Asymptotic stability of solitons for subcritical generalized KdV equation[END_REF][START_REF] Pego | Asymptotic stability of solitary waves[END_REF], is to introduce a description by a soliton ansatz where the parameters of the soliton fluctuate with time. In the case of an additive noise physicists use an approximation of the solution by a soliton ansatz of the form ϕ c ǫ (t) (x -x ǫ (t)) where c ǫ (t) and x ǫ (t) are random scalar processes evolving according to a system of coupled SDEs.

In the case where the noise is the time derivative of a one dimensional standard Brownian motion, it is easily seen (see [START_REF] Wadati | Stochastic Korteweg-de Vries equations[END_REF]) that the solution can be written as a modulated soliton plus a Brownian motion. However, proving such a result is more involved when the noise is a function of space as well. A mathematical justification is given in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] where the following result is proved.

Theorem 1.1 For ǫ > 0 and c 0 > 0, there exists α 0 > 0 such that for every α ∈ (0, α 0 ] there exists a stopping time τ ǫ α > 0 a.s. and semi-martingales c ǫ (t) and x ǫ (t) defined a.s. for t ≤ τ ǫ α with values in (0, ∞) and R respectively such that if we set ǫη ǫ (t) = u ǫ,ϕc 0 (t,

• + x ǫ (t)) -ϕ c ǫ (t) then R η ǫ (t, x)ϕ c0 (x)dx = (η ǫ , ϕ c0 ) = 0, ∀t ≤ τ ǫ α a.s., (1.9
) R η ǫ (t, x)∂ x ϕ c0 (x)dx = (η ǫ , ∂ x ϕ c0 ) = 0, ∀t ≤ τ ǫ α a.s. (1.10)
and for all t ≤ τ ǫ α , ǫη ǫ (t)

H 1 ≤ α (1.11) and |c ǫ (t) -c 0 | ≤ α. (1.12)
Moreover, there exists C > 0 such that for all T > 0 and α ≤ α 0 there exists ǫ 0 > 0 such that for all ǫ < ǫ 0 ,

P (τ ǫ α ≤ T ) ≤ Cǫ 2 T Φ L 0,1 2 α 4 . (1.13)
The proof uses the Lyapunov functional Q c as a central tool. The equations (1.9) and (1.10) are such that restricted to this subspace, the Lyapunov functional is coercive, i.e. the operator Q ′′ c0 is positive. It allows to keep |c ǫ (t) -c 0 | and ǫη ǫ (t) H 1 small on a longer time interval. Also, these two conditions together with the implicit function theorem allow to obtain c ǫ and x ǫ . Other results in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] give the asymptotic distribution of η ǫ as ǫ goes to zero as well as coupled equations for the evolution of the random scalar parameters. The parameters and remainder do not depend on α ≤ α 0 . In the upper bound (1.13) the product ǫ 2 T appears. The theorem says that the solution stays in a neighborhood of the randomly modulated soliton ϕ c ǫ (t) (x -x ǫ (t)) with high probability at least for times small compared to ǫ -2 . The time spent in a neighborhood of a soliton-like wave, when the initial datum gives rise to a soliton for ǫ = 0, is called the persistence time, see [START_REF] Debussche | Numerical simulation of the stochastic Korteweg-de Vries equation[END_REF] for numerical confirmations that the above order is the right order.

In this paper we first study the exit time τ ǫ α and obtain that the time scale on which the solution stays close to the deterministic soliton is indeed at most of the order of ǫ -2/3 . We then revisit the upper bound (1.13) and prove a sharper exponential bound. This bound is supplemented with an exponential lower bound of the same order in the parameters T and ǫ. We thus obtain the right order of the cumulative distribution function (CDF) of the exit time off neighborhoods of the randomly modulated soliton. This gives a confirmation that the time scale on which the approximation of the solution by a randomly modulated soliton is valid is of the order ǫ -2 . Our main tools are large deviations along with a study of the associated variational problems. Similarly, factors T and T 3 have also been obtained in the study of the tails of the mass and arrival time for stochastic nonlinear Schrödinger equations (NLS) in [START_REF] Debussche | Small noise asymptotic of the timing jitter in soliton transmission[END_REF][START_REF] Gautier | Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications[END_REF] with the same techniques. These quantities are the main processes impairing soliton transmission in optical fibers. In that setting, physicists again use the approximation by a randomly modulated soliton. An analogue of Theorem 1.1 for stochastic NLS equations would allow to tell up to what length of the fiber line the approximation is licit. Large deviations are also known to be a useful tool to study the exit problem from an asymptotic equilibrium point or noised induced transition between several equilibrium points in the small noise limit (see [START_REF] Freidlin | Random Perturbations of Dynamical Systems[END_REF], and [START_REF] Gautier | Exit from a neighborhood of zero for weakly damped stochastic nonlinear Schrödinger equations[END_REF] for an exit problem for stochastic weakly damped nonlinear Schrödinger equation). Here we however study a simpler problem than the escape from the asymptotically stable central manifold for the KdV equation which we hope to study in future works.

Large deviations and escape from a neighborhood of the soliton

We use sample path large deviations in this article in order to obtain lower bounds of the asymptotic as n goes to infinity and ǫ goes to zero of probabilities P (τ n,ǫ ≤ T ) where τ n,ǫ is the exit time of a neighborhood of either the deterministic soliton or the randomly modulated soliton. The n recalls that we consider a sequence of operators Φ n , see (1.7). Large deviation techniques, see for example [START_REF] Dembo | Large deviation techniques and applications[END_REF][START_REF] Deuschel | Large Deviations; Pure and Appl. Math[END_REF], allow to quantify convergence to zero of rare events. For example, it is easy to check that on a finite time interval [0, T ], the paths of the solution of 1.5 starting form ϕ c0 converge in probability to the paths of the deterministic soliton solution. The probability that exit from a neighborhood of the soliton occurs before T goes to zero as ǫ goes to zero. Large deviations quantify the convergence to zero of such probabilities. Following Varadhan's formalism, large deviations could be stated as a sequence of inequalities called a large deviation principle (LDP for short). The convergence to zero of the logarithm of the probabilities of rare events is characterized by a speed, here ǫ 2 , and a deterministic functional I n depending on the operator Φ n considered, called rate function, to be minimized on the closure and interior of the set defining the rare event in the state space. In the small noise asymptotics and for sample path large deviations, the rate function could be expressed in terms of the mild solution of the control problem

∂ t u + ∂ 3 x u + ∂ x (u 2 ) = Φ n h, u(0) = ϕ c0 and h ∈ L 2 0, T ; L 2 . (2.1)
We denote the solution by S n,ϕc 0 (h). The mapping h → S n,ϕc 0 (h) is called the control map and (2.1) the control equation. We recall that a rate function I on the sample space (here the paths space C([0, T ]; H 1 )) is lower semicontinuous and that a good rate function is such that

I -1 ([0, R]) is compact for every R positive.
Theorem 2.1 The laws µ u n,ǫ,ϕc 0 ǫ>0 of the paths of the solutions of (1.5) for the operator Φ = Φ n on C([0, T ]; H 1 ) with initial datum ϕ c0 satisfy a LDP of speed ǫ 2 and good rate function

I n (w) = 1 2 inf h∈L 2 (0,T ;L 2 ): w=S n,ϕc 0 (h) h 2 L 2 (0,T ;L 2 ) .
It means that for every Borel set B of C [0, T ]; H 1 , we have the lower bound

-inf • w ∈ B I n (w) ≤ lim ǫ→0 ǫ 2 log P (u n,ǫ,ϕc 0 ∈ B)
and the upper bound

lim ǫ→0 ǫ 2 log P (u n,ǫ,ϕc 0 ∈ B) ≤ -inf w∈B I n (w).
The proof uses the LDP for the laws of the stochastic convolution ǫZ where Z(t) = t 0 S(t -s)dW (s) on the Banach path space X T ; it is a subspace of C [0, T ]; H 1 where the fixed point argument proving the local well-posedness is used, see [START_REF] De Bouard | On the stochastic Korteweg-de Vries equation[END_REF]. The stochastic convolution appears when we write the equation satisfied by the mild solution of (1.5). These laws are Gaussian measures and the LDP is a consequence of the general result on LDP for centered Gaussian measures on real Banach spaces, see [START_REF] Deuschel | Large Deviations; Pure and Appl. Math[END_REF]. The second step is to prove the continuity of the mapping which, to the perturbation Z in X T assigns the solution

u n,1,ϕc 0 := G(Z) in C [0, T ]; H 1 . It is obtained noting that G(Z) = v(Z) + Z where v(x) denotes the solution of ∂ t v + ∂ 3 x v + ∂ x (v + Z) 2 = 0 v(0) = ϕ c0 .
Then the continuity of G is a consequence of the continuity of v with respect to the perturbation Z. It could be proved as in [START_REF] De Bouard | The stochastic nonlinear Schrödinger equation in H 1[END_REF] where the stochastic NLS equation is considered. LDP for the paths of the mild solution of the SPDE is then obtained by a direct application of the contraction principle which states that we can push forward LDP for measures on a Hausdorff topological space to a LDP for direct image measures on another Hausdorff topological space when the mapping is continuous. More details on the proof of such LDP are given in [START_REF] Gautier | Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications[END_REF] where the stochastic NLS equation with additive noise is considered. The control map S H,ϕc 0 (h) for the theoretical stochastic equation with spatially homogeneous noise is defined as the mild solution of

∂ t u + ∂ 3 x u + ∂ x (u 2 ) = Φ H h, u(0) = ϕ c0 and h ∈ L 2 0, T ; L 2 , (2.2)
where Φ H = (I -∆) -1/2 . Note that though we cannot give a mathematical meaning to the stochastic equation with such homogeneous noise, the corresponding control map is well defined.

Let us now consider the exit times τ n,ǫ α . Note that we only consider here a lower bound of the probability since it is enough to prove the heuristic of Remark 2.3 in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF]. Recall that we want to check that the time scale on which the approximation by the deterministic soliton is licit is at most of the order of ǫ -2/3 . When studying the exit times τ n,ǫ α , however, we give both upper and lower bounds of the tail probabilities of the same order in the parameters ǫ and T .

Proposition 2.2 Take T , c 0 positive; then for α 0 small enough, for every α < α 0 , there exists a constant C(α, c 0 ) which depends on α and c 0 but not on T such that lim n→∞ lim ǫ→0 ǫ 2 log P (τ n,ǫ α ≤ T ) ≥ -C(α, c 0 ) T 3 . Proof. For fixed n, using Theorem 2.1, lim ǫ→0 ǫ log P (τ n,ǫ α ≤ T ) is larger than

- 1 2 inf h 2 L 2 (0,T ;L 2 ) , h : S n,ϕc 0 (h)(T ) -ϕ c0 (• -c 0 T ) H 1 > α .
In a first step, we consider the preceding variational problem in which the operator Φ n is replaced by the operator Φ H = (I -∆) -1/2 and α is replaced by 2α.

We give upper bounds on the infimum by minimizing on smaller and smaller sets of controls, until we are able to handle the variational problem. Note that up to now, the problem is a control problem for the KdV equation that we cannot handle. We will show that we can work on more restrictive classes of controls and still obtain a nice qualitative order. Using the Sobolev embedding of H 1 into L ∞ , with norm C ∞ , the infimum is found to be less than

inf h 2 L 2 (0,T ;L 2 ) , h : S H,ϕc 0 (h)(T ) -ϕ c0 (• -c 0 T ) L ∞ > 2C
∞ α since we then minimize on a smaller set. We consider controls h giving rise to modulated solitons of the form

ϕ c(t) x - t 0 c(s)
in the homogeneous case. They are such that c(0) = c 0 and

Φ H h(t, x) = c ′ (t) ∂ c ϕ c | c=c(t) x - t 0 c(s)ds ,
since the soliton profile ϕ c satisfies the equation

-c∂ x ϕ c + ∂ 3 x ϕ c + ∂ x (ϕ c ) 2 = 0.
Again, taking the infimum on a smaller set of controls, we obtain the lower bound

- 1 2 inf T 0 c ′ (t)Φ -1 H ∂ c ϕ c | c=c(t) x - t 0 c(s)ds 2 L 2 dt, c ∈ C 1 ([0, T ]; (0, +∞)) : c(0) = c 0 , ϕ c(T ) T 0 (c 0 -c(s))ds - 3c 0 2 > 2C ∞ α ,
where we have bounded from below the L ∞ norm of the function by its value at c 0 T . This is in turn bigger than

- 1 2 inf T 0 (c ′ (t)) 2 Φ -1 H ∂ c ϕ c | c=c(t) 2 L 2 dt, c ∈ C 1 ([0, T ]; (0, +∞)) : c(0) = c 0 , 3c 0 2 -ϕ c(T ) T 0 (c 0 -c(s))ds > 2C ∞ α ,
due to the fact that Φ H commutes with spatial translations. Let us fix α 0 < 3c0 4C∞ so that c 0 -4C∞α 3 > 0 for 0 < α < α 0 . A sufficient condition for the constraint on the terminal value to hold is As in [START_REF] Debussche | Small noise asymptotic of the timing jitter in soliton transmission[END_REF] for the tails of the arrival time of a pulse driven by a stochastic nonlinear Schrödinger equation where we obtained the order -CT -3 , the boundary condition is in integrated form. The integral to be minimized is of the form

c 0 - 4C ∞ α 3 > 4c (
T 0 (c ′ (t)) 2 g (c(t)) dt where g(c) = (I -∆) 1/2 ∂ c ϕ c 2 L 2 .
Instead of solving the problem of the calculus of variations with a nonstandard boundary condition, we make a guess and look for solutions of the form c(t) = c 0 -2γt/T 2 for some positive γ with c 0 -2γ/T > 0. Note that if γ = inf{ 3 2 δ(c 0 , α), c0 4 }, then the boundary condition is satisfied for T ≥ 1. Also the term g(c(t)) in the integral is then such that there exists a constant C(c 0 ) with

T 0 (c ′ (t)) 2 g (c(t)) dt ≤ C(c 0 ) T 0 (c ′ (t)) 2 dt, for T ≥ 1, since c 0 /2 ≤ c(t) ≤ c 0 for any T ∈ [0, T ]. Thus, for a new constant C(α, c 0 ), we obtain T 0 (c ′ (t)) 2 g (c(t)) dt ≤ C(α, c 0 ) T 3 , for T ≥ 1.
Let us now consider the case where the square root of the covariance operator of the noise is Φ n , and let us start from the path c(t) exhibited in the homogeneous noise case; we denote the corresponding control by

h c (t, x) = c ′ (t) (I -∆) 1/2 ∂ c ϕ c | c=c(t) • - t 0 c(s)ds .
Then, since for such

h c S H,ϕc 0 (h c )(T ) -ϕ c0 (• -c 0 T ) H 1 > 2α,
we deduce from the continuity of the mild-solution of the control map with respect to the convolution of the semi-group with the control, used to prove the LDP, and the continuity of this last quantity with respect to the control, that for sufficiently large n

S n,ϕc 0 (h c )(T ) -ϕ c0 (• -c 0 T ) H 1 = S H,ϕc 0 (I -∆) 1/2 (I -∆ + 1 n (x 2 I -∆) k ) -1/2 h c (T ) -ϕ c0 (• -c 0 T ) H 1 > α.
This ends the proof.

As a consequence, the time scale on which an exit from a neighborhood of the soliton occurs is at most ǫ -2/3 . In the next section we prove that the time scale on which the solution remains close to the randomly modulated soliton is exactly of the much longer order ǫ -2 . We provide upper and lower bounds for this result.

Escape from a neighborhood of the randomly modulated soliton

In the proof of Theorem 2.1 in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] a local parametrization u → (C(u), X (u)) is used, in order to obtain parameters of the soliton wave form such that u = ϕ C(u) (•-X (u))+R(u) with R satisfying some adequate orthogonality conditions. This parametrization is obtained using the implicit function theorem, imposing that the constraints (1.9) and (1.10) hold. Such a parametrization holds as long as u remains in a proper neighborhood of the spatial translates of ϕ c0 ; thus, setting c ǫ (t) = C(u ǫ,ϕc 0 (t)) and x ǫ (t) = X (u ǫ,ϕc 0 (t)), with u ǫ,ϕc 0 a solution of (1.5) with paths a.s. in C 1 (R + ; H 1 ) and initial datum ϕ c0 , the processes c ǫ (t) and x ǫ (t) are well defined adapted processes, up to a stopping time of the form

τ ǫ α = inf {t ≥ 0, |c ǫ (t) -c 0 | ≥ α or u ǫ,ϕc 0 (t, • + x ǫ (t)) -ϕ c0 H 1 ≥ α} .
We can indeed always replace a function u by the function u(•+ X (u)) and come back to the case where x ǫ (t) is close to 0, and it is not necessary to include a condition of the form |x ǫ (t)| ≥ α. It is also shown that this stopping time could be bounded above and below by

τ ǫ Cα = inf t ≥ 0, |c ǫ (t) -c 0 | ≥ α or u ǫ,ϕc 0 (t, • + x ǫ (t)) -ϕ c ǫ (t) H 1 ≥ α
for some constants C depending solely on c 0 and α 0 , with 0 < α ≤ α 0 . Hence, the two stopping times are equivalent and the qualitative behavior of P(τ ǫ α ≤ T ) with respect to ǫ and T should be the same as the original P(τ ǫ α ≤ T ). These new stopping times τ ǫ α prove to be more convenient to work with. Note that the implicit function theorem says that each function in H 1 sufficiently close to a translate of the soliton could be written as a translated soliton with slightly different parameters plus a remainder in the subspace of H 1 orthogonal in L2 to ϕ c0 and ∂ x ϕ c0 . Therefore the exit in terms of the H 1 norm is the exit of a proper open subset.

Let us first revisit the upper bound given in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] and prove that the upper bound is indeed exponential. The proof relies on exponential tail estimates. We denote by L -1,0 and L 0,1 the spaces of bounded operators from H -1 to L 2 (respectively from L 2 to H 1 ). Proposition 3.1 For T > 0 and 0 < α ≤ α 0 fixed, there exists a constant C(α, c 0 ), depending on α and c 0 but not on T , and ǫ 0 > 0 with ǫ 2 0 T sufficiently small depending on Φ L 0,1 2 and α, such that for every positive ǫ < ǫ 0 ,

P (τ ǫ α T ) ≤ exp - C(α, c 0 ) ǫ 2 T Φ 2 L 0,1 . (3.1)
Proof. Fix T positive. The estimate (1.13) relies on the two following inequalities. Let τ = τ ǫ α ∧T ; then for α 0 sufficiently small there exists a positive constant C independent of T such that

|c ǫ (τ ) -c 0 | 2 ≤ C ǫη ǫ (τ ) 4 L 2 + 4ǫ 2 τ 0 (u ǫ,ϕc 0 (s), dW (s)) L 2 2 + ǫ 4 τ 2 Φ 4 L 0,1 2 and ǫη ǫ (τ ) 2 H 1 ≤C ǫη ǫ (τ ) 4 L 2 + 4ǫ 2 τ 0 (u ǫ,ϕc 0 (s), dW (s)) L 2 2 + ǫ 4 τ 2 Φ 4 L 0,1 2 + ǫ τ 0 (∂ x u ǫ,ϕc 0 (s), ∂ x dW (s)) L 2 -ǫ τ 0 (u ǫ,ϕc 0 (s)) 2 , dW (s)
L 2 + c 0 ǫ τ 0 (u ǫ,ϕc 0 (s), dW (s)) L 2 + ǫ 2 2 τ Φ 2 L 0,1 2 +ǫ 2 Φ 2 L 0,1 2 τ 0 u ǫ,ϕc 0 (s) L 2 ds + c 0 ǫ 2 2 τ Φ 2 L 0,1 2 .
These are obtained from several manipulations of the Lyapunov functional and evolution equations for the mass and Hamiltonian evaluated on the solution of the stochastic KdV equation (see [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF]). The evolution of these quantities is obtained using the Itô formula and a smoothing procedure. We do not reproduce the proof here. We may also write

P (τ ǫ α ≤ T ) ≤ P |c ǫ (τ ) -c 0 | 2 ≥ α 2 or ǫη ǫ (τ ) 2 H 1 ≥ α 2 ≤ P |c ǫ (τ ) -c 0 | 2 ≥ α 2 + P ǫη ǫ (τ ) 2 H 1 ≥ α 2 . Note that when |c ǫ (τ ) -c 0 | 2 ≥ α 2 we have |c ǫ (τ ) -c 0 | 2 = α 2 and ǫη ǫ (τ ) L 2 ≤ α.
Thus for ǫ 0 sufficiently small, depending on Φ L 0,1 and α, and for ǫ < ǫ 0 ,

P |c ǫ (τ ) -c 0 | 2 ≥ α 2 ≤ P ǫ τ 0 (u ǫ,ϕc 0 (s), dW (s)) L 2 ≥ α 4 ≤ P ǫ sup t∈[0,T ] t 0 u ǫ,ϕc 0 ,τ ǫ α (s), dW (s) L 2 ≥ α 4 . (3.2)
where u ǫ,ϕc 0 ,τ ǫ α is the process stopped at time τ ǫ α . Similarly, using as well the following property (P ) When |c ǫ (τ ) -c 0 | ≤ α ≤ α 0 , then for some C depending only on c 0 and α 0 , ϕ c ǫ (t) -ϕ c0 H 1 ≤ Cα, for all t ≤ τ , we obtain that there exists ǫ 0 sufficiently small, depending on Φ L 0,1 2 , T (with ǫ 2 0 T small) and α, such that for all ǫ < ǫ 0 ,

P ǫη ǫ (τ ) 2 H 1 ≥ α 2 ≤P ǫ sup t∈[0,T ] t 0 u ǫ,ϕc 0 ,τ ǫ α (s), dW (s) L 2 ≥ 1 2 α 2 10 +P ǫ sup t∈[0,T ] t 0 ∂ x u ǫ,ϕc 0 τ ǫ α (s), ∂ x dW (s) L 2 ≥ α 2 10 +P ǫ sup t∈[0,T ] t 0 u ǫ,ϕc 0 τ ǫ α (s) 2 , dW (s) L 2 ≥ α 2 10 +P ǫ sup t∈[0,T ] t 0 u ǫ,ϕc 0 ,τ ǫ α (s), dW (s) L 2 ≥ α 2 10c 0 . (3.3)
Let us denote by Z i (t) for i = {1, 2, 3} the stochastic integrals arising in the right hand sides of (3.2) and (3.3). We obtain exponential tail estimates for each of the above probabilities in the usual way, see for example [START_REF] Peszat | Exponential tail estimates for infinite-dimensional stochastic convolutions[END_REF], Theorem 2.1. We introduce the function f l (x) = √ 1 + lx 2 , where l is a positive parameter. We then apply the Itô formula to f l (Z i (t)) and each process decomposes into 1 + E l,i (t) + R l,i (t) where

E l,i (t) = t 0 lZ i (t) 1 + lZ i (t) 2 dZ i (t) - 1 2 t 0 lZ i (t) 1 + lZ i (t) 2 2 d < Z i > t ,
and

R l,i (t) = 1 2 t 0 lZ i (t) 1 + lZ i (t) 2 2 d < Z i > t + 1 2 t 0 d < Z i > t (1 + lZ i (t) 2 ) 3/2 .
Let us for example consider Z 2 . Given (e j ) j∈N a complete orthonormal system of L 2 ,

< Z 2 > t = t 0 j∈N ∂ x u ǫ,ϕc 0 τ ǫ α , ∂ x Φe j 2 L 2 (s)ds;
thus, using the Hölder inequality and the property (P ) we have, for some constant C(α, c 0 ) = Cα 2 + ∂ x ϕ c0 H 1 , and any t,

< Z 2 > t ≤ C(α, c 0 ) Φ * 2 L -1,0 t. Then |R l,2 (t)| ≤ lC(α, c 0 ) Φ 2 L 0,1 T.
The same bound also holds for Z 1 and Z 3 . We may thus write for any i and constants δ i > 0,

P sup t∈[0,T ] |Z i (t)| ≥ δ i ǫ = P sup t∈[0,T ] exp (f l (Z i (t))) ≥ exp f l δ i ǫ ≤ P sup t∈[0,T ] exp (E l,i (t)) ≥ exp f l δ i ǫ -1 -lC(α, c 0 ) Φ 2 L 0,1 T .
The Novikov condition is also satisfied and E l,i (t) is such that (exp (E l,i (t))) t≥0 is a uniformly integrable martingale. The Doob inequality then gives

P sup t∈[0,T ] exp (E l,i (t)) ≥ exp f l δ i ǫ -1 -lC(α, c 0 ) Φ 2 L 0,1 T ≤ exp -f l δ i ǫ + 1 + lC(α, c 0 ) Φ 2 L 0,1 T E [exp (E l,i (T ))] .
Since exp (E l,i (T )) is an exponential martingale E [exp (E l,i (T ))] = E [exp (E l,i (0))] = 1. For ǫ 0 small enough we have for ǫ < ǫ 0

C(α, c 0 ) Φ 2 L 0,1 T < δ 2 i
2ǫ 2 which implies that the l-derivative at 0 of the function in the exponential bound is negative. Then, optimizing on the parameter l, we obtain the minimum in l of the upper bound, which has the form

exp 1 - C(α, c 0 ) ǫ 2 T Φ 2 L 0,1
, with possibly another constant C(α, c 0 ). Using the largest of all constants in the exponentials for each tail probabilities, and taking a constant slightly bigger and ǫ 0 smaller if necessary, the multiplicative constant in front of the exponential decay disappears and the result follows.

Remark 3.2 As will appear elsewhere, Theorem 1.1 also holds for a noise of multiplicative type, and an exponential upper bound holds as well.

If we now consider the sequence of operators Φ n mimicking the spatially homogeneous noise with covariance operator (I -∆) -1 , denoting the exit times τ n,ǫ α , we obtain the following statement: Corollary 3.3 For T > 0, 0 < α < α 0 and n fixed, there exists a constant C(α, c 0 ), depending on α and c 0 , and there exists ǫ 0 > 0 with ǫ 2 0 T sufficiently small with respect to Φ n L 0,1 2 and α, such that for every ǫ < ǫ 0 ,

P (τ n,ǫ α ≤ T ) ≤ exp - C(α, c 0 ) ǫ 2 T . (3.4)
In particular, we have the following double asymptotic result

lim n→∞ lim ǫ→0 ǫ 2 log P (τ n,ǫ α ≤ T ) ≤ - C(α, c 0 ) T .
Let us now prove that when considering the sequence of operators Φ n as above, one can obtain a lower bound of the same order in the parameters T and ǫ as the upper bound in Corollary 3.3. We make use of the approximation via a modulated soliton together with the LDP obtained in Theorem 2.1, and again minimize the rate function on a smaller set of controls giving rise to a set of parameterized exit paths. Proposition 3.4 For every T and α positive, there exists a constant C(α, c 0 ) which depends on c 0 and α but not on T , such that lim n→∞ lim ǫ→0 ǫ 2 log P (τ n,ǫ α ≤ T ) ≥ -C(α, c 0 ) T .

Proof. Let us denote by U α0 the open set

U α0 = ϕ c (• -y) + g, g ∈ H 1 : g H 1 < α 0 , y ∈ R, |c -c 0 | < α 0 .
We know from [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] that the velocity is obtained via a continuous mapping C from U α0 to R such that c ǫ (t) = C (u n,ǫ,ϕc 0 (t)) .

Also for fixed n, we write that for 0 < 2α < α 0 ,

P (τ n,ǫ α ≤ T ) ≥ P (u n,ǫ,ϕc 0 ∈ B) where B = u ∈ C([0, T ]; H 1 ) : ∀t ∈ [0, T ], u(t) ∈ U α0 , |C (u(T )) -c 0 | > α .
Theorem 2.1 then leads to the following lower bound for lim ǫ→0 ǫ log P (τ n,ǫ α ≤ T ): . Now, coming back to the case where Φ n is the square root of the covariance operator of the noise, we start from a path c obtained from v which say minimizes the objective function in the above problem of the calculus of variations, though following the above argument it does not really matter. Then, the control h c is such that C S H,ϕc 0 (h c )(T ) = c 0 + 2α.

- 1 2 inf h 2 L 2 (0,T ;L 2 ) , h : ∀t ∈ [0, T ], S n,ϕc 0 (h)(t) ∈ U α0 , |C (S n,ϕc 0 (h)(T )) -c 0 | > α . ( 3
From the continuity of the mild-solution of the control map with respect to the convolution of the semi-group with the control, used to prove the LDP, and the continuity of this last quantity with respect to the control, and using also the continuity of C with respect to u ∈ U α0 , we know that for sufficiently large n C (S n,ϕc 0 (h c )(T )) = C S H,ϕc 0 (I -∆) 1/2 (I -∆ + 1 n (x 2 I -∆) k ) -1/2 h c (T ) > α.

We deduce that the inf-limit as n goes to infinity of the infimum in (3.5) is again bounded above by C(α,c0) T and this ends the proof of Proposition 3.4.

As a consequence of our two bounds, the typical time scale on which the solution remains in the neighborhood of the modulated soliton is indeed 1/ǫ 2 .

.

  Noticing that the function λ defined by λ(x) = 4x 2 exp -x T 0 (c 0 -c(s))ds attains its maximum at x = 2/ T 0 (c 0 -c(s))ds for x ≥ 0, if T 0 (c 0 -c(s))ds ≥ 0, we obtain that it is enough to have T 0 (c 0 -c(s))ds >4e c 0 -4C ∞ α/3 := δ(c 0 , α).

. 5 ) 1 2 inf h 2 L 2 cC 1 (T 1 0 2 ∂ c ϕ c 2 L 2

 512211222 Let us, as in the proof of Proposition 2.2, replace in a first step the above variational problem by a variational problem for Φ = Φ H = (I -∆) -1/2 and α by 2α. Minimizing on a smaller set, we obtain-(0,T ;L 2 ) , h : ∀t ∈ [0, T ], S H,ϕc 0 (h)(t) ∈ U α0 , C S H,ϕc 0 (h)(T ) -c 0 = 3α .We minimize on an even smaller set, considering solutions of the controlled equation which are modulated solitons of the formϕ c(t) x -where the one dimensional paths c are assumed to belong to C 1 ([0, T ]; (0, +∞)). The boundary conditions are thus that c(0) = c 0 and c(T ) = c 0 + 2α. A control h c associated to such a solution is given byh c (t, x) = Φ -1 H ∂ t S H,ϕc 0 (h c ) + ∂ 3 x S H,ϕc 0 (h c ) + ∂ x S H,ϕc 0 (h c ) 2 (t, x) = c ′ (t) Φ -1 H ∂ c ϕ c c=c(t) x -t 0 c(s)ds .Note that this control is the same as in Proposition 2.2, only the terminal boundary condition changes. We thus obtain the lower bound-′ (t)(I -∆) 1/2 ∂ c ϕ c | c=c(t) x -[0, T ]; (0, +∞)) : c(0) = c 0 , c(T ) = c 0 + 2α .(3.6)We now have to solve a problem of the calculus of variations. Our aim is to find the optimal paths c among the set of constrained paths minimizing the path integral. The integral may be written, with the same function g as in Proposition 2.2, asT 0 (c ′ (t)) 2 g (c(t)) dt.Using successively the change of variables t = T u and the change of unknown function v(u) = c(T u), we obtain an upper bound of the form 1(v ′ (u)) 2 g(v(u))dufor functions v which are C 1 ([0, T ]; (0, +∞)) and satisfy the two boundary conditions v(0) = c 0 and v(1) = c 0 + 2α independent of T . We recall that g(c) = (I -∆) 1/so that s → g(v(s)) is bounded on [0, 1] for any v ∈ C 1 ([0, T ]; (0, +∞)). Hence we deduce that the infimum in (3.6) is bounded above by C(α,c0) T

, T (so that ǫ 2 0 T is small)

Acknowledgments. The authors are grateful to the Centro di Ricerca Matematica Ennio De Giorgi in Pisa for allowing us to participate to the spring 2006 research period when this research was initiated.


Georges Girardi

Richard Grimm

Benjamin Labonne

Jean Orloff

R Grimm Cpt

Correspondence between the 3-form and a non-minimal multiplet in supersymmetry

come

Introduction

The scalar multiplet [START_REF] Wess | Supergauge transformations in four dimensions[END_REF], commonly termed chiral multiplet, is the most popular realization of the (0, 1/2) representation (massive or massless) of supersymmetry in terms of local quantum fields. It contains as components a complex scalar, a Majorana spinor and a complex scalar auxiliary field. Another realization of the same representation (generally massless in this case) is provided by the linear multiplet [START_REF] Ferrara | Supergauge multiplets and superfields[END_REF], given in terms of a real scalar, a 2-index antisymmetric tensor gauge field, a Majorana spinor and no auxiliary field. Contrary to the previous one the linear multiplet is a gauge multiplet. In classical Lagrangian field theory one can establish [START_REF] Siegel | Gauge spinor superfield as scalar multiplet[END_REF] a certain correspondence between the chiral multiplet and the linear multiplet, sometimes referred to as chiral-linear multiplet duality, in particular in applications where the linear multiplet incarnates a dilaton-axion multiplet.

In this note we would like to draw attention to yet another couple of realizations of the (0, 1/2) representation, the 3-form multiplet [START_REF] Gates | Super p-form gauge superfields[END_REF] and a non-minimal (0, 1/2) multiplet [START_REF] Jr | Variant superfield representations[END_REF], including simple chiral multiplet couplings.

The 3-form multiplet made of a 3-index antisymmetric tensor gauge field, a complex scalar, a Majorana spinor and a real auxiliary field may be understood as a further constrained chiral multiplet. It is the basic ingredient in the context of gaugino condensation, but is also relevant in the theory of supersymmetric gauge anomalies and in the description of curvature squared terms and Chern-Simons forms in supersymmetry. The non-minimal (0, 1/2) multiplet, on the other hand, is less well known. In this note, we would like to outline a relation with the 3-form multiplet in very much the same vein as the above mentioned chiral-linear correspondence. To be definite, we shall exhibit here a very simple toy model, coupling the gaugino-condensate multiplet to a single generic chiral multiplet and suggest a corresponding coupling of the non-minimal (0, 1/2) multiplet.

It may be worthwhile to comment briefly on the notion of gaugino condensate multiplet. In a supersymmetric gauge theory the gauge field-strength tensor is promoted to a multiplet containing as superpartners the gaugino and a real bosonic auxiliary field. The corresponding gaugino superfield, denoted

W α , W α is chiral ( D αW α = 0, D α W α = 0) and subject to the additional constraint D α W α = D α W α.
Irrespectively of the mechanisms underlying gaugino condensation, the constraints on W α , W α imply that the condensate 1 superfields tr(W α W α ), tr( W α W α) are not only chiral but fulfill the additional condition

D 2 tr(W α W α ) -D2 tr( W α W α) = iε klmn tr F kl F mn .
Interestingly enough, this supermultiplet can be viewed as a particular realization of a generic 3-form gauge theory in superspace with the 3-form gauge potential related to the Chern-Simons form of the Yang-Mills theory.

The 3-Form Multiplet

In multiplets of supersymmetry different components may be assigned different Rweights, in relation to their supersymmetry transformations and the chiral properties of their generators [6]. As it seems reasonable to assign vanishing R-weight to gauge potential components, the R-weights of their supersymmetry partners are then determined correspondingly. Precisely in the case of C klm (x), the 3-form gauge potential of the gaugino condensate multiplet with vanishing R-weight, the weights of the other components are dictated by supersymmetry: in units where r(θ

) = r( D) = +1, r( θ) = r(D) = -1, the complex scalar Y (x), Y (x) has r(Y ) = +2, r(Y ) = -2.
The fermionic components η α (x), η α(x) acquire r(η) = +1, r(η) = -1, whereas H(x), the real auxiliary field has r(H) = 0. Therefore, H(x) may constitute by itself an R-inert supersymmetric Lagrangian in analogy with the Fayet-Iliopoulos D-term familiar in supersymmetric gauge theory.

In superfield language, the 3-form multiplet is characterized by the superfields Y, Y subject to the chirality conditions

D αY = 0, D α Y = 0, (2.1)
and the additional constraint

D 2 Y -D2 Y = -8i ∂C(x), (2.2)
with ∂C = -4 3 ǫ klmn ∂ k C lmn . These superfield relations have an interpretation as Bianchi identities in superspace geometry [START_REF] Gates | Super p-form gauge superfields[END_REF], [START_REF] Binétruy | The 3-form multiplet in supergravity[END_REF]. Component fields are identified as usual by projection to lowest superfield components

Y = Y (x), D αY = √ 2 η α(x), Y = Y (x), D α Y = √ 2 η α (x), (2.3)
D 2 Y + D2 Y = -8 H(x), C lmn = C lmn (x). (2.4)
Supersymmetry transformations for these components read

δ ξ C mlk = √ 2 16 ξ σn η -ξσ n η ε nmlk , (2.5
)

δ ξ Y = √ 2 ξ α η α , δ ξ Y = √ 2 ξ α η α, (2.6
)

δ ξ η α = √ 2ξ α (H + i∂C) + i √ 2( ξσ m ǫ) α ∂ m Y, (2.7
)

δ ξ η α = √ 2 ξ α (H -i∂C) + i √ 2(ξσ m ǫ) α∂ m Y , (2.8)
δ ξ H = i √ 2 ( ξσ m ∂ m η) + i √ 2 (ξσ m ∂ m η). (2.9)
Taking care of the overall R-weights of Y and Y , invariant component field Lagrangians may be obtained from ("D-term integration")

L Y Y = d 2 θd 2 θ (Y Y ) = -∂ m Y ∂ m Y + i 2 (∂ m ησ m η -ησ m ∂ m η) + H 2 + ∂C 2 , (2.10)
the kinetic Lagrangian density and ("F -term integration")

d 2 θ Y + d 2 θ Y = 2H(x), (2.11)
giving rise to the H-term referred to above. Let us consider, as a very simple example, the coupling to a single chiral superfield3 , φ, of vanishing R-weight, i.e. adding a kinetic density d 2 θd 2 θ φ φ, and generalizing (2.11) to

d 2 θ Y U(φ) + d 2 θ Y Ū ( φ),
with U(φ), Ū ( φ) at most quadratic in the renormalizable case.

In fact, using the explicit solutions of the constraints (2.1, 2.2)

Y = -4 D2 Ω, Y = -4D 2 Ω,
(2.12)

with Ω the real unconstrained pre-potential (undetermined up to a linear superfield pregauge transformation) and employing integration by parts in superspace, the complete action density may be written as a pure D-term integration

d 2 θd 2 θ Y Y + φ φ + 16Ω U(φ) + Ū ( φ) , (2.13)
with suitable superfield equations of motion.

A simple model

In this section, we consider a particular combination of the 3-form and a chiral multiplet in choosing U(φ) = α + µφ, µ ∈ R, giving rise to the superfield action density:

d 2 θd 2 θ φ φ + Y Y + d 2 θ (α + µφ) Y + d 2 θ ᾱ + µ φ Y . (3.1)
At the component field level, this action contains the kinetic terms for A, Ā, χ, χ (chiral multiplet) and Y, Y , η, η (3-form multiplet), mixing terms of these with F, F (chiral multiplet), H (3-form multiplet) and, last but not least, the terms containing ∂C, the field-strength of the 3-index antisymmetric gauge potential. In many cases, in supersymmetric field theories, elimination of auxiliary fields means rather diagonalization in terms of non propagating fields (no derivative terms in the action density) with trivial algebraic equations of motion. In the case at hand, this can be done easily for the part of the action density containing F, F and H, yielding

L = -∂ m A ∂ m Ā - i 2 (χσ m ∂ m χ + χσ m ∂ m χ) -∂ m Y ∂ m Y - i 2 (ησ m ∂ m η + ησ m ∂ m η) -µ (χη + χη) -| µ Y | 2 -|α + µ A| 2 + ∂C + i 2 α -ᾱ + µ (A -Ā) 2 + F F + HH, (3.2)
with diagonalized auxiliary fields

F = F + µY , F = F + µY, H = H + 1 2 (α + ᾱ) + µ 2 (A + Ā). ( 3.3)
The complex scalar Y, Y satisfies a Klein-Gordon equation with mass µ, the Weyl spinors η, χ, combine into a Dirac spinor of the same mass. The equations of motion for the fields A, Ā, C klm are most conveniently written using

A = A 1 + iA 2 , α = α 1 + iα 2 , so that 2A 1 -µ 2 A 1 + α 1 µ = 0, (3.4) 2A 2 -µ∂C = 0, (3.5)
∂ m (∂C -µ A 2 ) = 0. (3.6)
The last equation is compatible with a constant K 2 = ∂C -µ A 2 , giving rise to a shifted Klein-Gordon equation for A 2

(2 -µ 2 ) A 2 + α 2 + K 2 µ = 0. (3.7)
We would like to stress that these features arise necessarily in the context of models dealing with gaugino condensation.

4

The X -Y Correspondence Independently of supersymmetry, the 3-index antisymmetric gauge potential C klm has been employed in the context of the cosmological constant problem [START_REF] Hawking | The cosmological constant is probably zero[END_REF], [START_REF] Duff | The cosmological constant is possibly zero, but the proof is probably wrong[END_REF]. The derivative quadratic action density is proportional to (∂C) 2 . This density can be related to a constant considering the density X 2 + X∂C with X(x) a real field. Varying with respect to X and substituting back reproduces (∂C) 2 . On the other hand, varying with respect to C klm implies ∂ m X = 0, i.e. X a constant. This mechanism can be extended to the supersymmetric case, e.g. the 3-form multiplet. Here we consider the combination

d 2 θd 2 θ -XX -XY -Y X + 16Ω U(φ) + Ū( φ) + φ φ . (4.1)
X, X is a complex unconstrained superfield, Y, Y , the 3-form superfield introduced above and Ω its unconstrained real pre-potential. φ, φ are considered as spectator superfields.

Varying with respect to X, X just implies X = Y , X = Y and one recovers (2.13) upon substitution. As to variation with respect to the 3-form multiplet we shall use the solution (2.12) of the constraints and integration by parts in superspace to arrive at

d 2 θd 2 θ -XX + 4Ω D2 X + D 2 X + 4U(φ) + 4 Ū ( φ) + φ φ , (4.2)
where Ω may be considered as a Lagrange multiplier superfield giving rise to a constraint

D2 X + D 2 X + 4U(φ) + 4 Ū ( φ) = 0, (4.3)
that can be separated into two constraints

D2 X = -4U(φ) -4K, D 2 X = -4 Ū ( φ) -4 K, (4.4)
related by a constant K = -K = iK 2 , which might be absorbed in a redefinition of U(φ), Ū ( φ). In other words, in supersymmetry, the analogue of the constant mentioned above (in the non supersymmetric case) is given by a complex superfield, X, X.

The component field action is then obtained from

d 2 θd 2 θ -XX + φ φ . (4.5)
In the case U = 0, this multiplet has been presented in [START_REF] Jr | Variant superfield representations[END_REF]. We shall call it non-minimal in what follows and use the term coupled non-minimal in the case of non vanishing U, to be discussed in the next section.

The coupled non-minimal multiplet

The superfield constraints (4.4) determine a multiplet of 12 bosonic and 12 fermionic component field degrees of freedom, identified as usual by successive applications of covariant spinor derivatives. We define the component fields contained in X as

X = X, D α X = √ 2 ψ α , D αX = - √ 2 ω α, D αD α X = V α α, D αD 2 X = -4ρ α, D 2 X = -4E. (5.6)
Observe that the θ2 component is given in terms of A, Cf.(4.4). For X we define similarly

X = X, D α X = - √ 2 ω α , D αX = √ 2 ψ α, D α D αX = Vα α, D α D2 X = -4ρ α , D2 X = -4 Ē. (5.7)
Projecting 1 16 D D2 D(-XX) to lowest components gives the canonical component field action density 4

L X = -∂ m X∂ m X - i 2 (ωσ m ∂ m ω + ωσ m ∂ m ω) (5.8)
-| U| 2 -U ′ XF + ωχ -Ū′ X F + ω χ + 1 2 XU ′′ (χχ) + 1 2 X Ū′′ ( χ χ) - 1 2 V m V m -E Ē - i 2 ψ σ m ∂ m ψ + i √ 2ρ - i 2 ψ σm ∂ m ψ + i √ 2ρ ,
describing a complex scalar X and a Majorana spinor ω as physical fields. Auxiliary fields consist of a complex scalar E, a complex vector V m , and 2 Majorana spinors ψ, ρ. This action density is invariant under supersymmetry transformations:

δ ξ X = √ 2 (ξψ -ξ ω), δ ξ X = √ 2 ( ξ ψ -ξω), δ ξ ψ α = √ 2E ξ α - 1 √ 2 V m ( ξσ m ǫ) α , δ ξ ψ α = √ 2 Ē ξ α - 1 √ 2 Vm (ξσ m ǫ) α, δ ξ ω α = - √ 2U ξ α - 1 √ 2 (V m + 2i∂ m X)(ξσ m ǫ) α, δ ξ ω α = - √ 2 Ū ξ α - 1 √ 2 ( Vm + 2i∂ m X)( ξσ m ǫ) α , 4
Primes indicate derivatives with respect to A or Ā, as the case may be.

δ ξ V m = (ξσ m ρ) + √ 2i (ξσ n σm ∂ n ψ -ξσ m σ n ∂ n ω) -( ξ σm χ)U ′ , δ ξ Vm = ( ξσ m ρ) + √ 2i ( ξ σn σ m ∂ n ψ -ξσ m σn ∂ n ω) -(ξσ m χ) Ū′ , δ ξ ρ α = 2i ∂ m E (ξσ m ǫ) α + (2i∂ m V m -22X -U ′′ χχ + 2U ′ F ) ξ α, δ ξ ρ α = 2i ∂ m Ē ( ξσ m ǫ) α + (2i∂ m V m -22X -Ū′′ χ χ + 2 Ū′ F ) ξ α , δ ξ E = ξ ρ, δ ξ Ē = ξ ρ, δ ξ D2 X = -4 √ 2U ′ (ξχ), δ ξ D 2 X = -4 √ 2 Ū′ ( ξ χ).
(5.9)

Adding the kinetic Lagrangian for φ

L S = -∂ m A ∂ m Ā - i 2 (χσ m ∂ m χ + χσ m ∂ m χ) + F F , (5.10) the complete Lagrangian is L = -∂ m A ∂ m Ā - i 2 ( χσ m ∂ m χ + χσ m ∂ m χ) -∂ m X∂ m X - i 2 ( ωσ m ∂ m ω + ωσ m ∂ m ω) -U ′ (A) ωχ -Ū′ ( Ā) ω χ -|U ′ (A)| 2 XX -|U(A)| 2 + 1 2 XU ′′ (A)(χχ) + 1 2 X Ū′′ ( Ā)( χ χ) + F F - 1 2 V m V m -E Ē - i 2 ψ σ m ∂ m ψ + i √ 2ρ - i 2 ψ σm ∂ m ψ + i √ 2ρ , (5.11) with F = F -X Ū′ , F = F -XU ′ , (5.12)
and exhibiting the general scalar potential

V = |U ′ (A)| 2 XX + |U(A)| 2 .
(5.13)

In order to make contact with the simple model of section 3, we set U(A) = α + µA, Ū ( Ā) = ᾱ + µ Ā. Then (5.11) describes two complex scalar fields and a Dirac field with common mass µ, just like the Lagrangian (3.2). The difference between the two Lagrangians appears in the auxiliary field sector and, correspondingly, in the component field supersymmetry transformations. Moreover, it should be stressed that Y, Y represents a gauge multiplet, whereas X, X does not; this is also the case for the linear-chiral multiplet correspondence.

Conclusions

The main purpose of this short communication was to establish a correspondence between the 3-form multiplet and a non-minimal multiplet, in analogy to the well-known relation between the 2-form (i.e. linear) multiplet and the chiral multiplet. Observe that in both cases the correspondence can only be established under certain restrictive assumptions.

Although the 3-form multiplet and the non-minimal multiplet might be considered as exotic multiplets, they are not. As indicated in the introduction, the 3-form multiplet describes naturally the gaugino squared chiral superfield tr (W α W α ) and its complex conjugate. On the other hand, the non-minimal multiplet appears naturally in the context of the solution of the chiral superfield constraints, i.e. φ = D2 ϕ, φ = D 2 φ, in terms of unconstrained potentials ϕ, φ, defined up to pre-gauge transformations ϕ → ϕ + ξ, φ → φ + ξ. These superfields are themselves subject to the pre-constraints D2 ξ = 0, D 2 ξ = 0, leaving φ, φ invariant.

Let us mention as well that the above-mentioned 3-form constraints appear in an intriguing way in supergravity, in the framework of U(1) superspace. The chiral supergravity superfields R, R † are intertwined with the vector superfield G a through the relation D 2 R -D2 R † = 4iD a G a . Remarkably enough, here, the H-term of R, R † corresponds to a D-term of the U(1) supergravity sector.

The emphasis of the present note was to draw attention to the basic features of the correspondence between the 3-form multiplet and the non-minimal multiplet restricting ourselves to quite elementary considerations. More involved structures as well as the corresponding supergravity couplings will be the subject of forthcoming publications.

Component fields for φ are φ = A(x), D α φ = √ 2χ α (x), D 2 φ = -4F (x), likewise for φ.