Christophe Chesneau
email: christophe.chesneau@unicaen.fr.

Fabienne Comte
email: fabi-enne.comte@parisdescartes.fr.

Gwennaelle Mabon
email: gwennaelle.mabon@ensae-paristech.fr.

Fabien Navarro
email: fabien.navarro@univ-nantes.fr

Estimation of convolution in the model with noise

Keywords: Adaptive estimation, Convolution of densities, Measurement errors, Oracle inequality, Nonparametric estimator, AMS Subject Classifications: 62G07 -62G05 June 7, 2014

Motivations

In actuarial or financial sciences, quantities of interest often involve sums of random variables. For example, in the individual risk model, the total amount of claims on a portfolio of insurance contracts is described by the sum of all claims on the individual policies. Consequently, it is natural to aim at estimating the density of sums of random variables, and estimating these directly from observations of the variables. A solution to this question is proposed in [START_REF] Chesneau | Fast nonparametric estimation for convolutions of densities[END_REF]. But the measures associated to the variables are also often not so precise, and clearly, measurement errors can occur. Thus, we consider the model

Y j = X j + ε j , j = 1, . . . , n, (1)
where the Y j 's are i.i.d. with density f Y , X j 's are i.i.d. with density f and ε j 's i.i.d. with density f ε . We assume that X j and ε j are independent, and that f is unknown whereas, in a first step, f ε is known. Note that our assumptions imply that f Y (u) = (f ⋆ f ε )(u), where (h ⋆ k)(t) = h(tx)k(x)dx denotes the convolution product. Our aim is to estimate the ℓ-fold convolution of f for any integer ℓ ≥ 1, i.e. to estimate g ℓ defined by

(2)

g ℓ (u) =(f ⋆ • • • ⋆ f ) ℓ times (u).
ϕ 2 = |ϕ(u)| 2 du, by ϕ, ψ the scalar product ϕ, ψ = ϕ(u) ψ(u)du. We recall the inverse Fourier formula ϕ(x) = (2π) -1 e iux ϕ * (u)du, and that, for the convolution product (ϕ ⋆ ψ)(t) = ϕ(tx)ψ(x)dx, we have by Fourier transform (ϕ ⋆ ψ) * (u) = ϕ * (u)ψ * (u). Lastly ϕ, ψ = (2π) -1 ϕ * , ψ * .

Moreover, we work all along the paper under the assumption (A1) f * ε (u) = 0, ∀u ∈ R. 2.2. Estimators when the noise density is known. We define now our estimators, considering that the observations are drawn from model (1) and assuming that f ε is known and satisfies (A1).

The idea for this construction is based on the following relation. First under independence of X j and ε j in model (1), we have

f Y = f ⋆ f ε , which implies f * Y = f * f * ε .
Since in addition g * ℓ (u) = (f * (u)) ℓ , we have, using (A1),

g ℓ (x) = 1 2π g * ℓ (u)e iux du = 1 2π (f * (u)) ℓ e iux du = 1 2π f * Y (u) f * ε (u) ℓ e iux du.
Then f * Y can easily be estimated by its empirical counterpart

f * Y (u) = 1 n n j=1
e -iuY j .

But a pure plug-in approach leads to integrability problems, this is why a cutoff has to be introduced, which can be seen as a truncated version of the empirical estimator: f * Y (u)1 I [-πm,πm] (u). Thus, we propose the estimator, ∀m ∈ N,

ĝℓ,m (x) = 1 2π πm -πm f * Y (u) f * ε (u) ℓ e iux du. (3)
Naturally, the performance of ĝℓ,m depends on the choice of m. We make ĝℓ,m adaptive by considering the estimator: ĝℓ, m, where m is the integer defined by (4) m = arg min m∈{1...,n} ĝℓ,m 2 + pen ℓ (m) .

where, for numerical constant ϑ specified later,

(5) pen ℓ (m) = ϑ2 2ℓ-1 λ 2ℓ ℓ (m, ∆ ℓ (m))

∆ ℓ (m) n ℓ , ∆ ℓ (m) = 1 2π πm -πm 1 |f * ε (u)| 2ℓ du and (6) λ ℓ (m, D) = max 2 -1/ℓ+1 log(1 + m 2 D)) 1/2 , 2 2-1/(2ℓ) √ n log(1 + m 2 D) .
2.3. Risk bound for ĝℓ,m . Now we can prove the following risk bound for the MISE, a decomposition which explains our adaptive strategy and will allow to discuss rates.

Proposition 2.1. Consider data from model (1), g ℓ given by (2), and assume that (A1) is fulfilled and f is square integrable. Let ĝℓ,m be defined by (3). Then all m in N, we have

(7) E( ĝℓ,m -g ℓ 2 ) ≤ 1 2π |u|≥πm |f * (u)| 2ℓ du + 2 ℓ -1 2π ℓ k=1 ℓ k C k n k πm -πm |f * (u)| 2(ℓ-k) |f * ε (u)| 2k du,
with C k = (72k) 2k (2k/(2k -1)) k .

Inequality (7) deserves two comments. First, we note that this bound is non asymptotic, and so is the adaptive strategy inspired from it. Secondly, we can see the usual antagonism between the bias term (π) -1 |u|≥πm |f * (u)| 2ℓ du which is smaller for large m and the variance term which gets larger when m increases. Now, we can see how Inequality (7) allows us to explain the definition of the model selection given in (4). The bias term can also be written

g ℓ 2 -g ℓ,m 2 with g * ℓ,m (u) = g * ℓ (u)1 I [-πm,πm] (u); as g ℓ 2 is a constant, only -g ℓ,m
2 is estimated, byĝℓ,m 2 . Usually, the variance is replaced by its explicit bound, given here by the second right-hand-side term of (7). It is noteworthy that here, the variance bound is the sum of ℓ terms and only the last term is involved in the penalty. This is rather unexpected and specific to our particular setting. These considerations explain the definition of (4) and the selection procedure.

2.4. Rates of convergence for ĝℓ,m . Now, from an asymptotic point of view, we can investigate the rates of convergence of our estimators according to the smoothness assumptions on f and f ε . The following balls of function spaces are considered, for positive α, β,

A(β, L) = f ∈ L 1 ∩ L 2 (R); |f * (u)| 2 exp(2|u| β )du ≤ L, sup u∈R |f * (u)| 2 exp(2|u| β ) ≤ L and S(α, L) = f ∈ L 1 ∩ L 2 (R); |f * (u)| 2 (1 + u 2 ) α du ≤ L, sup u∈R |f * (u)| 2 (1 + u 2 ) α ≤ L .
As the noise density is assumed to be known, regularity assumptions on f * ε are standardly formulated in the following way, for positive a, b:

[OS(a) ] f * ε is OS(a) if ∀u ∈ R, c ε (1 + u 2 ) a ≤ |f * ε (u)| 2 ≤ C ε (1 + u 2 ) a . [SS(b) ] f * ε is SS(b) if ∀u ∈ R, c ε e -2|u| b ≤ |f * ε (u)| 2 ≤ C ε e -2|u| b .
Note that the lower bound for f * ε is the one used to obtain upper bound of the risk. These classical sets allow to compute rates, depending on the types of f and f ε and to get the following consequence of Proposition 2.1.

Corollary 2.1. Under the assumptions of Proposition 2.1, the following table exhibits the values of m and ϕ n such that ϕ n is the sharpest rates of convergence satisfying "there exists a constant

C * ℓ > 0 such that E( ĝℓ,m -g ℓ 2 ) ≤ C * ℓ ϕ n "
, according some smoothness assumptions on f and f ε .

f ∈ A(β, L), f ε OS(a) f ∈ S(α, L), f ε SS(b) f ∈ S(α, L), f ε OS(a) m opt 1 π log(n) 2ℓ 1/β 1 π log(n) 4ℓ 1/b n k 0 /(2ak 0 +2αk 0 +1) ϕ n 1 n (log(n)) -4ℓα/b n -2αℓ/(2a+2α+1) ∨ n -1
where k 0 is the integer such that a k 0 -1 < 2α < a k 0 where a 0 = 0,

a k = (2ak + 1)/(ℓ -k), k = 1, . . . , ℓ -1.
The parametric rate of convergence 1/n is attained for the case f ∈ A(β, L), f ε OS(a), and in the case f ∈ S(α, L), f ε OS(a) when ℓ > 1+ (a+ 1/2)/α. To the best of our knowledge, Theorem 2.1 is the first result showing (sharp) rates of convergence for an estimator in the context of (1) for different types of smoothness on f and f ε . The results correspond to classical deconvolution rates for ℓ = 1 (see [START_REF] Comte | Penalized contrast estimator for density deconvolution[END_REF]). Moreover, this completes and improves the rate of convergence determined in (Chesneau and Navarro, 2014, Proposition 5.1).

The case where both f and f ε are super smooth is not detailed. For ℓ = 1, the rates corresponding to this case, are implicit solutions of optimization equations (see [START_REF] Butucea | Sharp optimality in density deconvolution with dominating bias[END_REF]) and recursive formula are given in [START_REF] Lacour | Rates of convergence for nonparametric deconvolution[END_REF]. Explicit polynomial rates can be obtained, and also rates faster than logarithmic but slower than polynomial. We do not investigate them here for sake of conciseness. We can emphasize that the function f being unknown, its regularity is also unknown, and the asymptotic optimal choice of m can not be done in practice. This is what makes an automatic choice of m mandatory to obtain a complete estimation procedure. The automatic resulting trade-off avoids tedious computations of asymptotic rates and perform at best for all sample sizes. 2.5. Risk bound on the adaptive estimator ĝℓ, m. Theorem 2.1 determines risk bound for the MISE of the estimator ĝℓ, m.

Theorem 2.1. Consider Model (1) under Assumption (A1). Let ĝℓ, m be defined by ( 3)-( 4) and define

(8) V ℓ (m) := 2 ℓ -1 π ℓ k=1 ℓ k C k 1 n k πm -πm |f * (u)| 2(ℓ-k) |f * ε (u)| 2k du.
Then, there exists ϑ 0 such that for any ϑ ≥ ϑ 0 , we have

E( ĝℓ, m -g ℓ 2 ) ≤ 2 2ℓ min m∈{1,...,n} g -g ℓ 2 + V ℓ (m) + pen ℓ (m) + c ℓ n ℓ
where c ℓ is a constant independent of n. Note that ϑ 0 = 1 suits.

The result in Theorem 2.1 shows that the adaptive estimator ĝℓ, m performs an automatic trade-off between the bias and variance terms described in Proposition 2.1. Moreover, the bound on the risk given in Theorem 2.1 is non asymptotic, and holds for all sample size.

As a consequence, we can derive asymptotic results. Table 1 exhibits the sharpest rate of convergence ϕ n satisfying "there exists a constant C * * ℓ > 0 such that E( ĝℓ, mg ℓ 2 ) ≤ C * * ℓ ϕ n ", depending on smoothness assumptions on f and f ε . Theorem 2.1 implies thus that g ℓ, m attains similar rates of convergence to g ℓ,m with automatic "optimal" choices for m. The only difference is possibly a logarithmic term.

f ∈ A(β, L), f ε OS(a) f ∈ S(α, L), f ε SS(b) f ∈ S(α, L), f ε OS(a) ϕ n log(n)/n (log(n)) -4ℓα/b (n/ log(n)) -2αℓ/(2a+2α+1) ∨ (log(n)/n)

Estimation with unknown noise density

We consider now that f ε is unknown, but still satisfying (A1). For the problem to be identifiable, we assume that we have at hand a sample ε ′ 1 , . . . , ε ′ M of i.i.d. replications of ε 1 , independent of the first sample.

3.1. Risk bound for fixed m. We have to replace in (3) the noise characteristic function f * ε by an estimator, and to prevent it from getting too small in the denominator. Thus, we propose the estimator, ∀m ∈ N,

gℓ,m (x) = 1 2π πm -πm f * Y (u) f * ε (u) ℓ e iux du, (9)
where still f * Y (u) = n -1 n j=1 e -iuY j and now

(10) fε (u) = f * ε (u) if f * ε (u) ≥ k M (u) k M (u) otherwise where f * ε (u) = 1 M M j=1 e -iuε ′ j , and (11)
k M (u) = κs M (u)M -1/2 .
The definition of s M is simply 1 when considering one estimator in the collection, and more elaborated to deal with the adaptive case. To study the risk bound of the estimator, we recall the following key Lemma (see [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF] and [START_REF] Kappus | Adaptive nonparametric estimation for Lévy processes observed at low frequency[END_REF]).

Lemma 3.1. Assume that (A1) holds. Let s M (u) ≡ 1 in (11) for fε defined by (10), then for all p ≥ 1, there exists a constant τ p such that, ∀u ∈ R,

E 1 f * ε (u) - 1 f * ε (u) 2p ≤ τ p 1 |f * ε (u)| 2p ∧ (k M (u)) 2p |f * ε (u)| 4p .
To study the MISE of the gℓ,m , we write gℓ,mg ℓ 2 = g ℓg ℓ,m 2 + gℓ,mg ℓ,m 2 , which starts the bias-variance decomposition. Then we can split the integrated variance term of gℓ,m into three terms

gℓ,m -g ℓ,m 2 = 1 2π πm -πm f * Y (u) f * ε (u) - f * Y (u) f * ε (u) + f * Y (u) f * ε (u) ℓ - f * Y (u) f * ε (u) ℓ 2 du = 1 2π πm -πm ℓ k=1 ℓ k f * Y (u) f * ε (u) ℓ-k f * Y (u) f * ε (u) - f * Y (u) f * ε (u) k 2 du ≤ 2 ℓ -1 2π πm -πm ℓ k=1 ℓ k f * Y (u) f * ε (u) 2(ℓ-k) f * Y (u) f * ε (u) - f * Y (u) f * ε (u) 2k du ≤ 2 ℓ 2π (T 1 + T 2 + T 3 )
with

T 1 = πm -πm ℓ k=1 ℓ k 3 2k-1 f * Y (u) f * ε (u) 2(ℓ-k) f * Y (u) -f * Y (u) f * ε (u) 2k du T 2 = πm -πm ℓ k=1 ℓ k 3 2k-1 f * Y (u) f * ε (u) 2(ℓ-k) ( f * Y (u) -f * Y (u)) 1 f * ε (u) - 1 f * ε (u) 2k du T 3 = πm -πm ℓ k=1 ℓ k 3 2k-1 f * Y (u) f * ε (u) 2(ℓ-k) f * Y (u) 1 f * ε (u) - 1 f * ε (u) 2k du.
Clearly, T 1 is the term corresponding to known f * ε and we have E(T 1 ) ≤ 3 2ℓ-1 V ℓ . Using the first bound in Lemma 3.1 and the independence of the samples, we have also E(T 2 ) ≤ 3 2ℓ-1 V ℓ . The novelty comes from T 3 , where we use the second bound given in Lemma 3.1:

E(T 3 ) ≤ ℓ k=1 ℓ k 3 2k-1 πm -πm |f * Y (u)| 2ℓ |f * ε (u)| 2(ℓ-k) τ k M -k |f * ε (u)| 4k du ≤ ℓ k=1 ℓ k 3 2k-1 τ k 1 M k πm -πm |f * Y (u)| 2ℓ |f * ε (u)| 2(ℓ+k) du = ℓ k=1 ℓ k 3 2k-1 τ k 1 M k πm -πm |f * (u)| 2ℓ |f * ε (u)| 2k du := W ℓ . (12)
Therefore, the following result holds.

Proposition 3.1. Consider model (1) under (A1). Let gℓ,m be defined by ( 9)-( 11) with s M (u) ≡ 1. Then there exists constants

C * * ℓ , C * * * ℓ > 0 (independent of n) such that E( gℓ,m -g ℓ 2 ) ≤ 1 π |u|≥πm |f * (u)| 2ℓ du + C * * ℓ V ℓ + C * * * ℓ W ℓ ,
where V ℓ is defined by ( 8) and W ℓ by ( 12).

We can see that if M ≥ n, we have

1 M k πm -πm |f * (u)| 2ℓ |f * ε (u)| 2k du ≤ 1 n k πm -πm |f * (u)| 2(ℓ-k) |f * ε (u)| 2k du and thus W ℓ ≤ C (4) ℓ V ℓ .
Thus, for M ≥ n, the risk of gℓ,m has the same order as the risk of ĝℓ,m and the estimation of the noise characteristic function does not modify the rate of the estimator. The results given in Corollary 2.1 are thus still valid. Obviously, if the noise sample size M is smaller than n, it may imply deteriorations of the rates. Moreover, the above result is a generalization of [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] and [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF], which both correspond to the case ℓ = 1.

3.2. Adaptive procedure. Adaptation estimation here is a difficult task. We apply the methodology recently proposed by [START_REF] Kappus | Adaptive density estimation in deconvolution problems with unknown error distribution[END_REF] in the context ℓ = 1, improving the results of [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF]. In particular the estimator f * ε is now taken with s M in (11) defined as follows. Let for a given δ > 0,

∀u ∈ R, w(u) = (log(e + |u|)) -1 2 -δ ,
originally introduced in Neumann and Reiss ( 2009), and f * ε given by ( 10)-( 11) and by ( 13)

s M (u) = (log M ) 1/2 w -1 (u).
With this threshold, the bound given in Proposition 3.1 becomes

E( gℓ,m -g ℓ 2 ) ≤ 1 π |u|≥πm |f * (u)| 2ℓ du + V ℓ with (14) V ℓ = C * * ℓ V ℓ + C * * * * ℓ ℓ k=1 ℓ k 3 2k-1 τ k log k (M ) M k πm -πm w -2k (u)|f * (u)| 2ℓ |f * ε (u)| 2k du.
Clearly, the definition of s M here implies logarithmic loss if we compare the last term above with W ℓ given by ( 12). Then using k M defined by ( 11) and ( 13) and considering gℓ,m defined by ( 9)-( 10), we propose the following adaptive strategy. We set first

∆ℓ (m) = 1 2π πm -πm w(u) -2ℓ f * ε (u) 2ℓ du and ∆f ℓ (m) = 1 2π πm -πm w(u) -2ℓ | f * Y (u)| 2ℓ f * ε (u) 4ℓ du,
as estimators of ∆ ℓ (m) given by ( 5) and of

∆ f ℓ (m) = 1 2π πm -πm |f * (u)| 2ℓ |f * ε (u)| 2ℓ du.
We can now define the stochastic penalty associated with the adaptive procedure

qen ℓ (m) = qen ℓ,1 (m) + qen ℓ,2 (m) = 2 2ℓ+1 λ 2ℓ ℓ (m, ∆ℓ (m)) ∆ℓ (m) n ℓ + 2 2ℓ+1 κ 2ℓ log ℓ (M m) ∆f ℓ (m) M ℓ which estimates the deterministic quantities qen ℓ (m) = qen ℓ,1 (m) + qen ℓ,2 (m) = 2 2ℓ+1 λ 2ℓ ℓ (m, ∆ ℓ (m)) ∆ ℓ (m) n ℓ + 2 2ℓ+1 κ 2 log ℓ (M m) ∆ f ℓ (m)
M ℓ where the weights λ ℓ (m, D) are defined by ( 6). Then, we select the cutoff parameter m as a minimizer of the following penalized criterion

(15) m = arg min m∈{1,...,n} -gℓ,m 2 + qen ℓ (m) .
Theorem 3.1. Consider Model (1) under (A1), associated with an independent noise sample ε ′ 1 , . . . , ε ′ M . Let the estimator gℓ, m be defined by ( 9) and ( 15). Then there are positive constants C ad and C such that

(16) E g ℓ -gℓ, m 2 ≤ C ad inf m∈Mn g ℓ -g ℓ,m 2 + V ℓ (m) + qen ℓ (m) + C n ℓ + C M ℓ ,
where V ℓ is defined by ( 14).

This result is a generalization of [START_REF] Kappus | Adaptive density estimation in deconvolution problems with unknown error distribution[END_REF], which corresponds to the case ℓ = 1. As in the known density case, we do not use the whole variance for penalization, but only the last terms. Moreover, the final estimator automatically reaches the bias-variance compromise, up to logarithmic terms.

Illustration

We now illustrate the theoretical results by a simulation study within the context described in Section 2 and Section 3. That is, we consider the problem of estimating the density g ℓ , emphasizing the case ℓ = 2, for both known and unknown types of noises. All simulations have been implemented under Matlab.

The performance of the proposed method is studied for four sets of test distributions for X j representing different degrees of smoothness (see Figure 1),

(1) Kurtotic distribution:

2 3 N (0, 1) + 1 3 N (0, (1/10) 2 ), (2) Standard normal distribution: N (0, 1), (3) Uniform distribution: U (-1, 1), (4) Claw distribution: 1 2 N (0, 1) + 4 l=0 N l/2 -1, (1/2) 2 .
Note that the two Gaussian mixture densities are taken from [START_REF] Marron | Exact Mean Integrated Squared Error[END_REF].

We consider two types of error density f ε , with the same variance σ 2 ε = 0.25. The first one is Laplace distribution, with f * ε OS(2), and the second one is the Gaussian distribution with f * ε SS(2).

(a) Laplace error:

f ε (x) = 1 2σε exp -|x| σε , f * ε (x) = 1 1+σ 2 ε x 2 . (b) Gaussian error: f ε (x) = 1 σε √ 2π exp -x 2 2σ 2 ε , f * ε (x) = exp -σ 2 ε x 2 2 ,
In the case of an entirely known noise distribution, the constant ϑ in ( 5) is set to 1.8 for all the tests and in the unknown case the penalties are chosen according to Theorem 3.1. We propose the following penalty:

qen ℓ (m) = κ 1 λ 2ℓ ℓ (m, ∆ℓ (m)) ∆ℓ (m) n ℓ + κ 2 κ 2ℓ log ℓ (M m) ∆f ℓ (m) M ℓ
with the associated constants penalties κ 1 = κ 2 = 1 and κ = 1.8. In both cases, the normalized sinc kernel was used throughout all experiments. As in [START_REF] Chesneau | Fast nonparametric estimation for convolutions of densities[END_REF], the estimators proposed in this paper are well suited for FFT-based method. Thus, the resulting estimators ĝℓ, m and gℓ, m are simple to implement and fast which allows us to perform the penalized criterion in a reasonable time. For numerical implementation, we consider an interval [a, b] that covers the range of the data and the density estimates were evaluated at M = 2 r equally spaced points t i = a + (ba)/M , i = 0, 1, . . . , M -1, between a and b, with r = 8, b = -a = 5 and M is the number of discretization points. In each case, the grid of m values that we have considered consisted of 50 values from 0.5 m0 to 1.1 m0 where m0 = n -ℓ/5 denotes a pilot bandwidth (see e.g. [START_REF] Silverman | Density estimation: for statistics and data analysis[END_REF]). In order to illustrate Theorem 2.1, we study the influence of the noise type on the numerical performances of the proposed estimator. For each density, samples of size n = 1000 were generated, with aim to estimate g ℓ from X 1 , . . . , X n drawn from any of the test densities. The results are depicted in Figure 2 and Figure 3 for Laplace and Gaussian noise respectively. Figures 23show the results of the numerical simulation of our adaptive estimator ĝℓ,m . Figure 4(a) contains a plot of the penalized criterion function versus the kernel bandwidth m and Figure 4(b) the estimated MISE as a function of m. We give this plot for kurtotic distribution and Laplace noise only, but the behaviour is the same in all the other cases. It is clear from Figure 4(a), that the value of m is the unambiguous minimizer ofĝℓ,m

2 + pen ℓ (m). We also see that m provides a result close to m MISE : in the Laplace case, for the Kurtotic density, the bandwidth which minimizes MISE(m) in this case is m MISE = 0.1306 and m = 0.1257. This holds true for all test densities. In practice, the minimum of the MISE is close to the minimum of the penalized loss function, thus supporting the choice dictated by our theoretical procedure. Therefore, the proposed estimator provides effective results for the four test densities.

We also compare the performance of ĝℓ, m with that of gℓ, m. For each density, samples with size n = 1000 were generated and the MISE was approximated as an average of the Integrated Squared Error over 100 replications. Table 2 presents the MISE from 100 repetitions for the two [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF], estimating the characteristic function of the noise reduces the risk compared to knowing it. Indeed, for virtually all cases, gℓ, m consistently showed lower risk than ĝℓ, m, with the exception of the Gaussian density for which ĝℓ, m performs slightly better.

g * ℓ,m (u) = g * ℓ (u)1 [-πm,πm] (u) = (f * (u)) ℓ 1 [-πm,πm] (u) = f * Y (u) f ε (u) ℓ 1 [-πm,πm] (u).
The support of (g ℓg ℓ,m ) * being [-πm, πm] c , disjoint from the support of (g ℓ,mĝℓ,m ) * , we get

E( ĝℓ,m -g ℓ 2 ) = g ℓ -g ℓ,m 2 + E( g ℓ,m -ĝℓ,m 2 ). ( 17
)
The Parseval identity yields

g ℓ -g ℓ,m 2 = 1 2π |u|≥πm |g * ℓ (u)| 2 du = 1 2π |u|≥πm |f * (u)| 2ℓ du. ( 18
)
The same argument gives

E( g ℓ,m -ĝℓ,m 2 ) = 1 2π πm -πm E |( f * Y (u)) ℓ -(f * Y (u)) ℓ | 2 |f * ε (u)| 2ℓ
du.

Using the binomial theorem and the inequality

| m i=1 b i a i | 2 ≤ ( m i=1 |b i |) m i=1 |b i ||a i | 2 , we get E |( f * Y (u)) ℓ -(f * Y (u)) ℓ | 2 = E |( f * Y (u) -f * Y (u) + f * Y (u)) ℓ -(f * Y (u)) ℓ | 2 = E   ℓ k=1 ℓ k ( f * Y (u) -f * Y (u)) k (f * Y (u)) ℓ-k 2   ≤ (2 ℓ -1) ℓ k=1 ℓ k E | f * Y (u) -f * Y (u)| 2k |f * Y (u)| 2(ℓ-k) .
The Marcinkiewicz & Zygmund Inequality inequality applied to the n i.i.d. random variables U 1 , . . . , U n with U j = e -iuY j -E(e -iuY 1 ), |U j | ≤ 2 and the exponent p = 2k (see Appendix) yields

E | f * Y (u) -f * Y (u)| 2k ≤ C k 1 n k , with C k = (72k) 2k (2k/(2k -1)) k . Therefore E |( f * Y (u)) ℓ -(f * Y (u)) ℓ | 2 ≤ (2 ℓ -1) ℓ k=1 ℓ k C k 1 n k |f * Y (u)| 2(ℓ-k) . Since f * Y (u) = f * (u)f * ε (u), we obtain E( g ℓ,m -ĝℓ,m 2 ) ≤ 2 ℓ -1 2π ℓ k=1 ℓ k C k 1 n k πm -πm |f * (u)| 2(ℓ-k) |f * ε (u)| 2k du. (19)
Putting ( 17), ( 18) and ( 19) together, we get the desired result:

E( ĝℓ,m -g ℓ 2 ) ≤ 1 2π |u|≥πm |f * (u)| 2ℓ du + 2 ℓ -1 2π ℓ k=1 ℓ k C k 1 n k πm -πm |f * (u)| 2(ℓ-k) |f * ε (u)| 2k du.

Proof of Corollary 2.1.

Case f ∈ A(β, L) and f ε OS(a). The bias term is bounded by

L ℓ e -2ℓ(πm) β since |u|≥πm |f * (u)| 2ℓ du = |u|≥πm |f * (u)| 2 e -2|u| β |f * (u)| 2(ℓ-1) e 2(ℓ-1)|u| β e -2ℓ|u| β du ≤ L ℓ e -2ℓ(πm) β
All the terms containing powers of |f * | in the numerator are integrable as f ε is ordinary smooth. Therefore they are of order less than 1/n. Then we have the general bound

E( ĝℓ,m -g ℓ 2 ) ≤ L ℓ e -2ℓ(πm) β + 1 n ℓ πm -πm du |f ε (u)| 2ℓ + C n where πm -πm du/|f ε (u)| 2ℓ = O(m 2ℓa+1 ). Choosing m = π -1 (log(n)/(2ℓ)) 1/β gives the rate 1/n.
Case f ∈ S(α, L) and f ε SS(b). Applying Lemma 6.2, we obtain

E( ĝℓ,m -g ℓ 2 ) ≤ L ℓ (πm) -2ℓα + ℓ k=1 ℓ k C k L ℓ-k c ε n k πm -πm (1 + u 2 ) -(ℓ-k) e 2k|u| b du ≤ L ℓ (πm) -2ℓα + ℓ-1 k=1 ℓ k C k L ℓ-k c ε n k e 2k(πm) b (1 + u 2 ) -1 du + C 1 c ε n k m -b e 2ℓ(πm) b .
The choice m = π -1 (log(n)/(4ℓ)) 1/b makes all variance terms of order at most O(1/ √ n), so that it implies a rate (log(n)) -4ℓα/b , which is logarithmic, but faster when ℓ increases.

Case f ∈ S(α, L), f ε OS(a).

In this case, there exists a constant K ℓ > 0 such that

E( ĝℓ,m -g ℓ 2 ) ≤ K ℓ m -2αℓ + ℓ-1 k=1 m (2ak-2α(ℓ-k)+1) + log(m) 1 I 2ak-2α(ℓ-k)+1=0 n k + m 2aℓ+1 n ℓ .
Then α must be compared to the increasing sequence a k = (2ak + 1)/(ℓk).

First case: 2α < a 1 . Then the trade-off between m -2αℓ and m 2a-2α(ℓ-1)+1 /n implies the choice m opt = n 1/(2a+2α+1) and the rate n -2αℓ/(2a+2α+1) , where 2αℓ/(2a + 2α + 1) ∈ (0, 1). In this situation, the terms following have orders

ℓ-1 k=2 m 2ak-2α(ℓ-k)+1 opt n k + m 2aℓ+1 opt n ℓ
and we notice that

m 2ak-2α(ℓ-k)+1 opt n k = n 2ak-2α(ℓ-k)+1 2a+2α+1 -k = n -2αℓ-k+1 2a+2α+1 ≤ n -2αℓ 2a+2α+1 since k ≥ 2.
Therefore the rate is of order n -2αℓ/(2a+2α+1) . Notice that making compromise using another variance term can be checked to give larger choice of m and thus worse rate, when inserted in the first variance term. Note also that the resulting rate is the usual rate of deconvolution to the power ℓ.

Second case: 2α > a 1 . The term corresponding to k = 1 is of order O(1/n). Let k 0 ≥ 2 be such that a k 0 -1 < 2α < a k 0 . Then the risk bound is E( ĝℓ,m -g ℓ 2 ) ≤ C   m -2αℓ + 1 n + • • • + 1 n k 0 -1 + ℓ-1 k=k 0 m (2ak-2α(ℓ-k)+1) n k + m 2aℓ+1 n ℓ   .
Then the compromise is made between -2αℓ and m 2ak 0 -2α(ℓ-k 0 )+1 /n k 0 and implies the choice m opt = n k 0 /(2ak 0 +2αk 0 +1) and the rate n -2αℓk 0 /(2ak 0 +2αk 0 +1) . But clearly

- 2αℓk 0 2ak 0 + 2αk 0 + 1 ≤ -1 ⇔ 2α ≥ 2a ℓ -1 + k 0 -ℓ + 1 k 0 (ℓ -1) = 2a ℓ -1 + 1 ℓ -1 - ℓ -1 k 0 (ℓ -1)
and this last condition is fulfilled in our present case so that the rate is smaller than 1/n.

In the same way, we can check that plugging the above m opt in the other terms let them less than 1/n. Namely

m 2aℓ+1 opt n ℓ = n - 2αℓk 0 +ℓ-k 0 2ak 0 +2αk 0 +1 ≤ n -1 , m 2ak-2α(ℓ-k)+1 opt n k = n - 2αk 0 ℓ+k-k 0 2ak 0 +2αk 0 +1 ≤ n -1 , for k = k 0 + j, j ≥ 1.
Consequently, we have

if 2α > 2a ℓ -1 + 1 ℓ -1 , then E( ĝℓ,m -g ℓ 2 ) ≤ Cn -1 .
Gathering both cases, we get that the rate for f ∈ S(α, L), f ε OS(a) is n -2αℓ/(2a+2α+1) ∨ n -1 . 5.3. Proof of Theorem 2.1. The result of Proposition 2.1 can be written

(20) E( ĝℓ,m -g ℓ 2 ) ≤ g ℓ -g ℓ,m 2 + V ℓ (m),
where V ℓ (m) is defined by ( 8). Let us define the oracle

m ⋆ = arg min m∈Mn -g ℓ,m 2 + V ℓ (m) + pen(m)
and note that, since g ℓg ℓ,m 2 = g ℓ 2g ℓ,m 2 , we also have

m ⋆ = arg min m∈Mn g ℓ -g ℓ,m 2 + V ℓ (m) + pen(m) .
Now, we start by writing that ( 21)

g ℓ -ĝℓ, m 2 ≤ 2 g ℓ -ĝℓ,m ⋆ 2 + 2 ĝℓ,m ⋆ -ĝℓ, m 2 . i) We consider the random set Ω = { m ≤ m ⋆ }, on which it holds that (22) ĝℓ,m ⋆ -ĝℓ, m 2 1 I Ω = ( ĝℓ,m ⋆ 2 -ĝℓ, m 2 )1 I Ω .
Now, the definition of m implies that

(23) -ĝℓ, m 2 + pen ℓ ( m) ≤ -ĝℓ,m ⋆ 2 + pen ℓ (m ⋆ ),
and thus

ĝℓ,m ⋆ 2 -ĝℓ, m 2 ≤ pen ℓ (m ⋆ ) -pen ℓ ( m) ≤ pen ℓ (m ⋆ ). (24)
Then, ( 22) and ( 24) imply ĝℓ,m ⋆ĝℓ, m 2 1 I Ω ≤ pen ℓ (m ⋆ ). Plugging this in ( 21) yields

E g ℓ -ĝℓ, m 2 1 I Ω ≤ 2E( g ℓ -ĝℓ,m ⋆ 2 ) + 2pen ℓ (m ⋆ ) ≤ 2( g ℓ -g ℓ,m ⋆ 2 + V ℓ (m ⋆ )) + 2pen ℓ (m ⋆ ) with (20), = 2 min m∈Mn g ℓ -g ℓ,m 2 + V ℓ (m) + pen ℓ (m) by definition of m ⋆ .
Thus we have

(25) E g ℓ -ĝℓ, m 2 1 I Ω ≤ 4 min m∈Mn g ℓ -g ℓ,m 2 + V ℓ (m) + pen ℓ (m) ,
which ends step i).

ii) Now, we study the bound on Ω c . Let us define, for k > m

pen ℓ (m, k) = 2 2ℓ λ ℓ (m, k) (∆ ℓ (k) -∆ ℓ (m)), λ ℓ (m, k) = max 2 2ℓ-1 log(1 + (k -m) 2 (∆ ℓ (k) -∆ ℓ (m)) ℓ , 2 4ℓ-1 n ℓ log(1 + (k -m) 2 (∆ ℓ (k) -∆ ℓ (m)) 2ℓ . First we write ĝℓ,m ⋆ -ĝℓ, m 2 1 I Ω c = ĝℓ,m ⋆ -ĝℓ, m 2 -2 2ℓ-1 g ℓ, m -g ℓ,m ⋆ 2 -pen ℓ (m ⋆ , m) 1 I Ω c + 2 2ℓ-1 g ℓ, m -g ℓ,m ⋆ 2 + pen ℓ (m ⋆ , m) 1 I Ω c ≤ sup k≥m ⋆ ĝℓ,m ⋆ -ĝℓ,k 2 -2 2ℓ-1 g ℓ,k -g ℓ,m ⋆ 2 -pen ℓ (m ⋆ , k) + 1 I Ω c +2 2ℓ-1 g ℓ -g ℓ,m ⋆ 2 + k≥m ⋆ pen ℓ (m ⋆ , k)1 I k= m. (26) Now, Inequality (23) writes 1 2 pen ℓ ( m) ≤ ĝℓ, m 2 -ĝℓ,m ⋆ 2 + pen ℓ (m ⋆ ) - 1 2 pen ℓ ( m) and we notice that, if k = m > m ⋆ , then ĝℓ, m 2 -ĝℓ,m ⋆ 2 = ĝℓ, m -ĝℓ,m ⋆ 2 and pen ℓ (m ⋆ , k) ≤ 2pen ℓ (k), so that 1 4 pen ℓ (m ⋆ , k) ≤ ĝℓ,k -ĝℓ,m ⋆ 2 - 1 4 pen ℓ (m, k) + pen ℓ (m ⋆ ) + 1 4 pen ℓ (m, k) - 1 2 pen ℓ (k) ≤ ĝℓ,k -ĝℓ,m ⋆ 2 - 1 4 pen ℓ (m, k) -2 2ℓ-1 g ℓ,m ⋆ -g ℓ,k 2 +pen ℓ (m ⋆ ) + 2 2ℓ-1 g ℓ,m ⋆ -g ℓ,k 2 ≤ sup k≥m ⋆ ĝℓ,k -ĝℓ,m ⋆ 2 - 1 4 pen ℓ (m, k) -2 2ℓ-1 g ℓ,m ⋆ -g ℓ,k 2 +pen ℓ (m ⋆ ) + 2 2ℓ-1 g ℓ,m ⋆ -g ℓ 2 .
It follows that, on Ω c ,

1 4 k>m ⋆ pen ℓ (m ⋆ , k)1 I m=k ≤ sup k>m ⋆ ĝℓ,k -ĝℓ,m ⋆ 2 - 1 4 pen ℓ (m, k) -2 2ℓ-1 g ℓ,m ⋆ -g ℓ,k 2 +pen ℓ (m ⋆ ) + 2 2ℓ-1 g ℓ,m ⋆ -g ℓ 2 .
Inserting this in (26

) implies ĝℓ,m ⋆ -ĝℓ, m 2 1 I Ω c ≤ 3 sup k≥m ⋆ ĝℓ,m ⋆ -ĝℓ,k 2 -2 2ℓ-1 g ℓ,k -g ℓ,m ⋆ 2 -pen ℓ (m ⋆ , k) + 1 I Ω c +3 2 2ℓ-1 g ℓ -g ℓ,m ⋆ 2 + 2pen ℓ (m ⋆ ). ( 27
)
We can prove the following proposition, see Section 5.4. Proposition 5.1. Under the assumptions of Theorem 2.1, there exists a constant K ℓ such that

E sup k>m ⋆ ĝℓ,m ⋆ -ĝℓ,k 2 -2 2ℓ-1 g ℓ,k -g ℓ,m ⋆ 2 -pen ℓ (m ⋆ , k) + ≤ K ℓ n ℓ .
Then Inequality ( 27) and Proposition 5.1 imply

E( ĝℓ,m ⋆ -ĝℓ, m 2 1 I Ω c ) ≤ 3 2 2ℓ-1 g ℓ -g ℓ,m ⋆ 2 + 2pen ℓ (m ⋆ ) + 3K ℓ n ℓ ≤ 4[ g ℓ -g ℓ,m ⋆ 2 + V ℓ (m ⋆ ) + pen ℓ (m ⋆ )] + 3K ℓ
n ℓ which gives the result on Ω c , by using again the definition of m ⋆ and plugging this in (21). Gathering the two steps i) and ii) gives the result of Theorem 2.1.

The rates of convergence follow from those of Theorem 2.1 and Corollary 2.1.

5.4. Proof of Proposition 5.1. The proof of Proposition 5.1 relies on the following Lemma:

Lemma 5.1. Let X 1 , . . . , X n be centered independent and identically distributed random variables, with |X i | ≤ b, ∀i a.s. and Var(X i ) ≤ v 2 . Then, for

S n = X 1 + • • • + X n , and H > 0, E S n n p -H p + ≤ 2 p Γ(p/2 + 1) n p/2 exp(-n2 2/p-2 H 2 /v 2 ) + 2 p Γ(p + 1) n p exp(-n2 1/p-2 H/b).
Proof of Lemma 5.1. The proof of Lemma 5.1 relies on Bernstein Inequality, which gives

P S n n ≥ x ≤ 2 exp - nx 2 /2 v 2 + bx ≤ 2 exp - nx 2 2v 2 + exp - nx 2b . Thus E S n n p -H p + ≤ +∞ 0 P S n n p -H p ≥ z dz ≤ +∞ 0 P S n n ≥ [H p + z] 1/p dz ≤ 2 +∞ 0 exp - n [H p + z] 2/p 2v 2 + exp - n [H p + z] 1/p 2b dz
Now we use that by concavity inequality, we have that, for q ≥ 1 and x, y ≥ 0, (x + y) 1/q ≥ 2 1/q-1 (x 1/q + y 1/q ), which implies for p ≥ 2,

E S n n p -H p + ≤ 2 exp(-n2 2/p-2 H 2 /v 2 ) +∞ 0 exp -n 2 2/p-1 z 2/p 2v 2 dz +2 exp(-n2 1/p-2 H/c) +∞ 0 exp -n 2 1/p-1 z 1/p 2b dz = 2 p Γ(p/2 + 1) n p/2 exp(-n2 2/p-2 H 2 /v 2 ) + 2 p Γ(p + 1) n p exp(-n2 1/p-2 H/b), which is the announced result. . Proof of Proposition 5.1. For k > m ⋆ , we have ĝℓ,k -ĝℓ,m ⋆ 2 = πm ⋆ ≤|u|≤πk f * Y (u) f * ε (u) 2ℓ du ≤ πm ⋆ ≤|u|≤πk 2 2ℓ-1 | f * Y (u) -f * Y (u)| 2ℓ |f * ε (u)| 2ℓ du + πm ⋆ ≤|u|≤πk 2 2ℓ-1 |f * Y (u)| 2ℓ |f * ε (u)| 2ℓ du = πm ⋆ ≤|u|≤πk 2 2ℓ-1 | f * Y (u) -f * Y (u)| 2ℓ |f * ε (u)| 2ℓ du + 2 2ℓ-1 g ℓ,k -g ℓ,m ⋆ 2 . Therefore E sup k>m ⋆ ĝℓ,m ⋆ -ĝℓ,k 2 -2 2ℓ-1 g ℓ,k -g ℓ,m ⋆ 2 -pen ℓ (m ⋆ , k) + ≤ E sup k>m ⋆ πm ⋆ ≤|u|≤πk 2 2ℓ-1 | f * Y (u) -f * Y (u)| 2ℓ |f * ε (u)| 2ℓ du -pen ℓ (m ⋆ , k) + ≤ k>m ⋆ E πm ⋆ ≤|u|≤πk 2 2ℓ-1 | f * Y (u) -f * Y (u)| 2ℓ |f * ε (u)| 2ℓ du -pen ℓ (m ⋆ , k) + = k>m ⋆ E πm ⋆ ≤|u|≤πk 2 2ℓ-1 |f * ε (u)| 2ℓ | f * Y (u) -f * Y (u)| 2ℓ - λ ℓ (m ⋆ , k) n ℓ + du . (28)
We apply now Lemma 5.1 with b = v = 1 and

H 2ℓ ≥ 2 2ℓ-1 n ℓ log(1 + (k -m ⋆ ) 2 (∆ ℓ (k) -∆ ℓ (m ⋆ ))
ℓ in the first exponential term of the bound, and

H 2ℓ ≥ 2 4ℓ-1 n 2ℓ log(1 + (k -m ⋆ ) 2 (∆ ℓ (k) -∆ ℓ (m ⋆ )) 2ℓ
in the second term. We obtain

E | f * Y (u) -f * Y (u)| 2ℓ - λ ℓ (m ⋆ , k) n ℓ + ≤ 2 2ℓ ℓ! n ℓ 1 (k -m ⋆ ) 2 (∆ ℓ (k) -∆ ℓ (m ⋆ )) + 2 2ℓ (2ℓ)! n 2ℓ 1 (k -m ⋆ ) 2 (∆ ℓ (k) -∆ ℓ (m ⋆ ))
.

Plugging this into (28) yields

E sup k>m ⋆ ĝℓ,m ⋆ -ĝℓ,k 2 -2 2ℓ-1 g ℓ,k -g ℓ,m ⋆ 2 -pen ℓ (m ⋆ , k) + ≤ 2 2ℓ-1 2 2ℓ ℓ! n ℓ + 2 2ℓ (2ℓ)! n 2ℓ k>m ⋆ 1 (k -m ⋆ ) 2 ≤ π 2 6 2 4ℓ (2ℓ)! n ℓ .
This ends the proof Proposition 5.1.

5.5. Proof of Theorem 3.1. Let us first introduce some notations.

For k > m, set ∆(m, k) = ∆(k) -∆(m) and ∆f (m, k) = ∆f (k) -∆f (m), qen(m, k) := 2 2ℓ+1 λ2 1 (m, k) ∆(m, k) n ℓ + 2 2ℓ+1 κ 2ℓ log ℓ (M (k -m)) ∆f (m, k) M ℓ with λ1 (m, k) = max 2 -1/ℓ+2 log 1 + ∆(m, k)(k -m) 2 , 2 -1/2ℓ+2 √ n log 1 + ∆(m, k)(k -m) 2 .
Now we can start the proof of Theorem 3.1. We denote by m ⋆ the oracle cutoff defined by

m ⋆ = arg min m∈Mn -g ℓ,m 2 + V ℓ (m) + qen ℓ (m) ,
where V ℓ (m) is defined by ( 14). We have

(29) g ℓ -gℓ, m 2 ≤ 2 g ℓ -gℓ,m ⋆ 2 + 2 gℓ,m ⋆ -gℓ, m 2 • Let us notice on the set G = { m ≤ m ⋆ } : gℓ,m ⋆ -gℓ, m 2 1 G = gℓ,m ⋆ 2 -gℓ, m 2 1 G .
Besides according to the definition of m, one has the following inequality:

(30) -gℓ, m 2 + qen ℓ ( m) ≤ -gℓ,m ⋆ 2 + qen ℓ (m ⋆ ), which implies -gℓ, m 2 ≤ -gℓ,m ⋆ 2 + qen ℓ (m ⋆ ). Thus gℓ,m ⋆ -gℓ, m 2 1 G = gℓ,m ⋆ 2 -gℓ, m 2 1 G ≤ qen ℓ (m ⋆ ).
Taking expectation, we apply the following Lemma Lemma 5.2. There exists a positive constant C such that for any arbitrary m ∈ M n

(31) E [ qen ℓ (m)] ≤ Cqen ℓ (m).
It yields for some positive constant C

E g ℓ -gℓ, m 2 1 G ≤ 2E g ℓ -gℓ,m ⋆ 2 + 2E [ qen ℓ (m ⋆ )] ≤ 2 g ℓ -g ℓ,m ⋆ 2 + 2V ℓ (m ⋆ ) + 2Cqen ℓ (m ⋆ ).
We just proved the result on G

(32) E g ℓ -gℓ, m 2 1 G ≤ C inf m∈Mn g ℓ -g ℓ,m 2 + V ℓ (m) + qen ℓ (m) .
• We now consider the set

G c = { m > m ⋆ }. gℓ, m -gℓ,m ⋆ 2 1 G c = gℓ, m -gℓ,m ⋆ 2 -4 g ℓ, m -g ℓ,m ⋆ 2 - 1 2 qen(m ⋆ , m) 1 G c + 4 g ℓ, m -g ℓ,m ⋆ 2 + 1 2 qen(m ⋆ , m) 1 G c ≤ sup k≥m ⋆ k∈Mn gℓ,k -gℓ,m ⋆ 2 -4 g ℓ,k -g ℓ,m ⋆ 2 - 1 2 qen(m ⋆ , k) + + 4 g ℓ -g ℓ,m ⋆ 2 + 1 2 k≥m ⋆ k∈Mn qen(m ⋆ , k)1 { m = k} . (33)
Let us first notice the following inequality

∀k > m, qen(m, k) ≤ qen ℓ (k). (34)
Besides by definition of m (see ( 15)), on the set { m = k} ∩ G c and applying (30), we have

1 2 ( qen ℓ (k) -qen ℓ (m ⋆ )) ≤ gℓ, m -gℓ,m ⋆ 2 - 1 2 ( qen ℓ (k) -qen ℓ (m ⋆ )) so that 1 2 qen ℓ (k) ≤ gℓ, m -gℓ,m ⋆ 2 - 1 2 qen ℓ (m ⋆ , k) + 1 2 qen(m ⋆ ) ≤ gℓ, m -gℓ,m ⋆ 2 -4 g ℓ, m -g ℓ,m ⋆ 2 - 1 2 qen(m ⋆ , k) + 4 g ℓ, m -g ℓ,m ⋆ 2 + 1 2 qen ℓ (m ⋆ ) ≤ gℓ, m -gℓ,m ⋆ 2 -4 g ℓ, m -g ℓ,m ⋆ 2 - 1 2 qen(m ⋆ , k) + 4 g ℓ -g ℓ,m ⋆ 2 + 1 2 qen ℓ (m ⋆ ) (35)
Now using Inequalities (34) and ( 35)

1 2 k≥m ⋆ k∈Mn qen(m ⋆ , k) ≤ sup k≥m ⋆ k∈Mn gℓ,k -gℓ,m ⋆ 2 -4 g ℓ, m -g ℓ,m ⋆ 2 - 1 2 qen(m ⋆ , k) + + 4 g ℓ -g ℓ,m ⋆ 2 + 1 2 qen ℓ (m ⋆ ).
From Inequality (33), we now have

gℓ, m -gℓ,m ⋆ 2 1 G c ≤ 2 sup k≥m ⋆ k∈Mn gℓ,k -gℓ,m ⋆ 2 -4 g ℓ,k -g ℓ,m ⋆ 2 - 1 2 qen(m ⋆ , k) + +8 g ℓ -g ℓ,m ⋆ 2 + 1 2 qen ℓ (m ⋆ ).
Taking expectation the first summand is negligible by applying the following Proposition.

Proposition 5.2. Under (A1) and (A2), there is a positive constant C such that for any arbitrary m ∈ M n (36)

E    sup k≥m k∈Mn gℓ,k -gℓ,m 2 -2 2ℓ g ℓ,k -g ℓ,m 2 - 1 2 qen(m, k) +    ≤ C n ℓ + C M ℓ .
Finally gathering the last result with (29), we have

E g ℓ -gℓ, m 2 1 G c ≤ C g ℓ -g ℓ,m ⋆ 2 + V ℓ (m ⋆ ) + qen(m ⋆ ) + C ′ 1 n ℓ + 1 M ℓ (37)
This combining with (32) complete the proof. 5.6. Proof of Lemma 5.2. Before proving Lemma 5.2, we first need to prove two auxiliary lemmas. In the sequel, C will always denote some universal positive constant, but the value may vary from line to line.

Lemma 5.3. For an estimator of f * ε defined by ( 10), assume κ > √ c 1 p. Let τ ≥ 2κ and x ≥ 1.

Then for some positive constant C

P ∃u ∈ R : | f * ε (u) -f * ε (u)| > τ (log(M x)) 1/2 w(u) -1 M -1/2 ≤ Cx -p M -p
Proof of Lemma 5.3. We write the decomposition

f * ε (u) -f * ε (u) ≤ f * ε (u) -f * ε (u) + f * ε (u) -f * ε (u) ≤ 2k M (u) + f * ε (u) -f * ε (u) .
Using the previous inequality and applying Lemma 5.5 in [START_REF] Kappus | Adaptive nonparametric estimation for Lévy processes observed at low frequency[END_REF], we have

P ∃u ∈ R : | f * ε (u) -f * ε (u)| > τ (log(M x)) 1/2 w(u) -1 M -1/2 ≤ P ∃u ∈ R : | f * ε (u) -f * ε (u)| + 2k M (u) > τ (log(M x)) 1/2 w(u) -1 M -1/2 ≤ P ∃u ∈ R : | f * ε (u) -f * ε (u)| > (τ -2κ) (log(M x)) 1/2 w(u) -1 M -1/2 ≤ Cx -p M -p .
Proof of Lemma 5.2. For q = 1/2 or 1, using Cauchy-Schwarz's inequality, we have

E log q 1 + ∆(m)m 2 ∆(m) ≤ E log 2q 1 + ∆(m)m 2 E ∆2 (m) Let A p (x) = 1 f * ε (x) - 1 f * ε (x) 2p 1 |f * ε (x)| 2p ∧ k 2p M (x) |f * ε (x)| 4p ∆(m) = 1 2π πm -πm w(u) -2 | f * ε (u)| 2ℓ du = 1 2π πm -πm w(u) -2ℓ 1 f * ε (u) - 1 f * ε (u) + 1 f * ε (u) 2ℓ du ≤ 2∆(m) + 1 π πm -πm w(u) -2ℓ 1 f * ε (u) - 1 f * ε (u) 2ℓ du ≤ 2∆(m) + 2∆(m) sup u∈R A ℓ (u) ≤ 2∆(m)(1 + sup u∈R A ℓ (u))
and applying Lemma 4.1 in [START_REF] Kappus | Adaptive density estimation in deconvolution problems with unknown error distribution[END_REF], we get E ∆2 (m) ≤ 4C∆ 2 (m). By Jensen inequality (since log is concave)

E log 2q 1 + ∆(m)m 2 ≤ log 2q E 1 + ∆(m)m 2 ≤ log 2q 1 + E ∆(m) m 2 ≤ log 2q 1 + 2∆(m) 1 + E sup u∈R A ℓ (u) m 2 ≤ log 2q 1 + 2∆(m) (1 + C) m 2 ≤ C log 2q 1 + ∆(m)m 2 So E log q 1 + ∆(m)m 2 ∆(m) ≤ C log q 1 + ∆(m)m 2 ∆(m)
which means E[ qen 1 (m)] ≤ Cqen 1 (m). Consider now qen 2 (m). Another application of Lemma 4.1 in [START_REF] Kappus | Adaptive density estimation in deconvolution problems with unknown error distribution[END_REF] yields

1 M ℓ E 1 2π πm -πm w(u) -2ℓ | f * Y (u)| 2ℓ | f * ε (u)| 4ℓ du ≤ 2 M ℓ E 1 2π πm -πm w(u) -2ℓ |f * Y (u)| 2ℓ | f * ε (u)| 4ℓ du + 2 M ℓ E 1 2π πm -πm w(u) -2ℓ | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 4ℓ du ≤ 4 M ℓ E 1 2π πm -πm w(u) -2ℓ |f * (u)| 2ℓ |f * ε (u)| 2ℓ du E 1 + sup u∈R A 2ℓ (u) + 2 M ℓ E 1 2π πm -πm w(u) -2ℓ | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 4ℓ du ≤ 4 M ℓ ∆ f (m) E 1 + sup u∈R A 2ℓ (u) + 2 M ℓ E 1 2π πm -πm w(u) -2ℓ | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 4ℓ du
We use the fact that f * ε (u) ≥ M -1/2 (log M ) 1/2 w(u) -1 , w(u) ≤ 1 as well as the independence of f * Y and f * ε to find

1 M ℓ E 1 2π πm -πm w(u) -2ℓ | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 4ℓ du = 1 M ℓ 1 2π πm -πm E | f * Y (u) -f * Y (u)| 2ℓ E w(u) -2ℓ | f * ε (u)| 4ℓ du ≤ c ℓ n ℓ E 1 2π πm -πm w(u) -2ℓ | f * ε (u)| 2ℓ
38) E sup k≥m 1 2π A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du - 1 2 2ℓ+1 qen 1 (m, k) + ≤ C n ℓ Proof of Lemma 5.4. E    sup k≥m k∈Mn 1 2π A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du - 1 2 2ℓ+1 qen 1 (m, k) +    ≤ E     k≥m k∈Mn E 1 2π A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du - 1 2 2ℓ+1 qen 1 (m, k) + f * ε     ≤ E     k≥m k∈Mn 1 2π A(m,k) E | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ - 1 2 2ℓ+1 λ2 (m, k) n ℓ | f * ε (u)| 2ℓ + f * ε du     Now f * Y (u)/ f * ε (u) (conditional on f * ε (u)) is the sum of n i.i.d. random variables with variance v 2 ≤ 1/| f * ε (u)| 2 which are bounded by 2/ f * ε (u). Thus Lemma 5.1 gives E | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ - 1 2 2ℓ+1 λ2 (m, k) n ℓ | f * ε (u)| 2ℓ + f * ε ≤ 2 ℓ Γ(2 ℓ-1 + 1) n -ℓ | f * ε (u)| 2ℓ exp -λ2 (m, k) + 2 ℓ Γ(2 ℓ + 1) n -2ℓ | f * ε (u)| 2ℓ exp -n 1/2 λ(m, k) ≤ 2 ℓ Γ(2 ℓ-1 + 1) n -ℓ | f * ε (u)| 2ℓ (k -m) -2 ∆(m, k) -1 + 2 ℓ Γ(2 ℓ + 1) n -2ℓ | f * ε (u)| 2ℓ (k -m) -2 ∆(m, k) -1 (
where we used the fact that

λ(m, k) ≤ max 2 -1/ℓ+2 log 1 + ∆(m, k)(k -m) 2 , 2 -1/2ℓ+2 √ n log 1 + ∆(m, k)(k -m) 2 .
We have thus shown for a universal positive constant C that, for any m

, k ∈ M n A(m,k) E | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ - 1 2 ℓ+1 λ2 (m, k) n ℓ | f * ε (u)| 2ℓ + f * ε du ≤ C n ℓ (k -m) -2 ∆(m, k) -1 A(m,k) du | f * ε (u)| 2ℓ ≤ C n ℓ (k -m) -2 . Finally E sup k≥m {(2π) -1 A(m,k) (| f * Y (u) -f * Y (u)| 2ℓ )/| f * ε (u)| 2ℓ du -2 -2ℓ+1 qen 1 (m, k)} + ≤ C/n ℓ ,
= 1 2π A(m,k) | f * Y (u))| 2ℓ | f * ε (u)| 2ℓ du = 1 2π A(m,k) | f * Y (u))| 2ℓ | f * ε (u)| 2ℓ 1 {| f * ε (u)|>|f * ε (u)|} du + 1 2π A(m,k) | f * Y (u))| 2ℓ | f * ε (u)| 2ℓ 1 {| f * ε (u)|≤|f * ε (u)|} du := S 1 + S 2 Let us consider S 1 . S 1 ≤ 2 2ℓ-1 π A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du + 2 2ℓ-1 π A(m,k) |f * Y (u)| 2ℓ |f * ε (u)| 2ℓ du 2 2ℓ g ℓ,k -g ℓ,m 2 , ( 39
) now S 2 , S 2 = 1 2π A(m,k) | f * Y (u)| 2ℓ 1 f * ε (u) - 1 f * ε (u) + 1 f * ε (u) 2ℓ 1 {| f * ε (u)|>|f * ε (u)|} du ≤ 2 2ℓ-1 π A(m,k) | f * Y (u)| 2ℓ 1 f * ε (u) - 1 f * ε (u) 2ℓ 1 {| f * ε (u)≤|f * ε (u)|} du + 2 2ℓ-1 π A(m,k) f * Y (u)) f * ε (u) 2ℓ 1 {| f * ε (u)≤|f * ε (u)|} du := S 2,1 + S 2,2 . (40)
Yet for the second term of (40), we can notice

(41) S 2,2 ≤ 2 4ℓ-2 π A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du + 2 4ℓ-1 g ℓ,k -g ℓ,m 2 .
For the first term of (40) we can write

S 2,1 = 2 2ℓ-1 π A(m,k) | f * Y (u)| 2ℓ | f * ε (u) -f * ε (u)| 2ℓ | f * ε (u)| 2ℓ |f * ε (u)| 2ℓ 1 {| f * ε (u)≤|f * ε (u)|} du ≤ 2 2ℓ-1 π A(m,k) | f * Y (u)| 2ℓ | f * ε (u) -f * ε (u)| 2ℓ | f * ε (u)| 4ℓ 1 {| f * ε (u)≤|f * ε (u)|} du. ( 42
)
Let us introduce the set following set

(43) C(m, k) = ∀u ∈ R : | f * ε (u) -f * ε (u)| 2 ≤ 4κ 2 log (M (k -m)) w(u) -2 M -1
On C(m, k), the following inequalities can be deduced

1 2π A(m,k) | f * Y (u)| 2ℓ | f * ε (u) -f * ε (u)| 2ℓ | f * ε (u)| 4ℓ 1 {| f * ε (u)≤|f * ε (u)|} du ≤ 4 ℓ κ 2ℓ log ℓ (M (k -m)) M -ℓ 1 2π A(m,k) w(u) -2ℓ | f * Y (u)| 2ℓ | f * ε (u)| 4ℓ 1 {| f * ε (u)|≤|f * ε (u)|} du ≤ 4 ℓ κ 2ℓ log ℓ (M (k -m)) M -ℓ ∆f (m, k) ≤ 1 2 2ℓ+1 qen 2 (m, k).
Gathering Equations ( 39), ( 41) and ( 42), we have gℓ,kgℓ,m 2 = 2 4ℓ-2 π

A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du + 2 4ℓ-1 g ℓ,k -g ℓ,m 2 + 2 2ℓ-1 π A(m,k) | f * Y (u)| 2ℓ | f * ε (u) -f * ε (u)| 2ℓ | f * ε (u)| 4ℓ 1 {| f * ε (u)≤|f * ε (u)|} du [1 C(m,k) + 1 C(m,k) c ] ≤ 2 4ℓ-2 π A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du + 2 4ℓ-1 g ℓ,k -g ℓ,m 2 + 1 2 qen 2 (m, k) + 2 2ℓ-1 π A(m,k) | f * Y (u)| 2ℓ | f * ε (u) -f * ε (u)| 2ℓ | f * ε (u)| 4ℓ 1 {| f * ε (u)≤|f * ε (u)|} du 1 C(m,k) c . ( 44
)
Starting from (44), we can now write the following inequalities gℓ,kgℓ,m 2 -2 4ℓ-1 g ℓ,kg ℓ,m 2 -

1 2 qen(m, k) ≤ 2 4ℓ-2 π A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du - 1 2 qen 1 (m, k) + 2 2ℓ-1 π A(m,k) | f * Y (u)| 2ℓ | f * ε (u) -f * ε (u)| 2ℓ | f * ε (u)| 4ℓ 1 {| f * ε (u)≤|f * ε (u)|} du 1 C(m,k) c ≤ 2 2ℓ 1 2π A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du - 1 2 2ℓ+1 qen 1 (m, k) + + 2 2ℓ-1 π A(m,k) | f * Y (u)| 2ℓ | f * ε (u) -f * ε (u)| 2ℓ | f * ε (u)| 4ℓ 1 {| f * ε (u)≤|f * ε (u)|} du 1 C(m,k) c .
Taking expectation we get

E    sup k≥m k∈Mn gℓ,k -gℓ,m 2 -2 4ℓ-1 g ℓ,k -g ℓ,m 2 - 1 2 qen(m, k) +    ≤ k≥m k∈Mn E 2 2ℓ 1 2π A(m,k) | f * Y (u) -f * Y (u)| 2ℓ | f * ε (u)| 2ℓ du - 1 2 2ℓ+1 qen 1 (m, k) + + 2 2ℓ 2π k≥m k∈Mn E A(m,k) | f * Y (u)| 2ℓ | f * ε (u) -f * ε (u)| 2ℓ | f * ε (u)| 4ℓ 1 {| f * ε (u)≤|f * ε (u)|} du 1 C(m,k) c .
We can notice that on C(m, k) c defined by (43) following Lemma 5.3 for x = km and p = 3ℓ, we have P [C(m, k) c ] ≤ M -3ℓ (km) -3ℓ and we get This completes the proof.

k≥m k∈Mn E A(m,k) | f * Y (u)| 2ℓ | f * ε (u) -f * ε (u)| 2ℓ | f * ε (u)| 4ℓ 1 {| f * ε (u)≤|f * ε (u)|} du 1 C(m,k) c ≤ k≥m k∈Mn E A(m,k) | f * Y (u)| 2ℓ 2 2ℓ |f * ε (u)| 2ℓ k 4ℓ M (u) 1 {| f * ε (u)≤|f * ε (u)|} du 1 C(m,k) c

Appendix

We recall the following result, which can be found in [START_REF] Shiryaev | Probability. Translated from the first[END_REF]

U j | p ) ≤ B p E(| n j=1 |U j | 2 | p/2 ),
We also give the following useful lemma, see Lemma 2 p. 35 in [START_REF] Butucea | Sharp optimality in density deconvolution with dominating bias[END_REF]. For two functions u, v, we denote u(x) v(x) if there exists a positive constant C not depending on x such that u(x) ≤ Cv(x) and u(x) ≈ v(x) if u(x) v(x) and v(x) u(x). Lemma 6.2. Consider c, s nonnegative real numbers, and γ a real such that 2γ > -1 if c = 0 or s = 0. Then, for all m > 0,

• m -m (x 2 + 1) γ exp(c|x| s )dx ≈ m 2γ+1-s e cm s , and if in addition 2γ > 1 if c = 0 or s = 0, • ∞ m (x 2 + 1) -γ exp(-c|x| s )dx ≈ m -2γ+1-s e -cm s .

Figure 1 .

 1 Figure1. Test densities f (solid) and g 2 (dashed).

Figure 2 .

 2 Figure 2. Laplace noise. True density (dotted), density estimates (gray) and sample of 20 estimates (thin gray) out of 50 proposed to the selection algorithm obtained with a sample of n = 1000 data.

Figure 3 .

 3 Figure 3. Gaussian noise. True density (dotted), density estimates (gray) and sample of 20 estimates (thin gray) out of 50 proposed to the selection algorithm obtained with a sample of n = 1000 data.

Figure 4 .

 4 Figure 4. Laplace noise, Kurtotic density. (a): Graph of the function ĝℓ,m 2 + pen ℓ (m) against the smoothing parameter m. (b): MISE(m). The gray diamond represents the global minimizer of MISE(m) and the gray circle represents the global minimizer ofĝℓ,m 2 + pen ℓ (m). 5. Proofs 5.1. Proof of Proposition 2.1. Upper bound on the MISE of ĝℓ,m . Let us introduce the function g ℓ,m defined by

  du where c ℓ is the constant of the Rosenthal inequality. Applying the same arguments as for the bounding of E [ qen 1 (m)], we get E du ≤ C∆(m). This completes the proof. 5.7. Proof of Proposition 5.2. For k > m, let us introduce the following notation : A(m, k) := {u ∈ R, |u| ∈ [πm, πk]}. We need to prove auxiliary lemmas before proving Proposition 5.2. Lemma 5.4. There is a positive constant C such that for any arbitrary m ∈ M n

2

  2ℓ κ -4ℓ (log M ) -2ℓ w(u) 4ℓ M 2ℓ du 1 C(m,k) c ≤ k≥m k∈Mn 2 2ℓ κ -4ℓ (log M ) -2ℓ M 2ℓ (km)P [C(m, k) c ]

Table 1 .

 1 Examples of rates of convergence ϕ n of the estimator ĝℓ,

	m

Table 2 .

 2 1000×MISE values from 100 replications of sample sizes n = 1000

			Noise
		Gaussian	Laplace
		known unknown known unknown
	Kurtotic 0.9200	0.4842	0.6678	0.4919
	Gaussian 0.0285	0.0397	0.0350	0.0389
	Uniform 1.4902	1.0301	1.1795	1.0300
	Claw	0.1130	0.0494	0.0556	0.0485
	types of errors. As noted in

  which ends the proof.

	Proof of Proposition 5.2. Applying Plancherel's formula we get
	gℓ,k -gℓ,m	2

  (inequality (26) p.498). Lemma 6.1 (Marcinkiewicz & Zygmund Inequality). Let n be a positive integer, p > 1 and U 1 , . . . , U n be n zero mean independent random variables such that sup j∈{1,...,n} E(|U j | p ) < ∞. Then, for B p = (18p)

p (p/(p -1)) p/2 , E(| n j=1


Caroline Bauzet
email: caroline.bauzet@univ-amu.fr

Guy Vallet
email: guy.vallet@univ-pau.fr

Petra Wittbold
email: petra.wittbold@uni-due.de

A degenerate parabolic-hyperbolic Cauchy problem with a stochastic force

Keywords: Stochastic PDE, degenerate parabolic-hyperbolic equation, Cauchy problem, multiplicative stochastic perturbation, Carrillo-Kruzhkov's entropy. MSC code: 35K65, 60H15, 35L65

In this paper we are interested in the Cauchy problem for a nonlinear degenerate parabolic-hyperbolic problem with multiplicative stochastic forcing. Using an adapted entropy formulation a result of existence and uniqueness of a solution is proved.

Introduction

In this paper, we are interested in the formal multi-dimensional (d ≥ 1) stochastic nonlinear degenerate parabolic problem of type:

(P ) : du -∆φ(u)dt -div( f (u))dt = g(x, u)dt + h(x, u)dw in Ω × Q, u(0, •) = u 0 in R d
where, in the sequel we assume that T is a positive number, Q =]0, T [×R d and that W = {w t , F t ; 0 ≤ t ≤ T } denotes a standard adapted one-dimensional continuous Brownian motion, defined on the classical Wiener space (Ω, F, P ). These assumptions on W are made for convenience.

Let us assume that H 1 : φ : R → R is a Lipschitz-continuous function and φ(0) = 0.

problem in R d modulo the existence of what they have called a "strong-entropy" solution 3 and the existence of such a solution in R. This study has been revisited by G.-Q. Chen, Q. Ding and K. H. Karlsen [START_REF] Chen | On nonlinear stochastic balance laws[END_REF] where they proved the existence of a strong-entropy solution in the multidimensional case by using BV information on the initial condition.

The first general result of existence and uniqueness has been proposed by A. Debussche and J. Vovelle [START_REF] Debussche | Scalar conservation laws with stochastic forcing[END_REF]. The problem is posed in a torus and the technique is based on the kinetic formulations associated to the equation. C. Bauzet, G. Vallet and P. Wittbold proposed in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] a similar result by using Feng and Nualart's entropy formulation for the Cauchy problem in R d in the framework of the Young measure theory. The same authors gave a similar result for the Dirichlet problem in [START_REF] Bauzet | The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation[END_REF].

To our knowledge, the only actual result concerning the case of a strongly degenerate parabolic-hyperbolic stochastic is a preprint of A. Debussche, M. Hofmanova and J. Vovelle extending the kinetic formulation in a torus of [START_REF] Debussche | Scalar conservation laws with stochastic forcing[END_REF]. In this present paper, we propose to extend the previous paper [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] to the context of a degenerate parabolic-hyperbolic problem in the spirit of J. Carrillo's work [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF] and revisited by G.-Q. Chen and K.-H. Karlsen [START_REF] Chen | Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients[END_REF]. Again, the existence of a solution is proved by using a vanishing viscosity method based on the compactness proposed by the theory of Young measures. The uniqueness of the solution is obtained via Kruzhkov's doubling variable method.

The paper is organized as follows. After this introductory part where we present some notations, we will present the entropy formulation, the definition of a solution and state the main result: the existence and uniqueness of the solution and some stability inequalities. Section 3 is devoted to the technical part of the paper where we show the existence of a solution and the uniqueness is presented in Section 4; followed by the last one containing technical lemmata.

Let us now introduce some notations and make precise the functional setting.

In the sequel we denote by H 1 (R d ) the usual Sobolev space.

We recall that H 1 (R d ) is also the closure of D(R d ), the space of C ∞ (R d )functions with compact support in R d . We denote by H -1 (R d ) the dual space of H 1 (R d ) which is also the space of derivatives of order less than one of elements of L 2 (R d ) in the common Gelfand-Lions identification

H 1 (R d ) ֒→ L 2 (R d ) ≡ L 2 (R d ) ′ ֒→ H 1 (R d ) ′ .
For any positive M , denote by Q M =]0, T [×B(0, M ) where B(0, M ) is the bounded open ball in R d of radius M .

In general, if G ⊂ R k , D(G) denotes the restriction to G of D(R k ) functions u such that support(u)∩G is compact. Then, D + (G) will denote the subset of non-negative elements of D(G).

For a given separable Banach space X we denote by N 2 w (0, T, X) the space of the predictable X-valued processes (cf. [START_REF] Da Prato | Encyclopedia of Mathematics and its Applications[END_REF] p.94 or [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF] p.28 for example). This space is the space L 2 (]0, T [×Ω, X) for the product measure dt⊗dP on P T , the predictable σ-field (i.e. the σ-field generated by the sets {0} × F 0 and the rectangles ]s, t] × A for any A ∈ F s ), with the L 2 (]0, T [×Ω, X)norm. If X = L 2 (R d ), one gets that N 2 w (0, T, L 2 (R d )) ⊂ L 2 (Q × Ω). We denote by E the set of non-negative even convex function in C 2,1 (R) approximating the absolute-value function, such that η(0) = 0 and that there exists τ > 0 such that η ′ (x) = 1 (resp. -1) if x > τ (resp. x < -τ ). Then, η ′′ has a compact support in [-τ, τ ]and η and η ′ are Lipschitz-continuous functions. A typical element of E is the function denoted by η τ such that

η ′ τ (r) = 1 + sin( π 2τ (2r -τ )) 2 if 0 ≤ r ≤ τ and η ′ τ (r) = 1 if r > τ .
For convenience, denote by sgn 0 (x) = x |x| if x = 0 and 0 otherwise;

F (a, b) = sgn 0 (a -b)[ f (a) -f (b)] and F η (a, b) = a b η ′ (σ -b) f ′ (σ)dσ.
Note, in particular, that F and F η are Lipschitz-continuous functions.

Denote also:

φ η (a, b) = a b η ′ (σ -b)φ ′ (σ)dσ and G(x) = x 0 φ ′ (s)ds.

Towards an entropy formulation and definition of a solution

Following the method proposed in G. Vallet [START_REF] Vallet | Stochastic perturbation of nonlinear degenerate parabolic problems[END_REF] 1 , for any ǫ > 0, there exists

a unique solution u in N 2 w (0, T, H 1 (R d )) with ∂ t (u - t 0 h(x, u)dw) in L 2 (Ω × (0, T ), H -1 (R d ))
, to Problem:

(P ǫ ) :    ∂ t u - t 0 h(x, u)dw -ǫ∆u -∆φ(u) -div f (u) = g(x, u) in Ω × Q u(0, •) = u 0 in R d Note that one has u ∈ L 2 (Ω, C([0, T ], L 2 (R d )).
Then, a slight modification of the Itô's formula proposed in D. Fellah and E. Pardoux [START_REF] Fellah | Une formule d'Itô dans des espaces de Banach et application[END_REF], for any ϕ

∈ D + ([0, T [×R d ), any reals v, k, η ∈ E and H(v, k) = η(v -k), yields (denote by Q t = (0, t) × R d ) R d H(u(t), k)ϕ(t)dx - R d H(u 0 , k)ϕ(0)dx + ǫ Qt ∇u∇[η ′ (u -k)ϕ]dxds + Qt ∇φ(u)∇[η ′ (u -k)ϕ]dxds + Qt f (u)∇[η ′ (u -k)ϕ]dxds = Qt H(u, k)∂ t ϕdxds + Qt η ′ (u -k)ϕh(x, u)dw(s)dx + 1 2 Qt η ′′ (u -k)[h(x, u)] 2 ϕdxds + Qt η ′ (u -k)g(x, u)ϕdxds.

Then, since

Qt

η ′′ (u -k) f (u)∇uϕdxds + Qt η ′ (u -k) f (u)∇ϕdxds = Qt F η (u, k)∇ϕdxds,
the following equality holds:

R d H(u(t), k)ϕ(t)dx + ǫ Qt η ′′ (u -k)|∇u| 2 ϕdxds + Qt η ′′ (u -k)φ ′ (u)|∇u| 2 ϕdxds = -ǫ Qt η ′ (u -k)∇u∇ϕdxds + R d H(u 0 , k)ϕ(0)dx + Qt H(u, k)∂ t ϕ -η ′ (u -k)∇φ(u)∇ϕ -F η (u, k)∇ϕ dxds + Qt η ′ (u -k)ϕh(x, u)dw(s)dx + Qt [η ′ (u -k)g(x, u) + 1 2 η ′′ (u -k)[h(x, u)] 2 ]ϕdxds. Since φ η (x, k) = x k η ′ (σ -k)φ ′ (σ)dσ and G(x) = x 0 φ ′ (s)ds, one gets that R d H(u(t), k)ϕ(t)dx + ǫ Qt η ′′ (u -k)|∇u| 2 ϕdxds + Qt η ′′ (u -k)|∇G(u)| 2 ϕdxds = -ǫ Qt η ′ (u -k)∇u∇ϕdxds + R d H(u 0 , k)ϕ(0)dx (1)
+ Qt H(u, k)∂ t ϕ + φ η (u, k)∆ϕ -F η (u, k)∇ϕ dxds + Qt η ′ (u -k)ϕh(x, u)dw(s)dx + Qt [η ′ (u -k)g(x, u) + 1 2 η ′′ (u -k)[h(x, u)] 2 ]ϕdxds.
Note that the second integral on the left hand side is non-negative. Moreover, one might expect that the first integral term on the right hand side of the equation tends to 0 as ǫ tends to 0.

Therefore, if we can show that the solutions of (P ǫ ) converge in an appropriate sense to a function u as ǫ tends to 0, the limit function will satisfy the entropy inequality [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] where ǫ = 0 and the equality sign is replaced by an inequality.

So we propose

Definition 1 A solution to Problem (P) is any

u ∈ N 2 w (0, T, L 2 (R d ))∩L ∞ (0, T, L 2 (Ω× R d )) such that G(u) ∈ L 2 ((0, T ) × Ω, H 1 (R d ))
and satisfying, a.s. the entropy formulation:

∀k ∈ R, ∀ϕ ∈ D + ([0, T [×R d ), ∀η ∈ E, Q H(u, k)∂ t ϕ + φ η (u, k)∆ϕ -F η (u, k)∇ϕ + η ′ (u -k)g(x, u)ϕ dxds + Q η ′ (u -k)ϕh(x, u)dw(s)dx + 1 2 Q η ′′ (u -k)[h(x, u)] 2 ϕdxds ≥ Q η ′′ (u -k)|∇G(u)| 2 ϕdxds - R d H(u 0 , k)ϕ(0)dx.
Let us first make some remarks on the definition.

Remark 1 1. Note that if G(u) ∈ L 2 ((0, T ) × Ω, H 1 (R d )), then φ(u) ∈ L 2 ((0, T ) × Ω, H 1 (R d ))
and, thanks to Lemma 3 (see Section 5), the entropy inequality is equivalent to

Q H(u, k)∂ t ϕ + φ η (u, k)∆ϕ -F η (u, k)∇ϕ + η ′ (u -k)g(x, u)ϕ dxds + Q η ′ (u -k)ϕh(x, u)dw(s)dx + 1 2 Q η ′′ (u -k)[h(x, u)] 2 ϕdxds. ≥ Q |∇ u 0 η ′′ (σ -k)G ′ (σ)dσ| 2 ϕdxds - R d H(u 0 , k)ϕ(0)dx 2.
Following Remark 2.6 in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF], one has that a solution in the sense of the above definition is also, a.s., a weak solution of (P).

3. Following now Remark 2.7 of the same paper, one gets that a solution u in the sense of the above definition satisfies ess lim

t→0 + E K |u -u 0 |dx = 0 for any compact K of R d , but also, ess lim t→0 + E R d |u -u 0 |ϕ(x)dx = 0 for any ϕ ∈ L 2 (R d ).
Let us also remark that any solution u belongs to

L 2 [(0, T ), L 2 (Ω × R d )],
and it is the same for u -t 0 h(x, u)dw(s) thanks to the properties of the Itô integral. As u is also a weak solution of (P),

∂ t [u - t 0 h(x, u)dw(s)] belongs to L 2 [(0, T ), L 2 (Ω, H -1 (R d ))]. Thus, u ∈ C([0, T ], L 2 (Ω, H -1 (R d ))). Since by definition u belongs to L ∞ (0, T, L 2 (Ω × R d ))
, Lemma 1.4 p.263 of [START_REF] Temam | Navier-Stokes equations[END_REF] yields: u is weakly continuous in time with values in L 2 (Ω × R d ).

Let us now present the main result of the paper.

Theorem 1

Under the assumptions H 1 to H 4 , there exists a unique solution in the sense of Definition 1.

Considering two initial conditions u 0,2 , u 0,1 , the corresponding solutions u 2 , u 1 and the weight α(x) = min(1, R a |x| a ) where R > 1 and a > d/2, there exists c > 0 such that for any positive t:

E R d |u 2 (t, x) -u 1 (t, x)|α(x)dx ≤ e ct R d |u 0,2 (x) -u 0,1 (x)|α(x)dx.
Moreover, if the initial conditions and g(•, 0) are also elements of L1 (R d ) and h(•, 0) = 0, then the solutions are in L ∞ (0, T, L 1 (Ω × R d )) and one has for any t,

(u 2 -u 1 )(t) L 1 (Ω×R d ) ≤ e ct u 0,2 -u 0,1 L 1 (R d ) .

Existence of a solution

Let us denote in the sequel u ǫ the solution of Problem (P ǫ ) with initial condition

u ǫ 0 ∈ D(R d ) that converges to a given u 0,2 in L 2 (R d ) and consider u δ a solution of Problem (P δ ) with initial condition u δ 0 ∈ D(R d ) that converges to a given u 0,1 in L 2 (R d ).
Based on the Kruzhkov's doubling variables method, our aim in this section is formally to pass to the limit when ǫ and δ go to 0 in a Kato's inequality. The compactness we use is the one given by the theory of Young measures and the classical uniqueness method for entropy solutions ensures the uniqueness of the limit point of the sequence of viscous approximation. This then yields the convergence of the whole sequence to an entropy solution in the sense of Definition 1. To prove such Kato's inequality, [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] used that ∆u δ ∈ L 2 (Ω × Q). In the present case, such a regularity is not possible to obtain and one needs to regularize u δ by convolution. Then, for a given mollifier-sequence ρ θ in R d , using in the equation satisfied by u δ the test function ϕ * ρ θ for any ϕ ∈ D(R d+1 ), one gets that u δ * ρ θ is a solution to the stochastic problem 1 :

u δ * ρ θ (t = 0) = u δ 0 * ρ θ and ∂ t [u δ * ρ θ - t 0 h(x, u δ ) * ρ θ dw] -[δ∆(u δ * ρ θ ) + ∆(φ(u δ ) * ρ θ ) + div f (u δ ) * ρ θ ] = g(x, u δ ) * ρ θ .
Note in particular that this problem is posed in L 2 (R d ) and not anymore in

H -1 (R d ).
Then, for any ϕ ∈ D([0, T [×R d ) (when needed in the sequel, one denotes by K the support of ϕ) any real k, the Itô formula applied to H(u δ * ρ θ (t, x), k)ϕ(t, x) where η ∈ E and H(v, k) = η(v -k), yields a.e.

H(u δ * ρ θ (t), k)ϕ(t) -H(u δ 0 * ρ θ , k)ϕ(0) -δ t 0 ∆[u δ * ρ θ ][η ′ (u δ * ρ θ -k)ϕ]ds - t 0 ∆[φ(u δ ) * ρ θ ][η ′ (u δ * ρ θ -k)ϕ]ds - t 0 div[ f (u δ ) * ρ θ ][η ′ (u δ * ρ θ -k)ϕ]ds = t 0 H(u δ * ρ θ , k)∂ t ϕds + t 0 η ′ (u δ * ρ θ -k)ϕ[h(x, u δ ) * ρ θ ]dw(s) + t 0 [η ′ (u δ * ρ θ -k)g(x, u δ ) * ρ θ + 1 2 η ′′ (u δ * ρ θ -k)[h(x, u δ ) * ρ θ ] 2 ]ϕds i.e., by integrating over Q δ Q η ′′ (u δ * ρ θ -k)|∇[u δ * ρ θ ]| 2 ϕdxds + δ Q η ′ (u δ * ρ θ -k)∇[u δ * ρ θ ]∇ϕdxds + Q η ′′ (u δ * ρ θ -k)∇[φ(u δ ) * ρ θ ]∇[u δ * ρ θ ]ϕdxds + Q η ′ (u δ * ρ θ -k)∇[φ(u δ ) * ρ θ ]∇ϕdxds + Q η ′′ (u δ * ρ θ -k)[ f (u δ ) * ρ θ ]∇[u δ * ρ θ ]ϕdxds + Q η ′ (u δ * ρ θ -k)[ f (u δ ) * ρ θ ]∇ϕdxds = Q H(u δ * ρ θ , k)∂ t ϕdxds + Q η ′ (u δ * ρ θ -k)ϕ[h(x, u δ ) * ρ θ ]dw(s)dx + R d H(u δ 0 * ρ θ , k)ϕ(0)dx + Q [η ′ (u δ * ρ θ -k)g(x, u δ ) * ρ θ + 1 2 η ′′ (u δ * ρ θ -k)[h(x, u δ ) * ρ θ ] 2 ]ϕdxds.
Or, if one agrees to denote, for any

v in L 2 (R d ), v * ρ θ by v θ , δ Q η ′′ (u δ θ -k)|∇u δ θ | 2 ϕdxds + Q η ′′ (u δ θ -k)∇φ(u δ ) θ ∇u δ θ ϕdxds = -δ Q η ′ (u δ θ -k)∇u δ θ ∇ϕdxds + R d H(u δ 0 * ρ θ , k)ϕ(0)dx + Q H(u δ θ , k)∂ t ϕ -η ′ (u δ θ -k)∇φ(u δ ) θ ∇ϕ -η ′ (u δ θ -k)[ f (u δ ) θ ]∇ϕdxds - Q η ′′ (u δ θ -k)[ f (u δ ) θ ]∇u δ θ ϕdxds + Q η ′ (u δ θ -k)ϕh(x, u δ ) θ dw(s)dx + Q [η ′ (u δ θ -k)g(x, u δ ) θ + 1 2 η ′′ (u δ θ -k)[h(x, u δ ) θ ] 2 ]ϕdxds. (2)
In the sequel of this section, unless for the two integrals with ǫ as a factor term2 , we will present the proofs in such a way that it can also be done for a entropy solution u (i.e. ǫ = 0). The main regularity difference between u ǫ and u is that

u ǫ ∈ H 1 (R d ) while u ∈ L 2 (R d ) with G(u) ∈ H 1 (R d ).
So we need to use carefully a chain-rule; instead of the classical one, we will use a generalized chain-rule (see Lemma 3). Let ψ be an element of

D + ([0, T [ 2 ×R 2d ). The idea in the sequel is to replace ψ(t, s, x, y) by ϕ(t, x)ρ n (t -s)ρ m (x -y) for a given ϕ ∈ D + ([0, T [×R) and mollifier sequences ρ n in time with supp ρ n ⊂ [ -2
n , 0] and ρ m in space with sufficiently large n and m. Thus, multiplying (1) at time t = T by ρ l [u δ * ρ θ (s, y) -k] and integrating the result over R × Q for the variables k, s and y, yields

ǫ R×Q 2 η ′′ (u ǫ -k)|∇u ǫ | 2 ψρ l [u δ θ (s, y) -k]dkdxdtdsdy + R×Q 2 η ′′ (u ǫ -k)|∇G(u ǫ )| 2 ψρ l [u δ θ (s, y) -k]dkdxdtdsdy = -ǫ R×Q 2 η ′ (u ǫ -k)∇u ǫ ∇ x ψρ l [u δ θ (s, y) -k]dkdxdtdsdy + R×R d ×Q H(u ǫ 0 , k)ψ(0)ρ l [u δ θ (s, y) -k]dkdxdsdy + R×Q 2 H(u ǫ , k)∂ t ψ + φ η (u ǫ , k)∆ x ψ -F η (u ǫ , k)∇ x ψ + η ′ (u ǫ -k)g(x, u δ ) θ ϕ ×ρ l [u δ θ (s, y) -k]dkdxdtdsdy + R×Q 2 η ′ (u ǫ -k)ψh(x, u ǫ )dw(t)ρ l [u δ θ (s, y) -k]dkdxdsdy + 1 2 R×Q 2 η ′′ (u ǫ -k)[h(x, u ǫ )] 2 ψρ l [u δ θ (s, y) -k]dkdtdxdsdy.
Similarly, considering (2) and multiplying by ρ l [u ǫ (t, x) -k] and integrating with respect to k, x and t, δ

R×Q 2 η ′′ (u δ θ -k)|∇u δ θ | 2 ψρ l [u ǫ (t, x) -k]dydsdkdxdt + R×Q 2 η ′′ (u δ θ -k)∇φ(u δ ) θ ∇u δ θ ρ l [u ǫ (t, x) -k]ψdydsdkdxdt = -δ R×Q 2 η ′ (u δ θ -k)∇u δ θ ∇ y ψρ l [u ǫ (t, x) -k]dydsdkdxdt + Q×R×R d H(u δ 0,θ , k)ψ(0)ρ l [u ǫ (t, x) -k]dydkdxdt + R×Q 2 H(u δ θ , k)∂ s ψ -η ′ (u δ θ -k)∇φ(u δ ) θ ∇ y ψ -η ′ (u δ θ -k)[ f (u δ ) θ ]∇ y ψ +η ′ (u δ θ -k)[g(y, u δ ) θ ]ψ ρ l [u ǫ (t, x) -k]dydsdkdxdt - R×Q 2 η ′′ (u δ θ -k)[ f (u δ ) θ ]∇u δ θ ψρ l [u ǫ (t, x) -k]dydsdkdxdt + R×Q 2 η ′ (u δ θ -k)ψh(y, u δ ) θ dw(s)ρ l [u ǫ (t, x) -k]dy + 1 2 R×Q 2 η ′′ (u δ θ -k)[h(y, u δ ) θ ] 2 ψρ l [u ǫ (t, x) -k]dsdydkdxdt.
Adding the two equations, by grouping similar terms together, we get:

ǫ R×Q 2 η ′′ (u ǫ -k)|∇u ǫ | 2 ψρ l [u δ θ (s, y) -k]dxdtdkdyds +δ R×Q 2 η ′′ (u δ θ -k)|∇u δ θ | 2 ψρ l [u ǫ (t, x) -k]dydsdkdxdt + Q×R×Q η ′′ (u ǫ -k)|∇G(u ǫ )| 2 ψρ l [u δ θ (s, y) -k]dxdtdkdyds + Q×R×Q η ′′ (u δ θ -k)[∇φ(u δ ) θ ∇u δ θ ]ψρ l [u ǫ (t, x) -k]dydsdkdxdt = -ǫ R×Q 2 η ′ (u ǫ -k)∇u ǫ ∇ x ψρ l [u δ θ (s, y) -k]dxdtdkdyds -δ R×Q 2 η ′ (u δ θ -k)∇u δ θ ∇ y ψρ l [u ǫ (t, x) -k]dydsdkdxdt + Q×R×R d H(u ǫ 0 , k)ψ(t = 0)ρ l [u δ θ (s, y) -k]dxdkdyds + Q×R×R d H(u δ 0,θ , k)ψ(s = 0)ρ l [u ǫ (t, x) -k]dydkdxdt + R×Q 2 H(u ǫ , k)∂ t ψ + φ η (u ǫ , k)∆ x ψ -F η (u ǫ , k)∇ x ψ ×ρ l [u δ θ (s, y) -k]dxdtdkdyds + R×Q 2 H(u δ θ , k)∂ s ψ -η ′ (u δ θ -k)∇φ(u δ ) θ ∇ y ψ -η ′ (u δ θ -k)[ f (u δ ) θ ∇ y ψ] ρ l [u ǫ (t, x) -k]dydsdkdxdt - R×Q 2 η ′′ (u δ θ -k)[ f (u δ ) θ ∇u δ θ ]ψρ l [u ǫ (t, x) -k]dydsdkdxdt + 1 2 R×Q 2 η ′′ (u ǫ -k)[h(x, u ǫ )] 2 ψρ l [u δ θ (s, y) -k]dtdxdkdyds + 1 2 R×Q 2 η ′′ (u δ θ -k)[h(y, u δ ) θ ] 2 ψρ l [u ǫ (t, x) -k]dydsdkdxdt + R×Q 2 η ′ (u ǫ -k)ψh(x, u ǫ )dw(t)ρ l [u δ θ (s, y) -k]dxdkdyds + R×Q 2 η ′ (u δ θ -k)ψh(y, u δ ) θ dw(s)ρ l [u ǫ (t, x) -k]dydkdxdt + R×Q 2 η ′ (u ǫ -k)ψg(x, u ǫ )dtρ l [u δ θ (s, y) -k]dxdkdyds + R×Q 2 η ′ (u δ θ -k)ψg(y, u δ ) θ dsρ l [u ǫ (t, x) -k]dydkdxdt,
i.e., I 1 + I 2 = I 3 + I 4 + I 5 + I 6 + I 7 + I 8 , where each I j denotes a sum of two corresponding integrals of the same type in the above equality.

Let us now study each of the terms I 1 , • • • , I 8 in detail. Our aim is to pass to the limit, successively with first n to infinity, then θ to 0, l to infinity, then ǫ, δ to 0. Then, depending on the situation (1 to 3), we pass to the limit with respect to τ to 0 (i.e. with η = η τ to the absolute-value function) and m to infinity, in an appropriate order.

In the sequel, we adopt the following notation: lim a,b means lim b lim a , also with lim sup or lim inf.

1) Since η is a convex function,

I 1 := ǫ Q 2 ×R η ′′ (u ǫ -k)|∇u ǫ | 2 ψρ l [u δ θ (s, y) -k]dxdtdkdyds +δ Q 2 ×R η ′′ (u δ θ -k)|∇u δ θ | 2 ψρ l [u ǫ (t, x) -k]dydsdkdxdt ≥ 0,
so, this term can be omitted in the sequel.

2) Remind that G(x) =

x 0 φ ′ (σ)dσ. Consider now

I 2 := Q×R×Q η ′′ (u ǫ -k)|∇G(u ǫ )| 2 ψρ l [u δ θ (s, y) -k]dxdtdkdyds + Q×R×Q η ′′ (u δ θ -k)[∇φ(u δ ) θ ∇u δ θ ]ψρ l [u ǫ (t, x) -k]dydsdkdxdt
Then, replacing ψ(t, s, x, y) by ϕ(t, x)ρ n (t -s)ρ m (x -y), classical properties of Lebesgue's points and convolution yield

lim n EI 2 = E Q×R×R d η ′′ (u ǫ -k)|∇G(u ǫ )| 2 ϕρ m (x -y)ρ l [u δ θ (t, y) -k]dxdtdkdy +E Q×R×R d η ′′ (u δ θ -k)[∇φ(u δ ) θ ∇u δ θ ]ϕρ m (x -y)ρ l [u ǫ (t, x) -k]dydkdxdt
Again, by properties of approximation by mollification,

G(u ǫ ), G(u δ ) ∈ L 2 (Ω × (0, T ); H 1 (R d ))
and since the nonlinear functions are bounded, one has lim n,θ

EI 2 = E Q×R×R d η ′′ (u ǫ -k)|∇G(u ǫ )| 2 ϕρ m (x -y)ρ l [u δ (t, y) -k]dxdtdkdy +E Q×R×R d η ′′ (u δ -k)[∇φ(u δ )∇u δ ]ϕρ m (x -y)ρ l [u ǫ (t, x) -k]dydkdxdt,
and, lim n,θ,l

EI 2 = E Q×R d η ′′ (u ǫ -u δ )|∇G(u ǫ )| 2 ϕρ m (x -y)dxdtdy +E Q×R d η ′′ (u δ -u ǫ )[∇φ(u δ )∇u δ ]ϕρ m (x -y)dydxdt = E Q×R d η ′′ (u ǫ -u δ )[|∇G(u ǫ )| 2 + |∇G(u δ )| 2 ]ϕρ m (x -y)dxdtdy
Now, following the idea of [START_REF] Chen | Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients[END_REF], one gets

Ĩ2 := E Q×R d η ′′ (u ǫ -u δ )[|∇G(u ǫ )| 2 + |∇G(u δ )| 2 ]ϕρ m (x -y)dxdtdy ≥ 2E Q×R d η ′′ (u ǫ (t, x) -u δ (t, y))∇G(u ǫ ).∇G(u δ )ϕρ m (x -y)dydxdt = 2E Q×R d η ′′ (u ǫ (t, x) -u δ (t, y)) φ ′ (u δ )∇ x G(u ǫ ).∇ y u δ ϕρ m (x -y)dydxdt = 2E Q×R d ∇ x G(u ǫ ).∇ y Ψ(u ǫ , u δ )ϕρ m (x -y)dydxdt := I 2 where Ψ(a, b) = b a η ′′ (a -σ) φ ′ (σ)dσ. Thus, Ĩ2 ≥ I 2 = -2E Q×R d Ψ(u ǫ , u δ )∇ x G(u ǫ ).∇ y [ϕρ m (x -y)]dydxdt.
Note that, for a fixed b, one has that

|Ψ(a, b)| ≤ φ ′ ∞ η ′ (|a -b|) is bounded by assumptions and Ψ(a, b) -Ψ(a 0 , b) ≤ b a [η ′′ (σ -a 0 ) -η ′′ (σ -b)] φ ′ (σ)dσ + a a0 η ′′ (σ -a 0 ) φ ′ (σ)dσ ≤ C(1 + |a -b|)|a -a 0 |.
Thus, for a fixed b, a → Ψ(a, b) is a continuous and bounded function, so, Lemma 3 and Green's formula yield:

I 2 = -2E Q×R d ∇ x u ǫ u δ Ψ(µ, u δ ) φ ′ (µ)dµ .∇ y [ϕρ m (x -y)]dydxdt = 2E Q×R d u ǫ u δ Ψ(µ, u δ ) φ ′ (µ)dµ div x ∇ y [ϕρ m (x -y)]dydxdt = 2E Q×R d u ǫ u δ u δ µ η ′′ (µ -σ) φ ′ (σ)dσ φ ′ (µ)dµ div x ∇ y [ϕρ m (x -y)]dydxdt.
In the sequel, we pass to the limit with δ and ǫ to zero in the sense of Young measures as in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF]. This Young measure can be written as a function of the same variables, plus an additional one living in (0, 1). To keep in mind the origin of the sequence, we denote by u 1 (•, δ) the first limit and by u 2 (•, ǫ) the second one. lim δ,ǫ

I 2 = 2E Q×R d ×(0,1) 2 u2(t,x,ǫ) u1(t,y,δ) u1(t,y,δ) µ η ′′ (µ -σ) φ ′ (σ)dσ φ ′ (µ)dµ ×div x ∇ y [ϕρ m (x -y)]dǫdδdydtdx.
3) Next, let us consider

I 3 := -ǫ Q×R×Q η ′ (u ǫ -k)∇u ǫ ∇ x ψρ l [u δ θ (s, y) -k]dxdtdkdyds -δ Q×R×Q η ′ (u δ θ -k)∇u δ θ ∇ y ψρ l [u ǫ (t, x) -k]dydsdkdxdt. |I 3 | ≤ ǫ Q×R×Q η ′ (u ǫ -k)∇u ǫ ∇ x ψρ l [u δ θ (s, y) -k]dxdtdkdyds +δ Q×R×Q η ′ (u δ θ -k)∇u δ θ ∇ y ψρ l [u ǫ (t, x) -k]dydsdkdxdt ≤ ǫ Q×R×Q ∇u ǫ ∇ x ψ ρ l [u δ θ (s, y) -k]dxdtdkdyds +δ Q×R×Q ∇u δ θ ∇ y ψ ρ l [u ǫ (t, x) -k]dydsdkdxdt = ǫ Q×Q ∇u ǫ ∇ x ψ dxdtdyds + δ Q×Q ∇u δ θ ∇ y ψ dydsdxdt ≤ ǫ Q ∇u ǫ (t, x) Q ∇ x ψ(t, x, s, y) dydsdxdt +δ Q ∇u δ θ (s, y) Q ∇ y ψ(t, x, s, y) dxdtdyds
Thus, replacing ψ(t, x, s, y) by ϕ(t, x)ρ n (t -s)ρ m (x -y),

|EI 3 | ≤ E|I 3 | ≤ ǫE K ∇u ǫ (t, x) Q ρ n (t -s) ϕ∇ x ρ m (x -y) + ρ m (x -y)∇ϕ dydsdxdt +δE K ∇u δ θ (s, y) Q ρ n (t -s) ϕ∇ y ρ m (x -y) dxdtdyds ≤ C(m, K) ǫ ∇u ǫ L 2 (Ω×Q) + δ ∇u δ L 2 (Ω×Q) ≤ C(m, K)[ √ ǫ + √ δ]
thanks to the a priori estimates (see Lemma 4). Therefore, lim n,θ,l,δ,ǫ EI 3 = 0.

4) Now let us consider the integrals coming from the initial conditions, i.e.

I 4 : = Q×R×R d H(u ǫ 0 , k)ψ(t = 0)ρ l [u δ θ (s, y) -k]dxdkdyds + Q×R×R d H(u δ 0,θ , k)ψ(s = 0)ρ l [u ǫ (t, x) -k]dydkdxdt.
If ψ(t, x, s, y) = ϕ(t, x)ρ n (t -s)ρ m (x -y), then

I 4 = Q×R×R d H(u ǫ 0 , k)ϕ(0, x)ρ n (-s)ρ m (x -y)ρ l [u δ θ (s, y) -k]dxdkdyds + Q×R×R d H(u δ 0,θ , k)ϕρ n (t)ρ m (x -y)ρ l [u ǫ (t, x) -k]dydkdxdt = Q×R×R d H(u ǫ 0 , k)ϕ(0, x)ρ n (-s)ρ m (x -y)ρ l [u δ θ (s, y) -k]dxdkdyds
as supp ρ n ⊂ [-2/n, 0], and then a slight modification of similar arguments in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] yields lim n,θ,l,δ,ǫ

EI 4 = R 2d ϕ(0, x)η(u 0,1 -u 0,2 )ρ m (x -y)dxdy.

5) Consider now

I 5 := Q×R×Q H(u ǫ , k)∂ t ψ + φ η (u ǫ , k)∆ x ψ -F η (u ǫ , k)∇ x ψ ×ρ l [u δ θ (s, y) -k]dxdtdkdyds + Q×R×Q H(u δ θ , k)∂ s ψ -η ′ (u δ θ -k)∇φ(u δ ) θ ∇ y ψ -η ′ (u δ θ -k)[ f (u δ ) θ ∇ y ψ] ×ρ l [u ǫ (t, x) -k]dydsdkdxdt - Q×R×Q η ′′ (u δ θ -k)[ f (u δ ) θ ∇u δ θ ]ψρ l [u ǫ (t, x) -k]dydsdkdxdt Since H(x, k) = η(x -k)
with an even function η,

I 5 = Q×R×Q H(u ǫ , u δ θ (s, y) -ζ)∂ t ψ + φ η (u ǫ , u δ θ (s, y) -ζ)∆ x ψ -F η (u ǫ , u δ θ (s, y) -ζ)∇ x ψ ρ l [ζ]dxdtdζdyds + Q×R×Q H(u δ θ , u ǫ (t, x) + ζ)∂ s ψ -η ′ (u δ θ -u ǫ (t, x) -ζ)∇φ(u δ ) θ ∇ y ψ -η ′ (u δ θ -u ǫ (t, x) -ζ)[ f (u δ ) θ ∇ y ψ] ρ l [u ǫ (t, x) -u ǫ (t, x) -ζ]dydsdζdxdt - Q 2 ×R η ′′ (u δ θ -u ǫ (t, x) -ζ)[ f (u δ ) θ ∇u δ θ ]ψρ l [u ǫ (t, x) -u ǫ (t, x) -ζ]dydsdζdxdt
Replacing ψ(t, s, x, y) by ϕ(t, x)ρ n (t -s)ρ m (x -y), one gets

I 5 = Q×R×Q H(u ǫ , u δ θ (s, y) -ζ)[∂ t ϕ]ρ n (t -s)ρ m (x -y) +φ η (u ǫ , u δ θ (s, y) -ζ)∆ x [ϕρ n (t -s)ρ m (x -y)] -F η (u ǫ , u δ θ (s, y) -ζ)∇ x [ϕρ n (t -s)ρ m (x -y)] ρ l [ζ]dxdtdζdyds - Q×R×Q η ′ (u δ θ -u ǫ (t, x) -ζ)∇φ(u δ ) θ ∇ y [ϕρ n (t -s)ρ m (x -y)] +η ′ (u δ θ -u ǫ (t, x) -ζ)[ f (u δ ) θ ∇ y [ϕρ n (t -s)ρ m (x -y)]] ρ l [ζ]dydsdζdxdt - Q 2 ×R η ′′ (u δ θ -u ǫ (t, x) -ζ)[ f (u δ ) θ ∇u δ θ ][ϕρ n (t -s)ρ m (x -y)]ρ l [ζ]dydsdζdxdt
Thus, passing to the limit with respect to n,

lim n EI 5 = E Q×R×R d H(u ǫ , u δ θ (t, y) -ζ)[∂ t ϕ]ρ m (x -y) +φ η (u ǫ , u δ θ (t, y) -ζ)∆ x [ϕρ m (x -y)] -F η (u ǫ , u δ θ (t, y) -ζ)∇ x [ϕρ m (x -y)] ρ l [ζ]dxdtdζdy -E Q×R×R d η ′ (u δ θ -u ǫ (t, x) -ζ)∇φ(u δ ) θ ∇ y [ϕρ m (x -y)] +η ′ (u δ θ -u ǫ (t, x) -ζ)[ f (u δ ) θ ∇ y [ϕρ m (x -y)]] ρ l [ζ]dydζdxdt -E Q×R×R d η ′′ (u δ θ -u ǫ (t, x) -ζ)[ f (u δ ) θ ∇u δ θ ][ϕρ m (x -y)]ρ l [ζ]dydζdxdt
and, passing to the limit with respect to θ,

lim n,θ EI 5 = E Q×R×R d H(u ǫ , u δ (t, y) -ζ)[∂ t ϕ]ρ m (x -y) +φ η (u ǫ , u δ (t, y) -ζ)∆ x [ϕρ m (x -y)] -F η (u ǫ , u δ (t, y) -ζ)∇ x [ϕρ m (x -y)] ρ l [ζ]dxdtdζdy -E Q×R×R d η ′ (u δ -u ǫ (t, x) -ζ)∇φ(u δ )∇ y [ϕρ m (x -y)] +η ′ (u δ -u ǫ (t, x) -ζ)[ f (u δ )∇ y [ϕρ m (x -y)]] ρ l [ζ]dydζdxdt -E Q×R×R d η ′′ (u δ -u ǫ (t, x) -ζ)[ f (u δ )∇u δ ][ϕρ m (x -y)]ρ l [ζ]dydζdxdt
Then, formulas of Green's type give lim n,θ,l

EI 5 = E Q×R d η(u ǫ -u δ )[∂ t ϕ]ρ m (x -y) dxdtdy + E Q×R d φ η (u ǫ , u δ (t, y))∆ x [ϕρ m (x -y)] + φ η (u δ , u ǫ (t, x))∆ y [ϕρ m (x -y)] dydxdt - E Q×R d F η (u ǫ , u δ (t, y))∇ x [ϕρ m (x -y)] + F η (u δ , u ǫ (t, x))∇ y [ϕρ m (x -y)] dydxdt.
Passing to the limits with respect to δ and ǫ gives lim n,θ,l,δ,ǫ

EI 5 = E Q×R d ×(0,1) 2 η[u 2 (t, x, ǫ) -u 1 (t, y, δ)]∂ t ϕρ m (x -y)dǫdδdxdtdy +E Q×R d ×(0,1) 2 φ η (u 2 (t, x, ǫ), u 1 (t, y, δ))∆ x [ϕρ m (x -y)] +φ η (u 1 (t, y, δ), u 2 (t, x, ǫ))∆ y [ϕρ m (x -y)] dǫdδdydxdt -E Q×R d ×(0,1) 2 F η (u 2 (t, x, ǫ), u 1 (t, y, δ))∇ x [ϕρ m (x -y)] +F η (u 1 (t, y, δ), u 2 (t, x, ǫ))∇ y [ϕρ m (x -y)] dǫdδdydxdt.
6) Let us now consider the additional deterministic integrals coming from the Itô integral formula:

I 6 := 1 2 Q×R×Q η ′′ (u ǫ -k)[h(x, u ǫ )] 2 ψρ l [u δ θ (s, y) -k]dtdxdkdyds + 1 2 Q×R×Q η ′′ (u δ θ -k)[h(y, u δ ) θ ] 2 ψρ l [u ǫ (t, x) -k]dydsdkdxdt.
Passing to the limit with respect to n, θ, then l, one obtains lim n,θ,l

EI 6 = 1 2 E Q×R d |h(x, u ǫ )| 2 + |h(y, u δ )| 2 ϕρ m (x -y)η ′′ (u ǫ -u δ )dxdtdy
Then, like in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF], we need to add this term to the one in item 7).

7) Now let us consider the stochastic Itô integral terms:

I 7 := Q×R×Q η ′ (u ǫ -k)ψh(x, u ǫ )dw(t)ρ l [u δ θ (s, y) -k]dxdkdyds + Q×R×Q η ′ (u δ θ -k)ψh(y, u δ ) θ dw(s)ρ l [u ǫ (t, x) -k]dydkdxdt
Taking the expectation, replacing ψ(t, s, x, y) by ϕ(t, x)ρ n (t -s)ρ m (x -y) and since the support of ρ n is negative, as already remarked in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF], the second integral vanishes and one gets that

EI 7 = E Q×R×Q η ′ (u ǫ -k)ψh(x, u ǫ )dw(t)ρ l [u δ θ (s, y) -k]dxdkdyds = E Q R R d s s-2/n η ′ (u ǫ (t, x) -k)ϕ(t, x)h(x, u ǫ (t, x))ρ n (t -s)dw(t) ×ρ m (x -y)ρ l [u δ θ (s, y) -k]dxdkdyds = E Q R R d F(x, s, k)ρ m (x -y)ρ l [u δ θ (s, y) -k]dxdkdyds = -E Q R R d F(x, s, k)ρ m (x -y) ∂ ∂k Sgn l [u δ θ (s, y) -k]dxdkdyds = E Q R R d ∂ ∂k F(x, s, k)ρ m (x -y)Sgn l [u δ θ (s, y) -k]dxdkdyds = E T 0 R R d ∂ ∂k F(x, s, k) R d Sgn l [u δ θ (s, y) -k] -Sgn l [u δ θ (s -2/n, y) -k] ×ρ m (x -y)dydxdkds,
where, for convenience, one denotes by Sgn l an antiderivative of ρ l and

F(x, s, k) = s s-2/n η ′ (u ǫ (t, x) -k)ϕ(t, x)h(x, u ǫ (t, x))ρ n (t -s)dw(t).

Thanks to Itô's formula, if one denotes by

A δ (s, y) = δ∇u δ θ + ∇φ(u δ ) θ + f (u δ ) θ (remind : du δ θ -div A δ dt = g(y, u δ ) θ dt + h(y, u δ ) θ dw), we find Sgn l [u δ θ (s, y) -k] -Sgn l [u δ θ (s -2/n, y) -k] ρ m (x -y) = s s-2/n div A δ Sgn ′ l [u δ θ (σ, y) -k]ρ m (x -y)] dσ + 1 2 s s-2/n Sgn ′′ l [u δ θ (σ, y) -k]ρ m (x -y)(h(y, u δ ) θ ) 2 dσ + s s-2/n Sgn ′ l [u δ θ (σ, y) -k]ρ m (x -y)h(y, u δ ) θ dw(σ) Since ∂ ∂k F(x, s, k) = - s s-2/n η ′′ (u ǫ (t, x) -k)ϕ(t, x)h(x, u ǫ (t, x))ρ n (t -s)dw(t)
(thanks to [START_REF] Kunita | Stochastic differential equations and stochastic flows of diffeomorphisms[END_REF](Theorem 7.6, p. 180)), following [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF], one gets that lim n,θ,l

EI 7 = -E Q×R d η ′′ (u ǫ -u δ )h(x, u ǫ )h(y, u δ )ρ m (x -y)dxdtdy. Therefore, lim n,θ,l EI 6 + lim n,θ,l EI 7 = 1 2 E Q×R d |h(x, u ǫ )| 2 + |h(y, u δ )| 2 ϕρ m (x -y)η ′′ (u ǫ -u δ )dxdtdy -E Q×R d η ′′ (u ǫ -u δ )h(x, u ǫ )h(y, u δ )ρ m (x -y)dxdtdy = 1 2 E Q×R d |h(x, u ǫ ) -h(y, u δ )| 2 ϕρ m (x -y)η ′′ (u ǫ -u δ )
EI 7 ] = 1 2 E Q×R d ×(0,1) 2 |h(x, u 2 (t, x, ǫ)) -h(y, u 1 (t, y, δ))| 2 ϕρ m (x -y)
×η ′′ (u 2 (t, x, ǫ) -u 1 (t, y, δ))dǫdδdxdtdy.

8)

Finally, let us consider the reaction terms:

I 8 := R×Q 2 η ′ (u ǫ -k)ψg(x, u ǫ )ρ l [u δ θ (s, y) -k] +η ′ (u δ θ -k)ψg(y, u δ ) θ ρ l [u ǫ (t, x) -k] dydsdkdxdt.
Classical convergence arguments for integrals yield lim n,θ,l Coming back to the contributions, we started with

EI 8 = E R d ×Q η ′ (u ǫ -u δ )ϕ[g(x, u ǫ ) -g(y, u δ )]ρ m (x -y)dydxdt ≤ E R d ×Q ϕ|g(x, u ǫ ) -g(y, u δ )|ρ m (x -y)dydxdt,
I 1 + I 2 = I 3 + I 4 + I 5 + I 6 + I 7 + I 8
to get, in a first step

EI 2 ≤ C(m, K)[ √ ǫ + √ δ] + EI 4 + EI 5 + EI 6 + EI 7 + EI 8 .
Then we can estimate

I 2 ≤ Ĩ2 = lim n,θ,l EI 2 ≤ C(m, K)[ √ ǫ + √ δ] + lim n,
EI 8 ≤ R 2d ϕ(0, x)η(u 0,1 -u 0,2 )ρ m (x -y)dxdy +E Q×R d ×(0,1) 2 η[u 2 (t, x, ǫ) -u 1 (t, y, δ)]∂ t ϕρ m (x -y)dǫdδdxdtdy +E Q×R d ×(0,1) 2 φ η (u 2 (t, x, ǫ), u 1 (t, y, δ))∆ x [ϕρ m (x -y)] +φ η (u 1 (t, y, δ), u 2 (t, x, ǫ))∆ y [ϕρ m (x -y)] dǫdδdydxdt -E Q×R d ×(0,1) 2 F η (u 2 (t, x, ǫ), u 1 (t, y, δ))∇ x [ϕρ m (x -y)] +F η (u 1 (t, y, δ), u 2 (t, x, ǫ))∇ y [ϕρ m (x -y)] dǫdδdydxdt + 1 2 E Q×R d ×(0,1) 2 |h(x, u 2 (t, x, ǫ)) -h(y, u 1 (t, y, δ))| 2 ϕρ m (x -y) ×η ′′ (u 2 (t, x, ǫ) -u 1 (t, y, δ))dxdtdy +E R d ×Q×(0,1) 2 ϕ|g(x, u 2 (t, x, ǫ)) -g(y, u 1 (t, y, δ))|ρ m (x -y)dδdǫdydxdt.
Developing terms we find

0 ≤ R 2d ϕ(0, x)η(u 0,1 -u 0,2 )ρ m (x -y)dxdy +E Q×R d ×(0,1) 2 η(u 2 (t, x, ǫ) -u 1 (t, y, δ))∂ t ϕρ m (x -y)dǫdδdxdtdy +E Q×R d ×(0,1) 2 φ η (u 2 (t, x, ǫ), u 1 (t, y, δ))∆ϕ -F η (u 2 (t, x, ǫ), u 1 (t, y, δ))∇ϕ ×ρ m (x -y)dǫdδdydxdt +2E Q×R d ×(0,1) 2 φ η (u 2 (t, x, ǫ), u 1 (t, y, δ))∇ϕ∇ρ m (x -y)dǫdδdydxdt +E Q×R d ×(0,1) 2 ϕ∆ρ m (x -y) φ η (u 2 (t, x, ǫ), u 1 (t, y, δ)) +φ η (u 1 (t, y, δ), u 2 (t, x, ǫ)) dǫdδdydxdt -E Q×R d ×(0,1) 2 ϕ F η (u 2 (t, x, ǫ), u 1 (t, y, δ)) -F η (u 1 (t, y, δ), u 2 (t, x, ǫ)) ×∇ρ m (x -y)dǫdδdydxdt + 1 2 E Q×R d ×(0,1) 2 |h(x, u 2 (t, x, ǫ)) -h(y, u 1 (t, y, δ))| 2 ϕρ m (x -y) ×η ′′ (u 2 (t, x, ǫ) -u 1 (t, y, δ))dxdtdy +E R d ×Q×(0,1) 2 ϕ|g(x, u 2 (t, x, ǫ)) -g(y, u 1 (t, y, δ))|ρ m (x -y)dδdǫdydxdt +2E Q×R d ×(0,1) 2 u2(t,x,ǫ) u1(t,y,δ) u1(t,y,δ) µ η ′′ (µ -σ) φ ′ (σ)dσ φ ′ (µ)dµ ×[∇ϕ∇ρ m (x -y) + ϕ∆ρ m (x -y)]dǫdδdydtdx.
Then, thanks to Lemma 1 and assumptions on h,

0 ≤ R 2d ϕ(0, x)|u 0,1 -u 0,2 |ρ m (x -y)dxdy (3)
+E

Q×R d ×(0,1) 2 |u 2 (t, x, ǫ) -u 1 (t, y, δ)|∂ t ϕρ m (x -y)dǫdδdxdtdy +E Q×R d ×(0,1) 2 |φ(u 2 (t, x, ǫ)) -φ(u 1 (t, y, δ))|∆ϕ -F (u 2 (t, x, ǫ), u 1 (t, y, δ))∇ϕ ρ m (x -y)dǫdδdydxdt +E R d ×Q×(0,1) 2 ϕ|g(x, u 2 (t, x, ǫ)) -g(y, u 1 (t, y, δ))|ρ m (x -y)dδdǫdydxdt +τ R d ϕ(0, x)dx + τ Q |∂ t ϕ| + c(φ)|∆ϕ| + c( f )|∇ϕ|dxdt +c(h)E Q×R d ×(0,1) 2 |u 2 (t, x, ǫ) -u 1 (t, y, δ)| 2 ϕρ m (x -y)η ′′ (u 2 (t, x, ǫ) -u 1 (t, y, δ))dxdtdy +c(h)E Q×R d ×(0,1) 2 |ω h ( x -y )| 2 ϕρ m (x -y)η ′′ (u 2 (t, x, ǫ) -u 1 (t, y, δ))dxdtdy -E Q×R d ×(0,1) 2 ϕ F η (u 2 (t, x, ǫ), u 1 (t, y, δ)) -F η (u 1 (t, y, δ), u 2 (t, x, ǫ)) ×∇ρ m (x -y)dǫdδdydxdt +2E Q×R d ×(0,1) 2 φ η (u 2 (t, x, ǫ), u 1 (t, y, δ))∇ϕ∇ρ m (x -y)dǫdδdydxdt +E Q×R d ×(0,1) 2 ϕ∆ρ m (x -y) φ η (u 2 (t, x, ǫ), u 1 (t, y, δ)) + φ η (u 1 (t, y, δ), u 2 (t, x, ǫ)) ×dǫdδdydxdt +2E Q×R d ×(0,1) 2 I τ (u 1 (t, y, δ), u 2 (t, x, ǫ))[∇ϕ∇ρ m (x -y) + ϕ∆ρ m (x -y)]dǫdδdydtdx = A 1 + A 2 + A 3 + A 4 + A 5 + A 6 + A 7 + A 8 + A 9 + A 10
where one sets, for any a, b,

I τ (a, b) := b a a µ η ′′ (µ -σ) φ ′ (σ)dσ φ ′ (µ)dµ.
Note that η ′′ (x) ≤ C/τ in [-τ, τ ] for a given constant, so that

|A 5 + A 6 | = c(h)E Q×R d ×(0,1) 2 |u 2 (t, x, ǫ) -u 1 (t, y, δ)| 2 ϕρ m (x -y)η ′′ (u 2 (t, x, ǫ) -u 1 (t, y, δ))dxdtdy +c(h)E Q×R d ×(0,1) 2 |ω h ( x -y )| 2 ϕρ m (x -y)η ′′ (u 2 (t, x, ǫ) -u 1 (t, y, δ))dxdtdy ≤ c(h)τ Q×R d ϕρ m (x -y)dxdtdy + c(h) τ Q×R d |ω h ( x -y )| 2 ϕρ m (x -y)dxdtdy ≤ c(h)τ Q ϕdxdt + c(h)|ω h ( 1 m )| 2 τ Q ϕdxdt.
Moreover,

|A 7 | = E Q×R d ×(0,1) 2 ϕ F η (u 2 (t, x, ǫ), u 1 (t, y, δ)) -F η (u 1 (t, y, δ), u 2 (t, x, ǫ)) ×∇ρ m (x -y)dǫdδdydxdt ≤ Cτ Q×R d ϕ|∇ρ m (x -y)|dydxdt ≤ τ mC Q ϕdxdt • First situation: h(x, u) = h(u). Then, ω h = 0 and, m being fixed, lim τ →0 A 4 + A 5 + A 6 + A 7 = 0. Moreover, A 8 + A 9 + A 10 = 2E Q×R d ×(0,1) 2 [I τ (u 1 (t, y, δ), u 2 (t, x, ǫ)) + φ η (u 2 (t, x, ǫ), u 1 (t, y, δ))] ×∇ϕ∇ρ m (x -y)dǫdδdydxdt +E Q×R d ×(0,1) 2 [2I τ (u 1 (t, y, δ), u 2 (t, x, ǫ)) + φ η (u 2 (t, x, ǫ), u 1 (t, y, δ)) +φ η (u 1 (t, y, δ), u 2 (t, x, ǫ))]ϕ∆ρ m (x -y)dǫdδdydxdt.
Note that, thanks to Lemma 2-(6), each integrand goes to 0 with τ and is bounded, respectively by

c(φ ′ )|u 2 (t, x, ǫ) -u 1 (t, y, δ)||∇ϕ∇ρ m (x -y)| and c(φ ′ )|u 2 (t, x, ǫ) -u 1 (t, y, δ)||ϕ∆ρ m (x -y)|.
Thus, one concludes that lim τ →0 A 8 + A 9 + A 10 = 0 and one can pass to the limit over m.

• Second situation: assume that φ = 0 or linear and that there exists θ ∈ (0, 1) such that ω h (r) 1+θ r → r→0 0 (this is the case for example if ω h (r) = |r| β for a given β > 1/2 by setting 1 > θ > (1 -β)/β). Then, A 8 + A 9 + A 10 = 0 and by setting τ = ω h (1/m) 1+θ , one has

|A 4 + A 5 + A 6 + A 7 + A 8 + A 9 + A 10 | ≤ C[τ + ω h (1/m) 2 τ ] + τ m],
and one concludes that lim m A 4 + A 5 + A 6 + A 7 + A 8 + A 9 + A 10 = 0.

• Last situation: assume the same for h, that φ is not linear and that t → φ ′ (t) has a modulus of continuity ω φ such that

ω φ [ω h (r) 1+θ ] r
→ r→0 0 (this is the case for example if ω φ (r) = C|r|). By using the classical form of the mollifier sequence 4 , where P (t) = (8 -2d)t 4 + 4(d -1)t 2 -2d. Note that there exists a ∈ (0, 1) such that P (t) ≤ 0 in [0, a] and P (t) ≥ 0 in [a, 1], so that with (5) (see Lemma 2), A 8 + A 9 + A 10 ≤ B where

ρ m (x) = cm d ρ(m x ) with ρ(t) = e 1 t 2 -1 1 {|t|<1} , one gets that ∇ρ m (x) = mρ m (x) -2m x (m 2 x 2 -1) 2 x x and ∆ρ m (x) = m 2 ρ m (x) P (m x ) (m 2 x 2 -1)
B := 2E Q×R d ×(0,1) 2 [I τ (u 1 (t, y, δ), u 2 (t, x, ǫ)) + φ η (u 2 (t, x, ǫ), u 1 (t, y, δ))] ×∇ϕ∇ρ m (x -y)dǫdδdydxdt +E [Q×R d ×(0,1) 2 ]∩{m x-y ∈[a,1]} [2I τ (u 1 (t, y, δ), u 2 (t, x, ǫ)) + φ η (u 2 (t, x, ǫ), u 1 (t, y, δ)) +φ η (u 1 (t, y, δ), u 2 (t, x, ǫ))]ϕ∆ρ m (x -y)dǫdδdydxdt.
Then, thanks to Lemma 2-( 7), one has

|B| ≤ Cτ Q×R d |∇ϕ∇ρ m (x -y)|dydxdt +Cω φ (τ ) 2 E Q×R d ×(0,1) 2 |u 2 (t, x, ǫ) -u 1 (t, y, δ)|[|∇ϕ∇ρ m (x -y)| +ϕ|∆ρ m (x -y)|1 {m x-y ∈[a,1]} ]dǫdδdydxdt. ≤ Cmτ + Cmω φ (τ ) 2 E Q×R d ×(0,1) 2 |u 2 (t, x, ǫ) -u 1 (t, y, δ)| 2 ρ m (x -y)dǫdδdydxdt × Q×R d ρ m (x -y)| 4m 2 x -y 2 |∇ϕ| 2 (m 2 x -y 2 -1) 4 dydxdt +Cm 2 ω φ (τ ) 2 E Q×R d ×(0,1) 2 ϕ|u 2 (t, x, ǫ) -u 1 (t, y, δ)| 2 ρ m (x -y)dǫdδdydxdt × Q×R d ρ m (x -y)| [ϕP (m x -y )] 2 (m 2 x -y 2 -1) 8 1 {m x-y ∈[a,1]} dydxdt ≤ C[mτ + m 2 ω φ (τ ) 2 ].
With the configuration of the previous situation, setting τ = ω h (1/m) 1+θ

and

|B| ≤ Cmτ + C mω φ (ω h (1/m) 1+θ 2 .
Then, with the assumption on the modulus ω φ , B converges to 0 when m goes to +∞.

Finally, whatever the situation, passing to the limit with respect to m, the following Kato inequality holds, for any ϕ

∈ D + ([0, T [×R d ), 0 ≤ R d |u 0,2 -u 0,1 |ϕ(0, x)dx +E Q×(0,1) 2 |u 2 (ǫ, •) -u 1 (δ, •)|∂ t ϕ + |g(•, u 2 (ǫ, •)) -g(•, u 1 (δ, •))|ϕ dδdǫdxdt +E Q×(0,1) 2 φ(u 2 (ǫ, •)) -φ(u 1 (δ, •)) ∆ϕ -F (u 2 (ǫ, •), u 1 (δ, •))∇ϕ dδdǫdxdt or, similarly, for any ϕ ∈ D + ([0, T [, H 1 (R d )), 0 ≤ R d |u 0,2 -u 0,1 |ϕ(0, x)dx +E Q×(0,1) 2 |u 2 -u 1 |∂ t ϕ + |g(•, u 2 ) -g(•, u 1 )|ϕ dδdǫdxdt -E Q ∇ (0,1) 2 φ(u 2 ) -φ(u 1 ) dδdǫ + (0,1) 2 F (u 2 , u 1 )dδdǫ ∇ϕdxdt
Following the idea of [START_REF] Andreianov | A note on uniqueness of entropy solutions to degenerate parabolic equations in R N[END_REF] |x| 2 is in L 2 ({|x| > R}). Thus,

0 ≤ R d |u 0,2 -u 0,1 |γ(0)α(x)dx +E Q×(0,1) 2 |u 2 -u 1 |γ ′ (t) + |g(•, u 2 ) -g(•, u 1 )|γ(t) α(x)dδdǫdxdt +E T 0 {|x>R|}×(0,1) 2 φ(u 2 ) -φ(u 1 ) ∆α(x) -F (u 2 , u 1 )∇α(x) γ(t)dδdǫdxdt -E T 0 γ(t) ∂{|x|>R}×(0,1) 2 φ(u 2 ) -φ(u 1 ) ∇α. ndδdǫdσdt. Since ∇α(x). n = -a R x. n = a R > 0 on ∂{|x| > R}, this yields 0 ≤ R d |u 0,2 -u 0,1 |γ(0)α(x)dx +E Q×(0,1) 2 |u 2 -u 1 |γ ′ (t) + |g(•, u 2 ) -g(•, u 1 )|γ(t) α(x)dδdǫdxdt +E T 0 {|x>R|}×(0,1) 2 α(x)γ(t) a(2+2ǫ-a) |x| 2 φ(u 2 )-φ(u 1 ) + a |x| F (u 2 , u 1 ). x |x| dδdǫdxdt,
and as |x| > R in the last integral,

0 ≤ R d |u 0,2 -u 0,1 |γ(0)α(x)dx +E Q×(0,1) 2 |u 2 -u 1 |γ ′ (t) + |g(•, u 2 ) -g(•, u 1 )|γ(t) α(x)dδdǫdxdt +C(d) R + 1 R 2 E T 0 {|x>1|}×(0,1) 2 α(x)γ(t) φ(u 2 ) -φ(u 1 ) + |F (u 2 , u 1 )| dδdǫdxdt.
Using now that R ≥ 1 and the Lipschitz properties of φ, f and g,

0 ≤ R d |u 0,2 -u 0,1 |γ(0)α(x)dx +E Q×(0,1) 2 |u 2 -u 1 |α(x) γ ′ (t) + C(d, φ, f , g)γ(t) dδdǫdxdt.
Assume now, by an approximation argument, that γ(t) = e -ct min(1, n(T -t) + ) where c = C(d, φ, f , g) + 1, then

n T T -1/n E R d ×(0,1) 2 |u 2 -u 1 |α(x)e -ct dδdǫdxdt +E Q×(0,1) 2 |u 2 -u 1 |α(x)e -ct dδdǫdx min(1, n(T -t) + )dt ≤ R d |u 0,2 -u 0,1 |α(x)dx. (4)
Thus, if one assumes that u 0,2 = u 0,1 , passing to the limit over n yields,

E Q×(0,1) 2 |u 2 -u 1 |α(x)e -ct dδdǫdxdt ≤ 0.
This means on the one hand that u 1 and u 2 are the same functions, but also on the other hand that they are not functions of the additional variables ǫ and δ respectively. Thus, one is able to conclude that the whole sequence of viscous approximation converges, weakly in L 2 (Ω × Q) and strongly in L p (Ω × (0, T ), L p loc (R d )) for any p < 2 to a weak entropy solution u in the sense of our definition. Then, back to (4), one gets by passing to the limit over n,

lim inf n n T T -1/n E R d |u 2 -u 1 |α(x)e -ct dxdt +E Q |u 2 -u 1 |α(x)e -ct dxdt ≤ R d |u 0,2 -u 0,1 |α(x)dx. Thanks to Remark 1, t → u 2 -u 1 is weakly continuous with values in L 2 (Ω×R d ) and since u ∈ L 2 (Ω × R d ) → E R d
|u|αdx is a non-negative convex continuous function, it is l.s.c. for the weak topology and

E R d |u 2 -u 1 |(T )α(x)dx + E Q |u 2 -u 1 |(t)α(x)e c(T -t) dxdt ≤ e cT R d |u 0,2 -u 0,1 |α(x)dx.
Since the time T is arbitrary, this last assertion closes the proof of the existence of a solution, limit of the viscous approximation and the stability of such solutions in L 1 (R, αdx). After the proof of the uniqueness of the solution in the sense of Definition 1 (see next section) this will prove the first part of the theorem.

Assume now that the initial conditions and g(•, 0) are elements of L 1 (R d ) and also h(•, 0) = 0. Thus, thanks to Remark 2, the corresponding solutions are in L ∞ (0, T, L 1 (Ω × R d )). Then, the above estimate yields

E R d |u 2 -u 1 |(T ) min(1, R a |x| a )dx ≤ e cT R d |u 0,2 -u 0,1 | min(1, R a |x| a )dx ≤ e cT R d |u 0,2 -u 0,1 |dx.
Thus, by Beppo Levi's theorem, one concludes that (u 2 -u 1 )(t), a priori element of L 2 (Ω × R d ), is an element of L 1 (Ω × R d ) with the information that

E R d |u 2 -u 1 |(T )dx ≤ e cT R d |u 0,2 -u 0,1 |dx.

Uniqueness of the solution

Our aim is to prove, as in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF], that any solution in the sense of our definition is unique by proving that it is equal to the solution obtained by viscous approximation. The method used to prove this result is exactly the same as the one proposed in the section dedicated to the result of existence, considering a solution u (i.e. ǫ = 0) and u δ . Coming back to the proofs, the only difference lies in the terms I 1 and I 3 where one has to set ǫ = 0. In the other terms, the proofs are the same since we used intentionally the generalized chain-rule (Lemma 3) instead of the classical one for Sobolev-functions since u is in general not a Sobolev-function, but G(u) is. This remark allows us to prove the theorem.

Technical lemmata

Lemma 1 For any Lipschitz-continuous function g : R → R and any

η ∈ E, ∀x, k ∈ R, g η (x, k) -sgn 0 (x -k)[g(x) -g(k)] ≤ lip(g)τ.
Proof. This comes from the remark that ∀x, k ∈ R,

g η (x, k) -sgn 0 (x -k)[g(x) -g(k)] = x k [η ′ (σ -k) -sgn 0 (σ -k)]g ′ (σ)dσ .
Lemma 2 Set, for any a, b:

I τ (a, b) := b a a µ η ′′ (µ -σ) φ ′ (σ)dσ φ ′ (µ)dµ. Then, I τ (a, b) = I τ (b, a), 0 ≤ 2I τ (a, b) + φ η (a, b) + φ η (b, a) = 1 2 b a b a η ′′ (µ -σ)[ φ ′ (σ) -φ ′ (µ)] 2 dµdσ ≤ 2 φ ′ ∞ |b -a| (5
)

and 3 φ ′ ∞ |b -a| ≥ |I τ (a, b) + φ η (b, a)| + |I τ (a, b) + φ η (a, b)| → τ →0 0. ( 6
)
Moreover, if √ φ ′ admits a modulus of continuity ω φ , then

2I τ (a, b) + φ η (a, b) + φ η (b, a) ≤ w φ (τ ) 2 |b -a|, (7) max
[|I τ (a, b) + φ η (b, a)|, |I τ (a, b) + φ η (a, b)|] ≤ 1 2 w φ (τ ) 2 |b -a| + φ ′ ∞ min(τ, |b -a|).
Proof. Note that, by Fubini's theorem,

I τ (a, b) = - b a µ a η ′′ (µ -σ) φ ′ (σ) φ ′ (µ)dσdµ = - b a b σ η ′′ (σ -µ) φ ′ (σ) φ ′ (µ)dσdµ
and since η ′′ is even, one gets that I τ (a, b) = I τ (b, a) since the above remark yields

I τ (a, b) = - 1 2 b a b a η ′′ (µ -σ) φ ′ (σ) φ ′ (µ)dσdµ.
As η ′ is odd and η(r) ≤ |r| when |r| ≤ τ , we get thanks to Saks lemma and since G(D) is at most countable.

2I τ (a, b) + φ η (a, b) + φ η (b, a) = b a [η ′ (µ -a) -η ′ (µ -b)]φ ′ (µ) - b a η ′′ (µ -σ) φ ′ (σ) φ ′ (µ)dσ dµ = b a b a η ′′ (σ -µ)[ φ ′ (µ) -φ ′ (σ)] φ ′ (µ)dσdµ = b a b a η ′′ (µ -σ)[ φ ′ (σ) -φ ′ (µ)] φ ′ (σ)dµdσ = - b a b a η ′′ (σ -µ)[ φ ′ (µ) -φ ′ (σ)] φ ′ (σ)dµdσ = 1 2 b a b a η ′′ (µ -σ)[ φ ′ (σ) -φ ′ (µ)] 2 dµdσ ≤ φ ′ ∞ |b -a|. Therefore, sup I τ (a, b) + φ η (b, a), I τ (a, b) + φ η (a, b) ≤ 1 4 b a b a η ′′ (µ -σ)[ φ ′ (σ) -φ ′ (µ)] 2 dµdσ + 1 2 b a |η ′ (µ -a) + η ′ (µ -b)|φ ′ (σ)dσ ≤ 1 4 b a b a η ′′ (µ -σ)[ φ ′ (σ) -φ ′ (µ)] 2 dµdσ + φ ′ ∞ min(τ, |b -a|) ≤ 2 φ ′ ∞ |b -a| + φ ′ ∞ min(τ, |b -a|). Finally, if one assumes that b > a (a > b is similar), then b a b a η ′′ (µ -σ)[ φ ′ (σ) -φ ′ (µ)] 2 dµdσ ≤ b a τ -τ η ′′ (α)| φ ′ (µ) -φ ′ (µ + α)| 2 dαdµ = τ -τ η ′′ (α) b a | φ ′ (µ) -φ ′ (µ + α)| 2 dµdα. In particular, since lim α→0 b a | φ ′ (µ -α) -φ ′ (µ)|
a boundary), u ∈ L 2 (O), f ∈ L ∞ (R), G a Lipschitz-continuous function with G(0) = 0 and assume that G(u) ∈ H 1 (O). Then v = u b f (s)G ′ (s)ds ∈ H 1 (O)
The conclusion is then that, in L where η denotes the even convex function, defined for any positive x by: η(x) = Note that if u ∈ H 1 (R d ), there exists a sequence (u n ) ∈ D(R d ) that converges to u in H 1 (R d ), so that Remark 2 Assume that u 0 , g(•, 0) ∈ L 1 (R d ), h(•, 0) = 0 and that (u ǫ ) converges weakly to a given u in L 2 (Ω × Q). Since η is a convex continuous function, one gets at the limit that:

2 (O) d , f (u)∇G(u) ← f oG -1 ǫ (G(u))∇G(u) = ∇ u b f oG -1 ǫ (G(s))G ′ (s)ds → ∇ u b f (
R d f (u)∇η ′ (u)dx = n R d f (u n )∇η ′ (u n )dx = lim n R d f (u n )η ′′ (u n )∇u n dx = lim n R d
Ω×Q η(u)dxdt ≤ lim inf ǫ Ω×Q η(u ǫ )dxdt ≤ C[ R d η(u 0 )dx + 1] ≤ C[ u 0 L 1 (R d ) + 1].
Since η is monotone with respect to its parameter τ , Beppo Levi's theorem yields u ∈ L ∞ (0, T, L 1 (Ω × R d )).

  , u 2 (t, x, ǫ)) -g(y, u 1 (t, y, δ))|ρ m (x -y)dδdǫdydxdt.

  , denote by ϕ(t, x) = γ(t)α(x) where γ ∈ D + ([0, T [), α is the function defined by α(x) = min(1, R a |x| a ) where R ≥ 1 and a = d/2 + ǫ, ǫ > 0 in order to have α in L 2 (R d ). Thus, ∇α(x) = -aR a |x| -a-1 x |x| 1 {|x|>R} = -a α(x) |x| x |x| 1 {|x|>R} ∈ L 2 (R d ) d ; and, in the set {|x| > R}, one has that ∆α(x) = a(2+2ǫ-a) α(x)

  2 dµ = 0 by continuity of the translations in L 2 , one gets the convergence to 0 claimed in the lemma. Lemma 3 (A generalized chain rule) Consider O ⊂ R d a domain with a Lipschitz boundary (if there is

  where b ∈ R if O has a finite measure and b = 0 else, and, a.e.∇ u b f (s)G ′ (s)ds = f (u)∇G(u).

Lemma 4 sup t u ǫ 2 L 2 (

 422 s)G ′ (s)ds and the result holds in that first case: a non-decreasing Lipschitz function G and a bounded continuous function f . Note that the same proof yields the result for a decreasing Lipschitz function G and any pointwise limit of sequences of bounded continuous function (f n ), uniformly bounded by a same constant. Thus, the result holds for the Baire class of uniformly bounded continuous function, i.e., the bounded Borel functions f . Now, since, in the Lebesgue class of f ∈ L ∞ (R), there exists a Borel function f , bounded by the same value ||f || L ∞ (R) , one hasf (u)∇G(u) = ∇ u b f (s)G ′ (s)ds = ∇ u b f (s)G ′ (s)ds.If now D denotes the negligible set where f and f differs,[ f (u) -f (u)]∇G(u) ≤ f (u) -f (u) ∇G(u) 1 {G(u)∈G(D)} = 0 a.e.since G(D) is negligible (D is negligible and G Lipschitz, non-decreasing) and by using Saks Lemma.To finish the proof, just remind that any Lipschitz function is the difference of Lipschitz non-decreasing functions. The weak solution u ǫ to Problem (P ǫ ) satisfies the following estimates:Ω×R d ) (t) + ǫ ∇u ǫ 2 L 2 (Ω×Q) + ∇G(u ǫ ) 2 L 2 (Ω×Q ≤ C. If, moreover, u 0 , g(•, 0) ∈ L 1 (R d) and h(•, 0) = 0, sup t E R d η(u ǫ (t))dx ≤ e c(h,g)T [ R d η(u ǫ 0 )dx + T c(g)[τ + g(x, 0) L 1 (R d ) ]

x 2 2τ 1 {ηη(u ǫ ) 2 η

 212 x|<τ } + (x -τ 2 )1 { x|≥τ } .Proof. Denote by η a non-negative convex-function, with η ′ Lipschitz-continuous, and assume that |η(u)| ≤ C|u| 2 for a given constant C. Thanks to Itô's formula, for any t,′′ (u ǫ )[ǫ + φ ′ (u ǫ )]|∇u ǫ | 2 + η ′′ (u ǫ ) f (u ǫ )∇u ǫ dxds = u ǫ )η ′ (u ǫ )dxdw(s) , u ǫ ) 2 η ′′ (u ǫ )dxds + t 0 R d g(x, u ǫ )η ′ (u ǫ )dxds. ′′ (u ǫ )ǫ|∇u ǫ | 2 + η ′′ (u ǫ )|∇G(u ǫ )| 2 ′′ (u ǫ )dxds + 2E t 0 R d h(x, 0) 2 η ′′ (u ǫ )dxds +c(g)E t 0 R d (|u ǫ | + |g(x, 0)|)|η ′ (u ǫ )|dxds.

η(u ǫ ) 2 η 2 L 2 (Ω×Q) ≤ u ǫ 0 2 L 2 (0 u ǫ 2 L 2 (u ǫ 2 L 2 ( 2 L 2

 22222222222 ′′ (u ǫ )ǫ|∇u ǫ | 2 + η ′′ (u ǫ )|∇G(u ǫ )| 2 dxds ≤ ′′ (u ǫ )dxds + 2E t 0 R d h(x, 0) 2 η ′′ (u ǫ )dxds +c(g)E t 0 R d (|u ǫ | + |g(x, 0)|)|η ′ (u ǫ )|dxds.Assume first that η(x) = x 2 . Then, this yields (the constants c(h) and c(g) may change from one line to another)u ǫ 2 L 2 (Ω×R d ) (t) + ǫ ∇u ǫ 2 L 2 (Ω×Q + ∇G(u ǫ ) R d ) + [c(h) + c(g)] t Ω×R d ) (s)ds +c(g)T g(x, 0) 2 L 2 (R d ) + 4T h(x, 0) 2 L 2 (R d ) ,and Gronwall's lemma implies that supt Ω×R d ) (t) + ǫ ∇u ǫ 2 L 2 (Ω×Q) + ∇G(u ǫ ) (Ω×Q ≤ C.Assume now that u 0 , g(•, 0) ∈ L 1 (R d ) and that h(•, 0) = 0 and denote by η = η the classical even and convex approximation of the absolute value function introduced in the lemma. Note that 0 ≤ η(x) ≤ 4x 2 τ , |xη ′ (x)| ≤ 2η(x) + τ and that x 2 η′′ (x) ≤ η(x) for any x. Then, ǫ )dxds+T c(g)[τ + g(x, 0) L 1 (R d ) ],and thanks to Gronwall's lemma, sup t E R d η(u ǫ (t))dx ≤ e c(h,g)T [ R d η(u ǫ 0 )dx + T c(g)[τ + g(x, 0) L 1 (R d ) ].

Here we don't mean pathwise, nor martingale solutions.

To adapt the proof of the main result of this paper, one just needs to consider the resolvant (I -∆) -1 instead of (-∆) -1 .

One uses that ρ θ is even and the properties of the Itô integral with continuous linear operators.

integrals that will disappear when ǫ will go to 0

Acknowledgment. The authors would like to thank Boris Andreianov for his valuable suggestions.

Proof. First, assume that f is a continuous function and that G is a nondecreasing function.

ǫ is well-defined on R and it is a Lipschitzcontinuous function. Then, the classical chain rule yields

To this end we consider the two following possible cases: -Assume that x is such that:

ǫ (G(x)) ≥ y, and at the limit when y → x -, one gets

and, regarding the definition of a(x) and the previous case,

since, if a(s) = s, then G ′ (s) = 0, unless for a countable number of points. f and G ′ are bounded function, so that the Lebesgue theorem yields the convergence of