Tadahisa Funaki

Yueyuan Gao

Danielle Hilhorst
email: danielle.hilhorst@math.u-psud.fr

Existence and uniqueness of the entropy solution of a stochastic conservation law with a 𝑄-Brownian motion

Keywords: Stochastic first order conservation law, 𝑄-Brownian motion, Kato inequality, Existence and uniqueness of the entropy solution, Associated parabolic problem

In this paper, we prove the existence and uniqueness of the entropy solution for a first order stochastic conservation law with a multiplicative source term involving a 𝑄-Brownian motion.

After having defined a measure-valued weak entropy solution of the stochastic conservation law, we present the Kato inequality and as a corollary we deduce the uniqueness of the measure-valued weak entropy solution which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure-valued entropy solution of the stochastic conservation law, which proves the existence of the measure-valued entropy solution.

INTRODUCTION

In this paper we prove the existence and uniqueness of the entropy solution for the first order stochastic conservation law { 𝑑𝑢 + div(𝐯𝑓 (𝑢))𝑑𝑡 = 𝑔(𝑢)𝑑𝑊 (𝑥, 𝑡), in Ω × 𝕋 𝑑 × (0, 𝑇 ) , 𝑢(𝜔, 𝑥, 0) = 𝑢 0 (𝑥), 𝜔 ∈ Ω, 𝑥 ∈ 𝕋 𝑑 .

(

) 1
where 𝕋 𝑑 is the 𝑑-dimensional torus, 𝑊 (𝑥, 𝑡) is a 𝑄-Brownian motion, satisfying the two conditions (4) and ( 5) in section 2, with the following hypotheses (𝐻):

• 𝑢 0 ∈ 𝐿 2 (𝕋 𝑑 ).

• 𝑓 ∶ ℝ → ℝ is a Lipschitz continuous function with Lipschitz constant 𝐶 𝑓 such that 𝑓 (0) = 0.

• 𝑔 ∶ ℝ → ℝ is a Lipschitz continuous function with Lipschitz constant 𝐶 𝑔 ; moreover we suppose that g is bounded such that |𝑔| < 𝑀 𝑔 for a positive constant 𝑀 𝑔 .

• 𝐯 ∈ 𝐶 1 (𝕋 𝑑 × [0, 𝑇 ]) with div 𝐯 = 0 for all (𝑥, 𝑡) ∈ 𝕋 𝑑 × [0, 𝑇 ], so that there exists 𝑉 < ∞ such that |𝐯(𝑥, 𝑡)| ≤ 𝑉 for all (𝑥, 𝑡) ∈ 𝕋 𝑑 × [0, 𝑇 ].

Several articles have been devoted to the study of stochastic perturbations of nonlinear first order hyperbolic problems. Let us mention the article of Bauzet-Vallet-Wittbold 2 who prove the existence and uniqueness of a stochastic entropy solution of the Cauchy problem for a stochastic conservation law with a multiplicative one-dimensional noise in time. They apply an artificial viscosity method to prove the existence of a solution, and an adaptation of the Kruzhkov doubling of variables method to prove that any stochastic entropy solution coincides with the solution obtained by the artificial viscosity method. Then as an extension, Bauzet-Vallet-Wittbold 3 prove an existence and uniqueness result for the stochastic entropy solution of a Dirichlet problem for the stochastic conservation law with multiplicative noise in time.

As for the uniqueness result, Debussche-Vovelle 5 consider the 𝑑-dimensional problem with multiplicative noise 𝑑𝑢 + 𝑓 (𝑢) 𝑥 𝑑𝑡 = ℎ(𝑢)𝑑𝑊 . The noise term 𝑑𝑊 is a function of space and time. The authors prove the existence and uniqueness of the kinetic solution. We believe however that our present results are important since entropy solutions are much easier to approximate numerically than kinetic solutions.

Funaki-Gao-Hilhorst [START_REF] Funaki | Convergence of a finite volume scheme for a stochastic conservation law involving a 𝑄-Brownian motion[END_REF] prove the convergence of a finite volume scheme for the stochastic conservation law (1). This convergence result implies the existence of a measure-valued weak entropy solution. In this paper, we extend the proof of the existence and uniqueness of the weak entropy solution of Bauzet-Vallet-Wittbold [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF][START_REF] Bauzet | The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation[END_REF] to the case of equation (1) where the noise involves a 𝑄-Brownian motion.

This paper is organized as follows: in section 2, we recall the definitions of a weak entropy solution and of a measure-valued entropy solution of Problem (1) and we present the associated parabolic problem and an inequality satisfied by its strong solution. We prove that the solution of the associated parabolic problem converges to a measure-valued entropy solution of the stochastic conservation law when the diffusion coefficient tends to 0.

In section 3, we first present the Kato inequality and as a corollary, we first deduce the uniqueness of the measure-valued weak entropy solution and then the uniqueness of the weak entropy solution. As a consequence of the proofs, the measurevalued weak entropy solution is unique and coincides with the unique weak entropy solution. The Kato inequality is proved by a doubling of variables method for which we need entropy type inequalities for measure-valued weak entropy solutions and for the strong solutions of the associated parabolic problem.

In section 4, we show the well-posedness of the associated parabolic problem. We apply an implicit time discretization to obtain a semi-discrete solution of the associated stochastic nonlinear parabolic problem. By Itô's formula and a priori estimates, we prove that when the time step tends to zero, the limit of this discrete solution is a strong solution of the corresponding parabolic problem. We then prove the uniqueness of the strong solution.

ENTROPY FORMULATION

We recall the definitions of a 𝑄-Brownian motion and of an entropy solution of Problem (1).

Definition 1 (𝑄-Brownian motion in 𝐿 2 (𝕋 𝑑 )). Let 𝑄 ≥ 0 be a symmetric trace class operator on 𝐿 2 (𝕋 𝑑 ), let {𝑒 𝑗 } 𝑗≥1 be an orthonormal basis in 𝐿 2 (𝕋 𝑑 ) diagonalizing 𝑄 and {𝜆 𝑗 ≥ 0} 𝑗≥1 be the corresponding eigenvalues, such that 𝑄𝑒 𝑗 = 𝜆 𝑗 𝑒 𝑗 for all 𝑗 ≥ 1. We recall that 𝑄 is of trace class, namely that

Tr 𝑄 = ∞ ∑ 𝑗=1 ⟨𝑄𝑒 𝑗 , 𝑒 𝑗 ⟩ 𝐿 2 (𝕋 𝑑 ) = ∞ ∑ 𝑗=1 𝜆 𝑗 ≤ Λ 0 , (2)
for some positive constant Λ 0 . Let (Ω,  , P) be a probability space equipped with a filtration ( 𝑡 ) [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF] and {𝛽 𝑗 } 𝑗≥1 be a sequence of independent ( 𝑡 )-Brownian motions defined on (Ω,  , P). We recall that a Brownian motion 𝛽(𝑡) is called an ( 𝑡 )-Brownian motion if it is ( 𝑡 )-adapted and the increment 𝛽(𝑡) -𝛽(𝑠) is independent of  𝑠 for every 0 ≤ 𝑠 < 𝑡. The process 𝑊 defined by

𝑊 (𝑥, 𝑡) = ∞ ∑ 𝑗=1 𝛽 𝑗 (𝑡)𝑄 1 2 𝑒 𝑗 (𝑥) = ∞ ∑ 𝑗=1 √ 𝜆 𝑗 𝛽 𝑗 (𝑡)𝑒 𝑗 (𝑥) (3)
is a 𝑄-Brownian motion in 𝐿 2 (𝕋 𝑑 ), and the series defined by ( 3) is convergent in 𝐿 [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] (Ω, 𝐶([0, 𝑇 ], 𝐿 2 (𝕋 𝑑 ))) [cf. [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] ].

We suppose furthermore that 𝑒 𝑗 ∈ 𝐶(𝕋 𝑑 ) for 𝑗 = 1, 2... and that there exists a positive constant Λ 1 such that

∞ ∑ 𝑗=1 𝜆 𝑗 ‖𝑒 𝑗 ‖ 2 𝐿 ∞ (𝕋 𝑑 ) ≤ Λ 1 . ( 4
)
In the proof of the existence of a strong solution of the associated parabolic problem, we need the further assumption that there exists a positive constant Λ 2 , such that

∞ ∑ 𝑗=1 𝜆 𝑗 ‖∇𝑒 𝑗 ‖ 2 𝐿 ∞ (𝕋 𝑑 ) 𝑑 ≤ Λ 2 . ( 5
)
Definition 2 (Weak entropy solution of Problem (1)). A function 𝑢 ∈  2 𝜔 (0, 𝑇 , 𝐿 2 (𝕋 𝑑 )) ∩ 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω × 𝕋 𝑑 )) is a weak entropy solution of the stochastic scalar conservation law (1) with the initial condition 𝑢 0 ∈ 𝐿 2 (𝕋 𝑑 ), if P-a.s. in Ω,

∫ 𝕋 𝑑 𝜂(𝑢 0 (𝑥) -𝑘)𝜑(𝑥, 0)𝑑𝑥 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂(𝑢 -𝑘)𝜕 𝑡 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝐹 𝜂 (𝑢, 𝑘)𝐯 ⋅ ∇ 𝑥 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 + ∫ 𝕋 𝑑 𝑇 ∫ 0 𝜂 ′ (𝑢 -𝑘)𝑔(𝑢)𝜑(𝑥, 𝑡)𝑑𝑊 (𝑥, 𝑡)𝑑𝑥 + 1 2 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′′ (𝑢 -𝑘)𝑔 2 (𝑢)𝜑(𝑥, 𝑡)𝑄(𝑥, 𝑥)𝑑𝑥𝑑𝑡 ≥ 0,
where

𝐹 𝜂 (𝑎, 𝑏) = 𝑎 ∫ 𝑏 𝜂 ′ (𝜎 -𝑏)𝑓 ′ (𝜎)𝑑𝜎 and 𝑄(𝑥, 𝑦) = ∞ ∑ 𝑗=1 𝜆 𝑗 𝑒 𝑗 (𝑥)𝑒 𝑗 (𝑦), (6)
for all 𝜑 ∈  ∶= {𝜑 ∈ 𝐶 ∞ 0 (𝕋 𝑑 × [0, 𝑇 ]), 𝜑 ≥ 0, 𝜑(⋅, 𝑇 ) = 0}
and for all 𝜂 ∈  where  is the set of nonnegative 𝐶 2 (ℝ) convex functions 𝜂 such that the support of 𝜂 ′′ is compact and 𝜂(0) = 0.  2 𝜔 (0, 𝑇 ; 𝐿 2 (𝕋 𝑑 )) is the subclass of 𝐿 2 (Ω × 𝕋 𝑑 × (0, 𝑇 )) consisting of predictable 𝐿 2 (𝕋 𝑑 )-valued processes [cf. [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] , page 98].

We remark that for all 𝑥, 𝑦 ∈ 𝕋 𝑑

|𝑄(𝑥, 𝑦)| ≤ ∞ ∑ 𝑗=1 𝜆 𝑗 ‖𝑒 𝑗 ‖ 2 𝐿 ∞ (𝕋 𝑑 ) ≤ Λ 1 . ( 7
)
Definition 3 (Measure-valued weak entropy solution of Problem (1)). A function 𝐮 of  2 𝜔 (0, 𝑇 ; 𝐿 2 (𝕋 𝑑 × (0, 1))) ∩ 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω × 𝕋 𝑑 × (0, 1))) is a measure-valued weak entropy solution of the stochastic scalar conservation law (1) with the initial condition 𝑢 0 ∈ 𝐿 2 (𝕋 𝑑 ), if P-a.s. in Ω, for all 𝜂 ∈  and for all 𝜑 ∈  ∫

𝕋 𝑑 𝜂(𝑢 0 -𝑘)𝜑(𝑥, 0)𝑑𝑥 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 1 ∫ 0 𝜂(𝐮(𝑥, 𝑡, 𝛼) -𝑘)𝜕 𝑡 𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝑥𝑑𝑡 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 1 ∫ 0 𝐹 𝜂 (𝐮(𝑥, 𝑡, 𝛼), 𝑘)𝐯 ⋅ ∇ 𝑥 𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝑥𝑑𝑡 + ∫ 𝕋 𝑑 𝑇 ∫ 0 1 ∫ 0 𝜂 ′ (𝐮(𝑥, 𝑡, 𝛼) -𝑘)𝑔(𝐮(𝑥, 𝑡, 𝛼))𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝑊 (𝑥, 𝑡)𝑑𝑥 + 1 2 𝑇 ∫ 0 ∫ 𝕋 𝑑 1 ∫ 0 𝜂 ′′ (𝐮(𝑥, 𝑡, 𝛼) -𝑘)𝑔 2 (𝐮(𝑥, 𝑡, 𝛼))𝜑(𝑥, 𝑡)𝑄(𝑥, 𝑥)𝑑𝛼𝑑𝑥𝑑𝑡 ≥ 0. (8)
The main result of this article is the following theorem.

Theorem 1. Suppose that the hypotheses (𝐻) hold and that the basis functions {𝑒 𝑗 } 𝑗≥1 of the 𝑄-Brownian motion satisfy the hypotheses (4) and (5); then there exists a unique measure-valued weak entropy solution. Moreover, it coincides with the unique weak entropy solution.

In section 4, we will study the initial value problem

( 𝜖 ) { 𝑑𝑢 𝜖 -𝜖Δ𝑢 𝜖 𝑑𝑡 + div(𝐯𝑓 (𝑢 𝜖 ))𝑑𝑡 = 𝑔(𝑢 𝜖 )𝑑𝑊 in Ω × 𝕋 𝑑 × (0, 𝑇 ), 𝑢 𝜖 (𝑥, 0) = 𝑢 𝜖 0 (𝑥) for all 𝑥 ∈ 𝕋 𝑑
and prove that Problem ( 𝜖 ) possesses a unique strong solution.

Next, we let 𝜖 → 0 in Problem ( 𝜖 ) to deduce the existence of a measure-valued entropy solution of Problem (1). More precisely, we prove the following result.

Lemma 1 (Existence of a measure-valued entropy solution). Let 𝑢 𝜖 be the strong solution of the parabolic problem (c.f. Definition 4 in section 4). Then 𝑢 𝜖 tends to a measure-valued entropy solution of Problem (1) as 𝜖 tends to 0 in the sense of Young measures.

Proof. We define the functional of 𝑋 = 𝑋(𝑥)

(𝑋, 𝑡) = ∫ 𝕋 𝑑 𝜂(𝑋(𝑥) -𝑘)𝜑(𝑥, 𝑡)𝑑𝑥, so that  𝑋 (𝑋, 𝑡) = 𝜂 ′ (𝑋(𝑥) -𝑘)𝜑(𝑥, 𝑡)  𝑋𝑋 (𝑋, 𝑡) = 𝜂 ′′ (𝑋(𝑥) -𝑘)𝜑(𝑥, 𝑡)  𝑡 (𝑋, 𝑡) = ∫ 𝕋 𝑑

𝜂(𝑋(𝑥) -𝑘)𝜕 𝑡 𝜑(𝑥, 𝑡)𝑑𝑥

and apply Itô's formula [c.f. [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] ] for (𝑢 𝜖 (𝑡), 𝑡) to obtain

∫ 𝕋 𝑑 𝜂(𝑢 𝜖 (𝑇 ) -𝑘)𝜑(𝑇 )𝑑𝑥 = ∫ 𝕋 𝑑 𝜂(𝑢 𝜖 (0) -𝑘)𝜑(0)𝑑𝑥 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂(𝑢 𝜖 (𝑥, 𝑡) -𝑘)𝜕 𝑡 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)𝜑(𝑥, 𝑡)[𝜖Δ𝑢 𝜖 (𝑥, 𝑡) -div(𝐯𝑓 (𝑢 𝜖 (𝑥, 𝑡)))]𝑑𝑥𝑑𝑡 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)𝜑(𝑥, 𝑡)𝑔(𝑢 𝜖 (𝑥, 𝑡))𝑑𝑊 (𝑥, 𝑡)𝑑𝑥 + 1 2 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)𝜑(𝑥, 𝑡)𝑔 2 (𝑢 𝜖 (𝑥, 𝑡))𝑄(𝑥, 𝑥)𝑑𝑥𝑑𝑡.
We first consider the term involving Δ𝑢 𝜖 . Note that for all 𝜑 ∈ , by integration by parts

𝜖 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡))𝜑(𝑥, 𝑡)Δ𝑢 𝜖 (𝑥, 𝑡)𝑑𝑥𝑑𝑡 = -𝜖 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)∇ 𝑥 𝑢 𝜖 (𝑥, 𝑡) ⋅ ∇ 𝑥 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 -𝜖 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)|∇ 𝑥 𝑢 𝜖 (𝑥, 𝑡)| 2 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡.
As for the term involving div(𝐯𝑓 (𝑢 𝜖 (𝑥, 𝑡))), we use integration by parts, the chain rule and div 𝑣 = 0 to deduce that

- 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘) div(𝐯𝑓 (𝑢 𝜖 (𝑥, 𝑡)))𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 = 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝐹 𝜂 (𝑢 𝜖 (𝑥, 𝑡), 𝑘)𝐯 ⋅ ∇ 𝑥 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡.
Remembering that 𝜑(𝑇 ) = 0 and that 𝜂 is a nonnegative convex function, we deduce the following inequality

∫ 𝕋 𝑑 𝜂(𝑢 𝜖 0 (𝑥) -𝑘)𝜑(𝑥, 0)𝑑𝑥 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂(𝑢 𝜖 (𝑥, 𝑡) -𝑘)𝜕 𝑡 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 -𝜖 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)∇ 𝑥 𝑢 𝜖 (𝑥, 𝑡) ⋅ ∇ 𝑥 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝐹 𝜂 (𝑢 𝜖 (𝑥, 𝑡), 𝑘)𝐯 ⋅ ∇ 𝑥 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)𝑔(𝑢 𝜖 (𝑥, 𝑡))𝜑(𝑥, 𝑡)𝑑𝑊 (𝑥, 𝑡)𝑑𝑥 + 1 2 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)𝑔 2 (𝑢 𝜖 (𝑥, 𝑡))𝜑(𝑥, 𝑡)𝑄(𝑥, 𝑥)𝑑𝑥𝑑𝑡 ≥ 0. (9)
We remark that (38) below and the strong convergence of {𝑢 Δ𝑡 } to 𝑢 𝜖 in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))) [cf. Lemma 12] below imply that 𝑢 𝜖 is uniformly bounded in 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))). Thus there exist an entropy process 𝐮 ∈ 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω × 𝕋 𝑑 × (0, 1))) and a subsequence of 𝑢 𝜖 which we denote again by 𝑢 𝜖 such that 𝑢 𝜖 converges to 𝐮 in the sense of Young measures as 𝜖 tends to 0 [cf. [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] , Page 668]. We multiply (9) by 𝟏 𝐴 where 𝐴 ∈  is arbitrary and take the expectation to obtain Next we pass to the limit 𝜖 → 0 in (10); the estimate (11) ensures that

E ⎡ ⎢ ⎢ ⎣ 𝟏 𝐴 ∫ 𝕋 𝑑 𝜂(𝑢 𝜖 0 (𝑥) -𝑘)𝜑(𝑥, 0)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ 𝟏 𝐴 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂(𝑢 𝜖 (𝑥, 𝑡) -𝑘)𝜕 𝑡 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ -E ⎡ ⎢ ⎢ ⎣ 𝜖𝟏 𝐴 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)∇ 𝑥 𝑢 𝜖 (𝑥, 𝑡) ⋅ ∇ 𝑥 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ 𝟏 𝐴 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝐹 𝜂 (𝑢 𝜖 (𝑥, 𝑡), 𝑘)𝐯 ⋅ ∇ 𝑥 𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ 𝟏 𝐴 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)𝑔(𝑢 𝜖 (𝑥, 𝑡))𝜑(𝑥, 𝑡)𝑑𝑊 (𝑥, 𝑡)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ (10)
-E ⎡ ⎢ ⎢ ⎣ 𝜖𝟏 𝐴 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ (𝑢 𝜖 (𝑥, 𝑡) -𝑘)∇𝑢 𝜖 (𝑥, 𝑡) ⋅ ∇𝜑(𝑥, 𝑡)𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ → 0
as 𝜖 → 0. We suppose that 𝑢 𝜖 0 → 𝑢 0 strongly in 𝐿 2 (𝕋 𝑑 ) and use similar arguments as in [START_REF] Funaki | Convergence of a finite volume scheme for a stochastic conservation law involving a 𝑄-Brownian motion[END_REF] for the other terms to deduce that

E ⎡ ⎢ ⎢ ⎣ 𝟏 𝐴 ∫ 𝕋 𝑑 𝜂(𝑢 0 -𝑘)𝜑(𝑥, 0)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ 𝟏 𝐴 𝑇 ∫ 0 ∫ 𝕋 𝑑 1 ∫ 0 𝜂(𝐮(𝑥, 𝑡, 𝛼) -𝑘)𝜕 𝑡 𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ 𝟏 𝐴 𝑇 ∫ 0 ∫ 𝕋 𝑑 1 ∫ 0 𝐹 𝜂 (𝐮(𝑥, 𝑡, 𝛼), 𝑘)𝐯 ⋅ ∇ 𝑥 𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ 𝟏 𝐴 ∫ 𝕋 𝑑 𝑇 ∫ 0 1 ∫ 0 𝜂 ′ (𝐮(𝑥, 𝑡, 𝛼) -𝑘)𝑔(𝐮(𝑥, 𝑡, 𝛼))𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝑊 (𝑥, 𝑡)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + 1 2 E ⎡ ⎢ ⎢ ⎣ 𝟏 𝐴 𝑇 ∫ 0 ∫ 𝕋 𝑑 1 ∫ 0 𝜂 ′′ (𝐮(𝑥, 𝑡, 𝛼) -𝑘)𝑔 2 (𝐮(𝑥, 𝑡, 𝛼))𝜑(𝑥, 𝑡)𝑄(𝑥, 𝑥)𝑑𝛼𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ ≥ 0,
which shows that 𝐮 is a measure-valued entropy solution of Problem (1).

UNIQUENESS OF THE MEASURE-VALUED WEAK ENTROPY SOLUTION

The main part of this section is to prove the Kato inequality for measure-valued weak entropy solutions of Problem (1). The Kato inequality will imply the uniqueness of the measure-valued weak entropy solution and as a consequence the uniqueness of the weak entropy solution.

Proposition 1 (Kato inequality). Let 𝐮, û be 2 measure-valued weak entropy solutions to (1) with initial data 𝑢 0 , û0 ∈ 𝐿 2 (𝕋 𝑑 ) respectively. Then, for any nonnegative function 𝜑 with compact support belonging to 𝐻 1 (𝕋 𝑑 × [0, 𝑇 ]), there holds

∫ 𝕋 𝑑 |𝑢 0 -û0 |𝜑(0)𝑑𝑥 + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 1 ∫ 0 1 ∫ 0 |𝐮(𝑥, 𝑡, 𝛼) -û(𝑥, 𝑡, 𝛽)| 𝜕 𝑡 𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 1 ∫ 0 1 ∫ 0 𝐹 (𝐮(𝑥, 𝑡, 𝛼), û(𝑥, 𝑡, 𝛽)) 𝐯 ⋅ ∇𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ ≥ 0 . ( 12
)
where 𝑄 𝑇 = 𝕋 𝑑 × (0, 𝑇 ) and 𝐹 (𝑎, 𝑏) = sgn(𝑎 -𝑏)[𝑓 (𝑎) -𝑓 (𝑏)] with sgn(𝑎) = 𝑎 |𝑎| for all 𝑎 ≠ 0. Note that 𝐹 is a Lipschitz continuous function with constant 𝐶 𝑓 .

Proof of Theorem 1:

We use below the fact that the Kato inequality holds for all nonnegative test functions 𝜑 ∈ 𝐻 1 (𝑄 𝑇 ). Let 𝐾 be a positive constant, and denote by 𝜓 any smooth nonincreasing function such that 𝟏 (-∞,𝐾] (⋅) ≤ 𝜓(⋅) ≤ 𝟏 (-∞,𝐾+1] (⋅).

Setting 𝜑(𝑥, 𝑡) = 𝜓(|𝑥| -𝑉 𝐶 𝑓 𝑡)𝛾(𝑡) with 𝛾(𝑡) =

𝑇 -𝑡 𝑇 and substituting 𝜑 in (12) yields

E ⎡ ⎢ ⎢ ⎣ ∫ 𝐓 𝑑 | û0 -𝑢 0 |𝜓(|𝑥|) ⎤ ⎥ ⎥ ⎦ 𝛾(0) + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 1 ∫ 0 1 ∫ 0 𝜓(|𝑥| -𝑉 𝐶 𝑓 𝑡)|𝐮(𝑥, 𝑡, 𝛽) -û(𝑥, 𝑡, 𝛼)|𝛾 ′ (𝑡)𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ -E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 1 ∫ 0 1 ∫ 0 𝑉 𝐶 𝑓 𝜓 ′ (|𝑥| -𝑉 𝐶 𝑓 𝑡)|𝐮(𝑥, 𝑡, 𝛽) -û(𝑥, 𝑡, 𝛼)|𝛾(𝑡)𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 1 ∫ 0 1 ∫ 0 𝜓 ′ (|𝑥| -𝑉 𝐶 𝑓 𝑡)𝐹 (𝐮(𝑥, 𝑡, 𝛽), û(𝑥, 𝑡, 𝛼))𝐯 ⋅ 𝑥 |𝑥| 𝛾(𝑡)𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ ≥ 0. (13)
Next, we study the signs of the last two terms in (13). Since 𝜓 is a nonincreasing function, we have that the sum of the last two terms is bounded from above by

E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 1 ∫ 0 1 ∫ 0 (𝑉 𝐶 𝑓 -𝑉 𝐶 𝑓 )𝜓 ′ (|𝑥| -𝑉 𝐶 𝑓 𝑡)|𝐮(𝑥, 𝑡, 𝛽) -û(𝑥, 𝑡, 𝛼)|𝛾(𝑡)𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ = 0,
which combined with (13) yields

E ⎡ ⎢ ⎢ ⎣ ∫ 𝐓 𝑑 | û0 -𝑢 0 |𝜓(|𝑥|) ⎤ ⎥ ⎥ ⎦ 𝛾(0) + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 𝛾 ′ (𝑡) 1 ∫ 0 1 ∫ 0 𝜓(|𝑥| -𝑉 𝐶 𝑓 𝑡) |𝐮(𝑥, 𝑡, 𝛽) -û(𝑥, 𝑡, 𝛼)|𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ ≥ 0 .
We suppose that û0 = 𝑢 0 and choose 𝐾 = sup 𝑥∈𝕋 𝑑 |𝑥| -𝑉 𝐶 𝑓 𝑇 to deduce that 𝜓(|𝑥| -𝑉 𝐶 𝑓 𝑡) = 1. Also in view of the fact that 𝛾 ′ (𝑡) = -1 𝑇 , we deduce that

E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 1 ∫ 0 1 ∫ 0 |𝐮(𝑥, 𝑡, 𝛽) -û(𝑥, 𝑡, 𝛼)|𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ = 0 (14)
for all 𝛼, 𝛽 ∈ (0, 1). This implies that 𝐮(𝑥, 𝑡, 𝛽) = û(𝑥, 𝑡, 𝛼) for almost every 𝑥 ∈ 𝕋 𝑑 , 𝑡 ∈ (0, 𝑇 ), 𝜔 ∈ Ω, 𝛼, 𝛽 ∈ (0, 1). Setting 𝛼 = 𝛽, we deduce that 𝐮(𝑥, 𝑡, 𝛼) = û(𝑥, 𝑡, 𝛼) for almost every 𝑥 ∈ 𝕋 𝑑 , 𝑡 ∈ (0, 𝑇 ) so that Problem (1) has a unique measurevalued weak entropy solution, say 𝐮 = 𝐮(𝑥, 𝑡, 𝛼). Then setting 𝛼 ≠ 𝛽, we deduce that 𝐮(𝑥, 𝑡, 𝛼) = 𝐮(𝑥, 𝑡, 𝛽) which implies that the measure-valued weak entropy solution does not depend on the third parameter. Thus it is a weak entropy solution and (14) also implies the uniqueness of the weak entropy solution of Problem (1). □

In the sequel we prove the Kato inequality, Proposition 1.

Proof of Proposition 1:

We suppose that 𝜌 𝑚 and 𝜌 𝑛 are two mollifier sequences in 𝕋 𝑑 and ℝ respectively, with

supp 𝜌 𝑚 ⊂ [- 2 𝑚 , 0] 𝑑 and supp 𝜌 𝑛 ⊂ [- 2 𝑛 , 0]. Suppose that 𝜌 𝑙 is a mollifier sequence with supp 𝜌 𝑙 ⊂ [- 1 𝑙 , 1 𝑙 ]. Remark that |𝜌 𝑚 | ≤ 𝐶𝑚 𝑑 , |𝜌 𝑛 | ≤ 𝐶𝑛 and |𝜌 𝑙 | ≤ 𝐶𝑙
for some constant 𝐶 and that 𝜌 𝑙 is an even function for all 𝑙. Let 𝑢 𝜖 be the strong solution of the associated parabolic problem in the sense of Definition 4. We use the notations û = û(𝑥, 𝑡, 𝛼) for a measure-valued solution and û0 for the corresponding initial condition, and set  𝑙 𝑘 ∶= 𝜌 𝑙 (û(𝑥, 𝑡, 𝛼) -𝑘) and  𝑙 𝑘 ∶= 𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑠) -𝑘). Next, we deduce two inequalities based on ( 8) and ( 9) respectively. We choose 𝜂 = 𝜂 𝛿 ∈  such that

𝜂 ′ 𝛿 (𝑟) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 if 𝑟 > 𝛿 sin( 𝜋𝑟 2𝛿 ) if |𝑟| ≤ 𝛿 -1 if 𝑟 < -𝛿
in the inequality (8) as well as in the inequality (9). Note that 𝜂 ′′ 𝛿 is an even Lipschitz continuous function. In the inequality (8), we replace the test function 𝜑(𝑥, 𝑡) by 𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠) and then multiply it by  𝑙 𝑘 and integrate 𝑘 on ℝ, 𝑦 on 𝕋 𝑑 and 𝑠 on (0, 𝑇 ) and write last the term coming from the stochastic integral to obtain

E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝜂 𝛿 ( û0 (𝑥) -𝑘)𝜑(𝑦, 𝑠)𝜌 𝑛 (-𝑠)𝜌 𝑚 (𝑥 -𝑦)𝑑𝑥 𝑙 𝑘 𝑑𝑘𝑑𝑦𝑑𝑠 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 1 ∫ 0 𝜂 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝜌 𝑛 (𝑡 -𝑠)𝜕 𝑡 𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝑑𝛼𝑑𝑥𝑑𝑡 𝑙 𝑘 𝑑𝑘𝑑𝑦𝑑𝑠 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 1 ∫ 0 𝜂 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝜑(𝑦, 𝑠)𝜕 𝑡 𝜌 𝑛 (𝑡 -𝑠)𝜌 𝑚 (𝑥 -𝑦)𝑑𝛼𝑑𝑥𝑑𝑡 𝑙 𝑘 𝑑𝑘𝑑𝑦𝑑𝑠 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 1 ∫ 0 𝐹 𝜂 𝛿 (û(𝑥, 𝑡, 𝛼), 𝑘)𝐯 ⋅ 𝜌 𝑚 (𝑥 -𝑦)∇ 𝑥 𝜑(𝑦, 𝑠)𝜌 𝑛 (𝑡 -𝑠)𝑑𝛼𝑑𝑥𝑑𝑡 𝑙 𝑘 𝑑𝑘𝑑𝑦𝑑𝑠 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 1 ∫ 0 𝐹 𝜂 𝛿 (û(𝑥, 𝑡, 𝛼), 𝑘)𝐯 ⋅ ∇ 𝑥 𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠)𝜑(𝑦 -𝑠)𝑑𝛼𝑑𝑥𝑑𝑡 𝑙 𝑘 𝑑𝑘𝑑𝑦𝑑𝑠 ⎤ ⎥ ⎥ ⎦ + 1 2 E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔 2 (û(𝑥, 𝑡, 𝛼))𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠)𝜑(𝑦, 𝑠)𝑄(𝑥, 𝑥)𝑑𝛼𝑑𝑥𝑑𝑡 𝑙 𝑘 𝑑𝑘𝑑𝑦𝑑𝑠 ] + E [ ∫ 𝑄 𝑇 ∫ ℝ 𝑇 ∫ 0 ∫ 𝕋 𝑑 1 ∫ 0 𝜂 ′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)

𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠)𝑑𝑥𝑑𝑊 (𝑥, 𝑡) 𝑙 𝑘 𝑑𝑘𝑑𝑦𝑑𝑠

]

∶=𝐼 1 + 𝐼 2 + 𝐼 3 + 𝐼 4 + 𝐼 5 + 𝐼 6 + 𝐼 7 ≥ 0.
Next in (9) we change the variables (𝑥, 𝑡) to (𝑦, 𝑠), replace 𝜑(𝑦, 𝑠) by 𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠) and write last the term coming from 𝜖Δ𝑢 𝜖 , then multiply by  𝑙 𝑘 and integrate on 𝛼, 𝑘, 𝑥 and 𝑡, to obtain

E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝜂 𝛿 (𝑢 𝜖 0 (𝑦) -𝑘)𝜑(𝑦, 0)𝜌 𝑛 (𝑡)𝜌 𝑚 (𝑥 -𝑦)𝑑𝑦 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 𝜂 𝛿 (𝑢 𝜖 (𝑦, 𝑠) -𝑘)𝜌 𝑛 (𝑡 -𝑠)𝜕 𝑠 𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝑑𝑦𝑑𝑠 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 𝜂 𝛿 (𝑢 𝜖 (𝑦, 𝑠) -𝑘)𝜕 𝑠 𝜌 𝑛 (𝑡 -𝑠)𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝑑𝑦𝑑𝑠 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 𝐹 𝜂 𝛿 (𝑢 𝜖 (𝑦, 𝑠), 𝑘)𝐯 ⋅ 𝜌 𝑚 (𝑥 -𝑦)∇ 𝑦 𝜑(𝑦, 𝑠)𝜌 𝑛 (𝑡 -𝑠)𝑑𝑦𝑑𝑠 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 𝐹 𝜂 𝛿 (𝑢 𝜖 (𝑦, 𝑠), 𝑘)𝐯 ⋅ ∇ 𝑦 𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑠)𝜌 𝑛 (𝑡 -𝑠)𝑑𝑦𝑑𝑠 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + 1 2 E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 𝜂 ′′ 𝛿 (𝑢 𝜖 (𝑦, 𝑠) -𝑘)𝑔 2 (𝑢 𝜖 (𝑦, 𝑠))𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠)𝜑(𝑦, 𝑠)𝑄(𝑦, 𝑦)𝑑𝑦𝑑𝑠 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ] + E [ ∫ 𝑄 𝑇 ∫ ℝ 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ 𝛿 (𝑢 𝜖 (𝑦, 𝑠) -𝑘)𝑔(𝑢 𝜖 (𝑦, 𝑠))𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠)𝑑𝑊 (𝑦, 𝑠)𝑑𝑦 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ] -𝜖 E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 𝜂 ′ 𝛿 (𝑢 𝜖 (𝑦, 𝑠) -𝑘)∇ 𝑦 𝑢 𝜖 (𝑦, 𝑠)∇ 𝑦 [𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)]𝜌 𝑛 (𝑡 -𝑠)𝑑𝑦𝑑𝑠 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ] ∶=𝐽 1 + 𝐽 2 + 𝐽 3 + 𝐽 4 + 𝐽 5 + 𝐽 6 + 𝐽 7 + 𝐽 8 ≥ 0.
We will show that the sum of 𝐼 𝑖 + 𝐽 𝑖 for 𝑖 = 1, 2, ..., 7 and of the term 𝐽 8 converges to the left-hand-side of Kato inequality (12). More precisely, we denote the limit of 𝑢 𝜖 by 𝐮; we should prove that

lim 𝜖→0 lim 𝛿→0 lim 𝑚→∞ lim 𝑙→∞ lim 𝑛→∞ [𝐼 1 + 𝐽 1 ] = E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 |𝑢 0 -û0 |𝜑(0)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ , ( 15
) lim 𝜖→0 lim 𝛿→0 lim 𝑚→∞ lim 𝑙→∞ lim 𝑛→∞ [𝐼 2 + 𝐽 2 ] = E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 1 ∫ 0 1 ∫ 0 |𝐮(𝑥, 𝑡, 𝛼) -û(𝑥, 𝑡, 𝛽)|𝜕 𝑡 𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ , ( 16
)
𝐼 3 + 𝐽 3 = 0, ( 17
) lim 𝜖→0 lim 𝛿→0 lim 𝑚→∞ lim 𝑙→∞ lim 𝑛→∞ [𝐼 4 + 𝐽 4 ] = E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 1 ∫ 0 1 ∫ 0 𝐹 (𝐮(𝑥, 𝑡, 𝛽), û(𝑥, 𝑡, 𝛼))𝐯 ⋅ ∇𝜑(𝑥, 𝑡)𝑑𝛼𝑑𝛽𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ , ( 18
) lim 𝜖→0 lim 𝛿→0 lim 𝑚→∞ lim 𝑙→∞ lim 𝑛→∞ [ 𝐼 5 + 𝐽 5 ] = 0, (19)
lim 𝜖→0 lim 𝛿→0 lim 𝑚→∞ lim 𝑙→∞ lim 𝑛→∞ [ 𝐼 6 + 𝐽 6 + 𝐼 7 + 𝐽 7 ] = 0, (20)
and lim

𝜖→0 lim 𝛿→0 lim 𝑚→∞ lim 𝑙→∞ lim 𝑛→∞ 𝐽 8 = 0. ( 21
)
We refer to 2 for the detailed proofs of ( 15) -( 19) and of ( 21). Here we will only prove (20). Since we are dealing with a 𝑄-Brownian motion instead of a one-dimensional Brownian motion, the proof of ( 20) is novel with respect to the proofs given in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] .

However, let us remark some simple facts. The partial derivatives 𝜕 𝑡 [𝜑(𝑦, 𝑠)] and ∇ 𝑥 [𝜑(𝑦, 𝑠)] in the terms 𝐼 2 and 𝐼 4 respectively vanish. Therefore 𝐼 2 = 𝐼 4 = 0. Moreover, since 𝑡 ∈ (0, 𝑇 ) and since the support of 𝜌 𝑛 is included in ℝ -, the term 𝐽 1 vanishes.

Let us now prove (20). We start by showing the following result.

Lemma 2. 𝐽 7 = 0.

Proof. We set  𝑚 𝑛 = 𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠), and use the fact that supp 𝜌 𝑛 ⊂ ℝ -to obtain

𝐽 7 = E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ ℝ 𝑇 ∫ 0 ∫ 𝕋 𝑑 𝜂 ′ 𝛿 (𝑢 𝜖 -𝑘)𝑔(𝑢 𝜖 )𝜑(𝑦, 𝑠) 𝑚 𝑛 𝑑𝑊 (𝑦, 𝑠)𝑑𝑦 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ = ∫ 𝑄 𝑇 ∫ ℝ E ⎡ ⎢ ⎢ ⎣ 𝑇 ∫ 𝑡 ∫ 𝕋 𝑑 𝜂 ′ 𝛿 (𝑢 𝜖 -𝑘)𝑔(𝑢 𝜖 )𝜑(𝑦, 𝑠) 𝑚 𝑛 𝑑𝑊 (𝑦, 𝑠)𝑑𝑦 1 ∫ 0  𝑙 𝑘 𝑑𝛼 ⎤ ⎥ ⎥ ⎦ 𝑑𝑘𝑑𝑥𝑑𝑡.
The above expectation vanishes, since

1 ∫ 0  𝑙 𝑘 𝑑𝛼 is  𝑡 -measurable and 𝑡 ∫ 0 𝑔(𝑢 𝜖 )𝜂 ′ 𝛿 (𝑢 𝜖 -𝑘)𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠)𝑑𝑊 (𝑦, 𝑠) is
 𝑡 -martingale for each 𝑦. This yields that 𝐽 7 = 0.

Next we study the limits of the sum of the Itô correction terms 𝐼 6 + 𝐽 6 as 𝑛 and 𝑙 tend to infinity.

Lemma 3. We have the following limit properties:

𝐼 6 + 𝐽 6 ⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖⃗ 𝑛 → ∞ 1 2 E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔 2 (û(𝑥, 𝑡, 𝛼))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑄(𝑥, 𝑥)𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑡) -𝑘)𝑑𝛼𝑑𝑘𝑑𝑦𝑑𝑥𝑑𝑡 ] + 1 2 E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝜂 ′′ 𝛿 (𝑢 𝜖 (𝑦, 𝑡) -𝑘)𝑔 2 (𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑄(𝑦, 𝑦) 1 ∫ 0 𝜌 𝑙 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑑𝛼𝑑𝑥𝑑𝑡𝑑𝑘𝑑𝑦 ] ⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖ ⃗ 𝑙 → ∞ 1 2 E [ ∫ 𝕋 𝑑 ∫ 𝑄 𝑇 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝑔 2 (û(𝑥, 𝑡, 𝛼))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑄(𝑥, 𝑥)𝑑𝛼𝑑𝑥𝑑𝑡𝑑𝑦 ] + 1 2 E [ ∫ 𝕋 𝑑 ∫ 𝑄 𝑇 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝑔 2 (𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑄(𝑦, 𝑦)𝑑𝛼𝑑𝑥𝑑𝑡𝑑𝑦 ] .
Proof. We only justify the limit as 𝑛 → ∞. We define 𝐴 1 as the difference of 𝐼 6 + 𝐽 6 and the limit term and perform a change of variables from 𝑘 to 𝑢 𝜖 (𝑦, 𝑠) -𝑘 when 𝑦 and 𝑠 are fixed to obtain

𝐴 1 = 1 2 E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔 2 (û(𝑥, 𝑡, 𝛼))𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠)𝜑(𝑦, 𝑠)𝑄(𝑥, 𝑥)𝑑𝛼𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑠) -𝑘)𝑑𝑘𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡 ] - 1 2 E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔 2 (û(𝑥, 𝑡, 𝛼))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑄(𝑥, 𝑥)𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑡) -𝑘)𝑑𝛼𝑑𝑘𝑑𝑦𝑑𝑥𝑑𝑡 ] + 1 2 E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝑄 𝑇 𝜂 ′′ 𝛿 (𝑢 𝜖 (𝑦, 𝑠) -𝑘)𝑔 2 (𝑢 𝜖 (𝑦, 𝑠))𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑛 (𝑡 -𝑠)𝜑(𝑦, 𝑠)𝑄(𝑦, 𝑦)𝑑𝑦𝑑𝑠 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ] - 1 2 E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝜂 ′′ 𝛿 (𝑢 𝜖 (𝑦, 𝑡) -𝑘)𝑔 2 (𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑄(𝑦, 𝑦)𝑑𝑦 1 ∫ 0  𝑙 𝑘 𝑑𝛼𝑑𝑘𝑑𝑥𝑑𝑡 ] .
Therefore, since 𝜂 ′′ 𝛿 is bounded and Lipschitz-continuous, 𝜑 ∈ , |𝑄(𝑥, 𝑥)| ≤ Λ 1 for all 𝑥 ∈ 𝕋 𝑑 and 𝑔 is a bounded function, after some computations, we deduce that

|𝐴 1 | ≤ 𝑐(𝑚, 𝜑, Λ 1 , 𝑀 2 𝑔 ) 2 E [ ∫ 𝑄 𝑇 ∫ 𝕋 𝑑 𝑇 ∫ 0 1 ∫ 0 min(2‖𝜂 ′′ 𝛿 ‖ ∞ , |𝑢 𝜖 (𝑦, 𝑠) -𝑢 𝜖 (𝑦, 𝑡)|)𝜌 𝑛 (𝑡 -𝑠)𝑑𝛼𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝑠 ] + 𝑐(𝜑, Λ 1 , 𝑀 2 𝑔 ) 2 E ⎡ ⎢ ⎢ ⎣ 𝑇 ∫ 0 ∫ 𝑄 𝑇 min(2‖𝜂 ′′ 𝛿 ‖ ∞ , |𝑢 𝜖 (𝑦, 𝑠) -𝑢 𝜖 (𝑦, 𝑡)|)𝜌 𝑛 (𝑡 -𝑠)𝑑𝑦𝑑𝑠𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + 𝑐(𝜑, 𝜂 ′′ 𝛿 , Λ 1 ) 2 E ⎡ ⎢ ⎢ ⎣ 𝑇 ∫ 0 ∫ 𝑄 𝑇 [𝑔 2 (𝑢 𝜖 (𝑦, 𝑠)) -𝑔 2 (𝑢 𝜖 (𝑦, 𝑡))]𝜌 𝑛 (𝑡 -𝑠)𝑑𝑦𝑑𝑠𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + 𝑐(𝜑, 𝜂 ′′ 𝛿 , Λ 1 ) 2𝑛 𝑀 2 𝑔 |𝕋 𝑑 |.
One can then prove that the 4 terms above tend to 0 as 𝑛 → ∞. One can justify the limit as 𝑙 → ∞ in a similar way as in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] .

Next we study the stochastic integral term, namely 𝐼 7 . Proof. Using the same type of arguments as in the proof of Lemma 2, we deduce that

E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝑠 ∫ 𝑠-2 𝑛 1 ∫ 0 𝜂 ′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜌 𝑛 (𝑡 -𝑠)𝑑𝑥𝑑𝑊 (𝑥, 𝑡)𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑠 - 2 𝑛 ) -𝑘)𝑑𝑘𝑑𝑦𝑑𝑠 ] = 0. (22)
Therefore subtracting equality (22) from the expression of 𝐼 7 , we deduce that

𝐼 7 = E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝑠 ∫ 𝑠-2 𝑛 1 ∫ 0 𝜂 ′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜌 𝑛 (𝑡 -𝑠)𝑑𝑊 (𝑥, 𝑡)𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝑑𝑥𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑠) -𝑘)𝑑𝑘𝑑𝑦𝑑𝑠 ] -E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝑠 ∫ 𝑠-2 𝑛 1 ∫ 0 𝜂 ′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜌 𝑛 (𝑡 -𝑠)𝑑𝑥𝑑𝑊 (𝑥, 𝑡)𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑠 - 2 𝑛 ) -𝑘)𝑑𝑘𝑑𝑦𝑑𝑠 ] = E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝑠 ∫ 𝑠-2 𝑛 1 ∫ 0 𝜂 ′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜌 𝑛 (𝑡 -𝑠)𝑑𝑊 (𝑥, 𝑡)𝜌 𝑚 (𝑥 -𝑦)𝑑𝑥 × 𝜑(𝑦, 𝑠) ( 𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑠) -𝑘) -𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑠 - 2 𝑛 ) -𝑘) ) 𝑑𝑘𝑑𝑦𝑑𝑠 ] .
We define 𝐴 𝜖 = 𝜖Δ𝑢 𝜖 -div(𝐯𝑓 (𝑢 𝜖 )) and apply Itô's formula for (𝑢 𝜖 (𝑡), 𝑡) taking (𝑋, 𝑡) = 𝜌 𝑙 (𝑋). Then, noting

 𝑋 (𝑋, 𝑡) = 𝜌 ′ 𝑙 (𝑋),  𝑋𝑋 (𝑋, 𝑡) = 𝜌 ′′ 𝑙 (𝑋),  𝑡 (𝑋, 𝑡) = 0
we deduce that for almost all 𝑦 ∈ 𝕋 𝑑 and almost all 𝑠 ∈ (0, 𝑇 ), and

𝑛 ≥ 2 𝑠 𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑠) -𝑘) -𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑠 - 2 𝑛 ) -𝑘) = 𝑠 ∫ 𝑠-2 𝑛 𝜌 ′ 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝐴 𝜖 (𝑦, 𝜎)𝑑𝜎 + 𝑠 ∫ 𝑠-2 𝑛 𝜌 ′ 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝑔(𝑢 𝜖 (𝑦, 𝜎))𝑑𝑊 (𝑦, 𝜎) + 1 2 𝑠 ∫ 𝑠-2 𝑛 𝜌 ′′ 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝑔 2 (𝑢 𝜖 (𝑦, 𝜎))𝑄(𝑦, 𝑦)𝑑𝜎 = - 𝜕 𝜕𝑘 [ 𝑠 ∫ 𝑠-2 𝑛 𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝐴 𝜖 (𝑦, 𝜎)𝑑𝜎 + 𝑠 ∫ 𝑠-2 𝑛 𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝑔(𝑢 𝜖 (𝑦, 𝜎))𝑑𝑊 (𝑦, 𝜎) + 1 2 𝑠 ∫ 𝑠-2 𝑛 𝜌 ′ 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝑔 2 (𝑢 𝜖 (𝑦, 𝜎))𝑄(𝑦, 𝑦)𝑑𝜎 ] .
After some computation, we obtain

𝐼 7 = -E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝑠 ∫ 𝑠-2 𝑛 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜌 𝑛 (𝑡 -𝑠)𝑑𝑊 (𝑥, 𝑡)𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑠)𝑑𝑥 × [ 𝑠 ∫ 𝑠-2 𝑛 𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝐴 𝜖 (𝑦, 𝜎)𝑑𝜎 + 𝑠 ∫ 𝑠-2 𝑛 𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝑔(𝑢 𝜖 (𝑦, 𝜎))𝑑𝑊 (𝑦, 𝜎) + 1 2 𝑠 ∫ 𝑠-2 𝑛 𝜌 ′ 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝑔 2 (𝑢 𝜖 (𝑦, 𝜎))𝑄(𝑦, 𝑦)𝑑𝜎 ] 𝑑𝑘𝑑𝑦𝑑𝑠 ] =∶ 𝕀 1 + 𝕀 2 + 𝕀 3
In the sequel, we prove the limits lim 𝑛→∞ 𝕀 1 = 0 and lim

𝑛→∞ 𝕀 3 = 0; and lim 𝑙→∞ lim 𝑛→∞ 𝕀 2 = -∫ 𝑄 𝑇 ∫ 𝕋 𝑑 E ⎡ ⎢ ⎢ ⎣ 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝑔(û(𝑥, 𝑡, 𝛼))𝑔(𝑢 𝜖 (𝑦, 𝑡))𝑑𝛼 ⎤ ⎥ ⎥ ⎦ 𝜌 𝑚 (𝑥 -𝑦)𝑄(𝑥, 𝑦)𝜑(𝑦, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑡.
We start by estimating 𝕀 1 . Setting

𝐺 𝕀 1 ∶= 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜌 𝑛 (𝑡 -𝑠)𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡),
recalling the assumption that

∞ ∑ 𝑗=1 𝜆 𝑗 ‖𝑒 𝑗 ‖ 2 𝐿 ∞ (𝕋 𝑑 ) ≤ Λ 1
, and using the properties that |𝜌 𝑚 | ≤ 𝐶𝑚 𝑑 , |𝜌 𝑛 | ≤ 𝐶𝑛 and |𝜌 𝑙 | ≤ 𝐶𝑙 for some constant 𝐶, Cauchy-Schwarz inequality and the Itô isometry, we obtain after some computations that

|𝕀 1 | = | | | | | | | | ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 E ⎡ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜ ⎜ ⎜ ⎝ 𝑠 ∫ 𝑠-2 𝑛 𝐺 𝕀 1 𝑑𝑊 (𝑥, 𝑡) ⎞ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ 𝑠 ∫ 𝑠-2 𝑛 𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝐴 𝜖 (𝑦, 𝜎)𝑑𝜎 ⎞ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎦ ≤ 𝐶(𝜑, 𝜂 𝛿 )𝑙 √ Λ 1 √ 𝑛 [ 𝑀 2 𝑔 + ‖𝐴 𝜖 ‖ 2 𝐿 2 ((0,𝑇 )×Ω×𝕋 𝑑 )
] ,

which tends to 0 as 𝑛 tends to ∞. Lemma 14 in section 4 ensures that ‖𝐴 𝜖 ‖ 2 𝐿 2 ((0,𝑇 )×Ω×𝕋 𝑑 ) is bounded. Using a similar idea and the fact that |𝑄(𝑥, 𝑦)| ≤ Λ 1 for all 𝑥, 𝑦 ∈ 𝕋 𝑑 we prove that lim 𝑛→∞ 𝕀 3 = 0. We now consider the term 𝕀 2 . Applying the Itô isometry by observing that 𝑑𝑊 (𝑥, 𝑡)𝑑𝑊 (𝑦, 𝜎) = 𝑄(𝑥, 𝑦)𝛿(𝑡 -𝜎)𝑑𝑡, we obtain:

𝕀 2 = -E [ ∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 𝑠 ∫ 𝑠-2 𝑛 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜌 𝑛 (𝑡 -𝑠)𝑑𝑊 (𝑥, 𝑡)𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝑑𝑥 × ⎛ ⎜ ⎜ ⎜ ⎝ 𝑠 ∫ 𝑠-2 𝑛 𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝜎) -𝑘)𝑔(𝑢 𝜖 (𝑦, 𝜎))𝑑𝑊 (𝑦, 𝜎) ⎞ ⎟ ⎟ ⎟ ⎠ 𝑑𝑘𝑑𝑦𝑑𝑠 ] = -∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 E ⎡ ⎢ ⎢ ⎢ ⎣ 𝑠 ∫ 𝑠-2 𝑛 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜌 𝑛 (𝑡 -𝑠)𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑡) -𝑘)𝑔(𝑢 𝜖 (𝑦, 𝑡))𝑑𝑡 ⎤ ⎥ ⎥ ⎥ ⎦ × 𝜑(𝑦, 𝑠)𝜌 𝑚 (𝑥 -𝑦)𝑄(𝑥, 𝑦)𝑑𝑥𝑑𝑘𝑑𝑦𝑑𝑠 ⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖⃗ 𝑛 → ∞ -∫ 𝑄 𝑇 ∫ ℝ ∫ 𝕋 𝑑 E ⎡ ⎢ ⎢ ⎣ 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑘)𝑔(û(𝑥, 𝑡, 𝛼))𝑑𝛼𝜌 𝑙 (𝑢 𝜖 (𝑦, 𝑡) -𝑘)𝑔(𝑢 𝜖 (𝑦, 𝑡)) ⎤ ⎥ ⎥ ⎦ 𝜑(𝑦, 𝑡)𝜌 𝑚 (𝑥 -𝑦)𝑄(𝑥, 𝑦)𝑑𝑥𝑑𝑘𝑑𝑦𝑑𝑡 ⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖ ⃗ 𝑙 → ∞ -∫ 𝑄 𝑇 ∫ 𝕋 𝑑 E ⎡ ⎢ ⎢ ⎣ 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝑔(û(𝑥, 𝑡, 𝛼))𝑔(𝑢 𝜖 (𝑦, 𝑡))𝑑𝛼 ⎤ ⎥ ⎥ ⎦ 𝜑(𝑦, 𝑡)𝜌 𝑚 (𝑥 -𝑦)𝑄(𝑥, 𝑦)𝑑𝑥𝑑𝑦𝑑𝑡.
The limits of 𝑛 → ∞ and 𝑙 → ∞ are proved by using the fact that 𝑔 is a bounded function, |𝑄(𝑥, 𝑦)| ≤ Λ 1 , 𝜂 ′′ 𝜎 is a Lipschitz-continuous function and supp 𝜌 𝑙 ⊂ [- [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF] 𝑙 ,

+ 1 2 E [ ∫ 𝑄 𝑇 ∫ 𝕋 𝑑 1 ∫ 0 𝑔 2 (𝑢 𝜖 (𝑦, 𝑡))𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑄(𝑦, 𝑦)𝑑𝛼𝑑𝑦𝑑𝑥𝑑𝑡 ] -∫ 𝑄 𝑇 ∫ 𝕋 𝑑 E ⎡ ⎢ ⎢ ⎣ 1 ∫ 0 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝑔(û(𝑥, 𝑡, 𝛼))𝑔(𝑢 𝜖 (𝑦, 𝑡))𝑑𝛼 ⎤ ⎥ ⎥ ⎦ 𝜑(𝑦, 𝑡)𝜌 𝑚 (𝑥 -𝑦)𝑄(𝑥, 𝑦)𝑑𝑦𝑑𝑥𝑑𝑡 = 1 2 E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ 𝕋 𝑑 1 ∫ 0 ∞ ∑ 𝑗=1 𝜆 𝑗 [ 𝑔(û(
] 2 + 2𝑀 2 𝑔 ∞ ∑ 𝑗=1 𝜆 𝑗 [ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦) ] 2 , so that lim 𝑙→∞ lim 𝑛→∞ [ 𝐼 6 + 𝐽 6 + 𝐼 7 ] = 1 2 E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ 𝕋 𝑑 ∞ ∑ 𝑗=1 𝜆 𝑗 [ 𝑔(û(𝑥, 𝑡, 𝛼))𝑒 𝑗 (𝑥) -𝑔(𝑢 𝜖 (𝑦, 𝑡))𝑒 𝑗 (𝑦) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑑𝑦𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ ≤ E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ 𝕋 𝑑 𝑄(𝑦, 𝑦) [ 𝑔(û(𝑥, 𝑡, 𝛼)) -𝑔(𝑢 𝜖 (𝑦, 𝑡)) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑑𝑦𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ 𝕋 𝑑 𝑀 2 𝑔 ∞ ∑ 𝑗=1 𝜆 𝑗 [ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑑𝑦𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ . ( 23
)
First we estimate the first term on the right hand-side of (23); from (7) and the hypothesis (𝐻) for 𝑔 and noting that supp 𝜂 ′′ 𝛿 ⊂ [-𝛿, 𝛿] and 0 ≤ 𝜂 ′′ 𝛿 ≤ 2𝜋∕𝛿, it is bounded from above by

Λ 1 𝐶 2 𝑔 E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ 𝕋 𝑑 (û -𝑢 𝜖 ) 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑑𝑦𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ ≤ 2𝜋Λ 1 𝐶 2 𝑔 𝛿 E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ 𝕋 𝑑 𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑑𝑦𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦ ≤ 2𝜋Λ 1 𝐶 2 𝑔 𝛿 ∫ 𝑄 𝑇

𝜑(𝑦, 𝑡)𝑑𝑦𝑑𝑡,

which tends to 0 as 𝛿 tends to 0. In what follows, we prove that the second term on the right-hand-side of (23) tends to 0 as 𝑚 → ∞. Recalling (4) for each 𝜖 1 > 0, there exists 𝐾 0 , such that for all 𝐾 > 𝐾 0 ,

∞ ∑ 𝑗=𝐾+1 𝜆 𝑗 [ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦) ] 2 ≤ 2 ∞ ∑ 𝑗=𝐾+1 𝜆 𝑗 𝑒 2 𝑗 (𝑥) + 2 ∞ ∑ 𝑗=𝐾+1 𝜆 𝑗 𝑒 2 𝑗 (𝑦) ≤ 𝜖 1 , ( 24
)
we choose 𝐾 > 𝐾 0 to divide the infinite sum in the second term of (23) into 2 parts, forgetting the constant 𝑀 2

𝑔 𝑇 ∫ 0 ∫ 𝕋 𝑑 ∫ 𝕋 𝑑 𝜌 𝑚 (𝑥 -𝑦) ∞ ∑ 𝑗=1 𝜆 𝑗 [ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑡 = 𝑇 ∫ 0 ∫ 𝕋 𝑑 ∫ 𝕋 𝑑 𝜌 𝑚 (𝑥 -𝑦) 𝐾 ∑ 𝑗=1 𝜆 𝑗 [ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑡 + 𝑇 ∫ 0 ∫ 𝕋 𝑑 ∫ 𝕋 𝑑 𝜌 𝑚 (𝑥 -𝑦) ∞ ∑ 𝑗=𝐾+1 𝜆 𝑗 [ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑡. ( 25
)
We study the first term on the right-hand side of (25). Using the fact that supp

𝜌 𝑚 ⊂ [- 1 2𝑚 , 1 2𝑚
] 𝑑 , as well as the fact that 𝑒 𝑗 ∈ 𝐶(𝕋 𝑑 ), setting 𝑥 -𝑦 ∶= 𝑧 and (𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦)) [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF]

lim 𝑚→∞ 𝑇 ∫ 0 ∫ 𝕋 𝑑 ∫ 𝕋 𝑑 𝜌 𝑚 (𝑥 -𝑦) ∞ ∑ 𝑗=1 𝜆 𝑗 [ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑡 ≤ 𝜖 1 𝑇 |𝕋 𝑑 |‖𝜂 ′′ 𝛿 ‖ 𝐿 ∞ (𝕋 𝑑 ) ‖𝜑‖ 𝐿 ∞ (𝕋 𝑑 ) for all 𝜖 1 > 0, i.e. lim 𝑚→∞ 𝑇 ∫ 0 ∫ 𝕋 𝑑 ∫ 𝕋 𝑑 𝜌 𝑚 (𝑥 -𝑦) ∞ ∑ 𝑗=1 𝜆 𝑗 [ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑡 = 0.
Recall that

lim 𝑙→∞ lim 𝑛→∞ [ 𝐼 6 + 𝐽 6 + 𝐼 7 ] = 1 2 E ⎡ ⎢ ⎢ ⎣ ∫ 𝑄 𝑇 ∫ 𝕋 𝑑 1 ∫ 0 ∞ ∑ 𝑗=1 𝜆 𝑗 [ 𝑔(û(

EXISTENCE AND UNIQUENESS OF THE STRONG SOLUTION OF THE ASSOCIATED PARABOLIC PROBLEM

In this section, we prove the existence and uniqueness of the strong solution of the associated stochastic nonlinear parabolic problem; to that purpose we apply a time discretization method.

Definition 4. We say that 𝑢 𝜖 is a strong solution of the problem

( 𝜖 ) { 𝑑𝑢 𝜖 -𝜖Δ𝑢 𝜖 𝑑𝑡 + div(𝐯𝑓 (𝑢 𝜖 ))𝑑𝑡 = 𝑔(𝑢 𝜖 )𝑑𝑊 in Ω × 𝕋 𝑑 × (0, 𝑇 ), 𝑢 𝜖 (𝑥, 0) = 𝑢 𝜖 0 (𝑥) for all 𝑥 ∈ 𝕋 𝑑 (27) with initial condition 𝑢 𝜖 0 (𝑥) ∈ 𝐻 1 (𝕋 𝑑 ), if 𝑢 𝜖 satisfies 1. 𝑢 𝜖 ∈ 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))) ∩ 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))
) is an 𝐿 2 (𝕋 𝑑 )-adapted process to the filtration ( 𝑡 );

2. for almost all 𝑡 ∈ (0, 𝑇 ), P-a.s. there holds,

𝑢 𝜖 (⋅, 𝑡) = 𝑢 𝜖 0 + 𝜖 𝑡 ∫ 0 Δ𝑢 𝜖 𝑑𝑠 - 𝑡 ∫ 0 div(𝐯𝑓 (𝑢 𝜖 ))𝑑𝑠 + 𝑡 ∫ 0 𝑔(𝑢 𝜖 )𝑑𝑊 (𝑠), (28)
in 𝐿 2 (𝕋 𝑑 ).

We suppose that 𝑢 𝜖 0 → 𝑢 0 strongly in 𝐿 2 (𝕋 𝑑 ) as 𝜖 → 0, where 𝑢 0 is the initial condition of Problem (1). The following result is the analog of Proposition A.2 in 2 .

Theorem 2. Let 𝜖 > 0 be arbitrary. Problem ( 𝜖 ) admits a unique strong solution.

Proof. We perform an implicit time discretization with Δ𝑡 = 𝑇 ∕𝑁. The scheme is as follows: for a given small positive constant Δ𝑡 and 𝑢 𝜖 𝑛 in 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 )),  𝑛Δ𝑡 -measurable, find 𝑢 𝜖 𝑛+1 in 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 )),  (𝑛+1)Δ𝑡 -measurable, such that P-a.s. and for all 𝑣 in 𝐻 1 (𝕋 𝑑 )

∫ 𝕋 𝑑 [ (𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 )𝑣 + Δ𝑡(𝜖∇𝑢 𝜖 𝑛+1 ⋅ ∇𝑣 -𝑓 (𝑢 𝜖 𝑛+1 )𝐯 ⋅ ∇𝑣) ] 𝑑𝑥 = ∫ 𝕋 𝑑 𝑔(𝑢 𝜖 𝑛 )𝑣(𝑊 𝑛+1 -𝑊 𝑛 ) 𝑑𝑥 ( 29
)
where

𝑊 𝑛 = 𝑊 (𝑡 𝑛 ) = ∞ ∑ 𝑗=1 √ 𝜆 𝑗 𝛽 𝑗 (𝑡 𝑛 )𝑒 𝑗 (𝑥)
; we remark that 𝑢 𝜖 0 = 𝑢 𝜖 0 (𝑥). Before pursuing the proof of Theorem 2, we first show the existence of the sequence {𝑢 𝜖 𝑛 } 𝑛≥0 .

Lemma 6. If Δ𝑡 < 2𝜖 (𝑉 𝐶 𝑓 ) 2 , Problem (29) possesses a unique solution 𝑢 𝜖 𝑛+1 ∈ 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 )).
Proof. The proof is similar to that of [c.f. 2 , Lemma A.

3.]

Next, we prove some a priori estimates.

Lemma 7. We define

𝑢 Δ𝑡 (𝑡) = 𝑁-1 ∑ 𝑛=0 𝑢 𝜖 𝑛+1 ⋅ 𝟏 [𝑛Δ𝑡,(𝑛+1)Δ𝑡) (𝑡) (30)
where 𝑁 = 𝑇 ∕Δ𝑡; then the following hold

‖𝑢 Δ𝑡 ‖ 𝐿 ∞ (0,𝑇 ;𝐿 2 (Ω;𝐿 2 (𝕋 𝑑 ))) ≤ 𝐶 1 , ‖∇𝑢 Δ𝑡 ‖ 𝐿 2 (0,𝑇 ;(𝐿 2 (Ω;𝐿 2 (𝕋 𝑑 ))) 𝑑 ) ≤ 𝐶 2 ,
where the constant 𝐶 1 depends on 𝑢 0 , Λ 1 , 𝑀 𝑔 and 𝑇 , and the constant 𝐶 2 depends on 𝜖, 𝑢 0 , Λ 1 , 𝑀 𝑔 and 𝑇 . Thus,

‖𝑢 Δ𝑡 ‖ 𝐿 2 (0,𝑇 ;𝐿 2 (Ω;𝐻 1 (𝕋 𝑑 ))) ≤ 𝐶
for a constant 𝐶 which depends on 𝜖, 𝑢 0 , Λ 1 , 𝑀 𝑔 and 𝑇 . We define the linear interpolation function

ũΔ𝑡 = 𝑁 ∑ 𝑛=1 [ 𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 Δ𝑡 [𝑡 -(𝑛 -1)Δ𝑡] + 𝑢 𝜖 𝑛-1 ] ⋅ 𝟏 [(𝑛-1)Δ𝑡,𝑛Δ𝑡) . ( 31
)
The set {ũ Δ𝑡 } is bounded in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))) ∩ 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))), and ũΔ𝑡 -𝑢 Δ𝑡 converges to 0 in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))).

Proof. Setting 𝑣 = 𝑢 𝜖 𝑛+1 in (29) and taking the expectation yields

1 2 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ( |𝑢 𝜖 𝑛+1 | 2 -|𝑢 𝜖 𝑛 | 2 + |𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 | 2 ) 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + Δ𝑡𝜖 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 |∇𝑢 𝜖 𝑛+1 | 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ Δ𝑡 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑓 (𝑢 𝜖 𝑛+1 )𝐯 ⋅ ∇𝑢 𝜖 𝑛+1 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ = E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 )𝑔(𝑢 𝜖 𝑛 )(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 )𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 )𝑔(𝑢 𝜖 𝑛 )𝑢 𝜖 𝑛 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ . ( 32
)
Define 𝐹 (𝑠) = 𝑠 ∫ 0 𝑓 (𝜏)𝑑𝜏 and use the fact that div 𝐯 = 0; then for all functions 𝑣 ∈ 𝐻 1 (𝕋 𝑑 )

∫ 𝕋 𝑑 𝑓 (𝑣)𝐯 ⋅ ∇𝑣𝑑𝑥 = ∫ 𝕋 𝑑 𝐯 ⋅ ∇𝐹 (𝑣)𝑑𝑥 = -∫ 𝕋 𝑑 𝐹 (𝑣) div 𝐯𝑑𝑥 = 0.
Next we use ( 6), ( 7), the equality

E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 )𝑔(𝑢 𝜖 𝑛 )𝑢 𝜖 𝑛 𝑑𝑥 ⎤ ⎥ ⎥ ⎦
= 0, and the monotone convergence theorem to deduce that

E [ (𝑊 𝑛+1 -𝑊 𝑛 ) 2 ] = E ⎡ ⎢ ⎢ ⎣ ( ∞ ∑ 𝑗=0 √ 𝜆 𝑗 (𝛽 𝑗 (𝑡 𝑛+1 ) -𝛽 𝑗 (𝑡 𝑛 ))𝑒 𝑗 (𝑥) ) 2 ⎤ ⎥ ⎥ ⎦ = E [ ∞ ∑ 𝑗=0 𝜆 𝑗 (𝛽 𝑗 (𝑡 𝑛+1 ) -𝛽 𝑗 (𝑡 𝑛 )) 2 𝑒 2 𝑗 (𝑥) ] (33) = ∞ ∑ 𝑗=0 E [ 𝜆 𝑗 (𝛽 𝑗 (𝑡 𝑛+1 ) -𝛽 𝑗 (𝑡 𝑛 )) 2 𝑒 2 𝑗 (𝑥) ] = Δ𝑡𝑄(𝑥, 𝑥) ≤ Λ 1 Δ𝑡.
Applying Young's inequality and (33) yields

E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 )𝑔(𝑢 𝜖 𝑛 )(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 )𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ≤ 1 2 ε E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 ) 2 𝑔 2 (𝑢 𝜖 𝑛 )𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + ε 2 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 ) 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ≤ Δ𝑡 2 ε ∫ 𝕋 𝑑 E [ 𝑔 2 (𝑢 𝜖 𝑛 ) ] 𝑄(𝑥, 𝑥)𝑑𝑥 + ε 2 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 ) 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ , ( 34
)
for all ε > 0. We choose ε = 1 2 and deduce from (32) that

1 2 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ( |𝑢 𝜖 𝑛+1 | 2 -|𝑢 𝜖 𝑛 | 2 + |𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 | 2 ) 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + Δ𝑡𝜖 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 |∇𝑢 𝜖 𝑛+1 | 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ≤Δ𝑡 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑔 2 (𝑢 𝜖 𝑛 )𝑄(𝑥, 𝑥)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + 1 4 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 ) 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ≤Λ 1 Δ𝑡 ∫ 𝕋 𝑑 E [ 𝑔 2 (𝑢 𝜖 𝑛 ) ] 𝑑𝑥 + 1 4 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 ) 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ . ( 35
)
We denote by ‖⋅‖ the norm in 𝐿 2 (𝕋 𝑑 ), change the index in (35) to 𝑘 and sum from 0 to 𝑛 -1, for all 𝑛 ≥ 1, to obtain,

1 2 E [ ‖𝑢 𝜖 𝑛 ‖ 2 ] + 1 4 𝑛-1 ∑ 𝑘=0 E [ ‖𝑢 𝜖 𝑘+1 -𝑢 𝜖 𝑘 ‖ 2 ] + Δ𝑡𝜖 𝑛-1 ∑ 𝑘=0 E [ ‖∇𝑢 𝜖 𝑘+1 ‖ 2 ] ≤ 1 2 ‖𝑢 𝜖 0 ‖ 2 + Λ 1 Δ𝑡 𝑛-1 ∑ 𝑘=0 E [ ‖𝑔(𝑢 𝜖 𝑘 )‖ 2 ] ≤ 1 2 ‖𝑢 𝜖 0 ‖ 2 + Λ 1 𝑇 𝑀 2 𝑔 , ( 36
) so that E [ ‖𝑢 𝜖 𝑛 ‖ 2 ] ≤ 𝐶 1 , ( 37
)
where the constant 𝐶 1 depends on 𝑢 𝜖 0 , Λ 1 , 𝑀 𝑔 and 𝑇 , and does not depend on 𝑛. In view of (37), and the definition of

𝑢 Δ𝑡 we obtain ‖𝑢 Δ𝑡 ‖ 𝐿 ∞ (0,𝑇 ;𝐿 2 (Ω;𝐿 2 (𝕋 𝑑 ))) ≤ √ 𝐶 1 , (38)
and we deduce from (36) that

√ 𝜖‖∇𝑢 Δ𝑡 ‖ 𝐿 2 (0,𝑇 ;𝐿 2 (Ω;𝐿 2 (𝕋 𝑑 ))) ≤ 𝐶 2 , ( 39
)
where the constant 𝐶 2 depends on 𝐶 1 . In turn ( 38) and (39) imply that

‖𝑢 Δ𝑡 ‖ 𝐿 2 (0,𝑇 ;𝐿 2 (Ω;𝐻 1 (𝕋 𝑑 ))) ≤ 𝐶, ( 40
)
for a constant 𝐶 which depends on 𝐶 1 , 𝜖 and 𝑇 . In view of (36) and ( 37)

‖ũ Δ𝑡 -𝑢 Δ𝑡 ‖ 2 𝐿 2 (0,𝑇 ;𝐿 2 (Ω×𝕋 𝑑 )) = E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑁 ∑ 𝑛=1 𝑛Δ𝑡 ∫ (𝑛-1)Δ𝑡 (𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 ) 2 ( 𝑡 -𝑛Δ𝑡 Δ𝑡 ) 2 𝟏 [(𝑛-1)Δ𝑡,𝑛Δ𝑡] 𝑑𝑡𝑑𝑥 ⎤ ⎥ ⎥ ⎦ = 1 3 Δ𝑡 𝑁 ∑ 𝑛=1 E [ ‖𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 ‖ 2 𝐿 2 (𝕋 𝑑 ) ] ≤ 4 3 Δ𝑡( 1 2 ‖𝑢 𝜖 0 ‖ 2 + Λ 1 𝑇 𝑀 2 𝑔 ) (41)
which converges to 0 as Δ𝑡 tends to 0. Next we prove that

∇(𝑢 Δ𝑡 -ũΔ𝑡 ) ⇀ 0, weakly in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ) 𝑑 )).

Consider a test function 𝜙(𝑥, 𝑡)

= 𝑁 ∑ 𝑛=1 𝜑(𝑥) 𝑛Δ𝑡 -𝑡 Δ𝑡 𝟏 [(𝑛-1)Δ𝑡,𝑛Δ𝑡) with 𝜑(𝑥) ∈ 𝐻 1 (𝕋 𝑑 ) 𝑑 , so that 𝜙 ∈ 𝐿 2 (0, 𝑇 ; 𝐻 1 (𝕋 𝑑 ) 𝑑 ).
In view of the definitions of 𝑢 Δ𝑡 and of ũΔ𝑡 , we deduce that

E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑇 ∫ 0 ∇(𝑢 Δ𝑡 -ũΔ𝑡 )𝜙(𝑥, 𝑡)𝑑𝑡𝑑𝑥 ⎤ ⎥ ⎥ ⎦ = E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑇 ∫ 0 ∇ ( 𝑢 𝜖 𝑛 - 𝑁 ∑ 𝑛=1 𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 Δ𝑡 [𝑡 -(𝑛 -1)Δ𝑡] -𝑢 𝜖 𝑛-1
)

𝜑(𝑥)𝟏 [(𝑛-1)Δ𝑡,𝑛Δ𝑡) 𝑑𝑡𝑑𝑥 ⎤ ⎥ ⎥ ⎦ = E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑁 ∑ 𝑛=1 𝑛Δ𝑡 ∫ (𝑛-1)Δ𝑡 ( 1 - 𝑡 -(𝑛 -1)Δ𝑡 Δ𝑡 ) (∇𝑢 𝜖 𝑛 -∇𝑢 𝜖 𝑛-1 )𝜑(𝑥)𝑑𝑡𝑑𝑥 ⎤ ⎥ ⎥ ⎦ = E ⎡ ⎢ ⎢ ⎣ Δ𝑡 2 ∫ 𝕋 𝑑 𝑁 ∑ 𝑛=1 (∇𝑢 𝜖 𝑛 -∇𝑢 𝜖 𝑛-1 )𝜑(𝑥)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ = -E ⎡ ⎢ ⎢ ⎣ Δ𝑡 2 𝑁 ∑ 𝑛=1 ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 ) div 𝜑(𝑥)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ . Since by Cauchy-Schwarz inequality | | | | | | | E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 ) div 𝜑(𝑥)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ | | | | | | | ≤ E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 ) 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 |div 𝜑(𝑥)| 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ , it follows that | | | | | | | E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑇 ∫ 0 ∇(𝑢 Δ𝑡 -ũΔ𝑡 )𝜙(𝑥, 𝑡)𝑑𝑡𝑑𝑥 ⎤ ⎥ ⎥ ⎦ | | | | | | | = | | | | | | | E ⎡ ⎢ ⎢ ⎣ Δ𝑡 2 𝑁 ∑ 𝑛=1 ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 ) div 𝜑(𝑥)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ | | | | | | | ≤ Δ𝑡 2 𝑁 ∑ 𝑛=1 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 ) 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 |div 𝜑(𝑥)| 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦
which converges to 0 in view of (36). This implies 𝑢 Δ𝑡 -ũΔ𝑡 converges weakly to 0 in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))).

We deduce from Lemma 7 that there exist a subsequence of {Δ𝑡} which we denote by {Δ𝑡 𝑗 } and functions 𝑢 ∈ 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))), 𝑓 𝑢 ∈ 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))) and 𝑔 𝑢 ∈ 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))) such that 𝑢 Δ𝑡 𝑗 ⇀ 𝑢 and ũΔ𝑡 𝑗 ⇀ 𝑢 weakly in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))) as Δ𝑡 𝑗 → 0, and

𝑓 (𝑢 Δ𝑡 𝑗 ) ⇀ 𝑓 𝑢 and 𝑔(𝑢 Δ𝑡 𝑗 ) ⇀ 𝑔 𝑢 weakly in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))) as Δ𝑡 𝑗 → 0. Since 𝑔 is a bounded function, 𝑔(𝑢 Δ𝑡 ) is bounded in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))) uniformly in Δ𝑡. Moreover E [ ‖𝑓 (𝑢 𝑛 )‖ 2 ] = E [ ‖𝑓 (𝑢 𝑛 ) -𝑓 (0)‖ 2 ] ≤ 𝐶 2 𝑓 E [ ‖𝑢 𝑛 ‖ 2 ] ≤ 𝐶 2 𝑓 𝐶 1 , (42)
so that 𝑓 (𝑢 Δ𝑡 ) is bounded in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))) uniformly in Δ𝑡. In view of (40), there exist a function 𝑢 ∈ 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))) and a subsequence {𝑢 Δ𝑡 𝑗 } of {𝑢 Δ𝑡 } which converges weakly to 𝑢 in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))) as Δ𝑡 𝑗 tends to zero. Moreover (42) and the assumption on 𝑔 imply that there exist functions 𝑓 𝑢 and 𝑔 𝑢 and a subsequence of {𝑢 Δ𝑡 𝑗 }, which we denote again by {𝑢 Δ𝑡 𝑗 }, such that 𝑓 (𝑢 Δ𝑡 𝑗 ) converges weakly to 𝑓 𝑢 and 𝑔(𝑢 Δ𝑡 𝑗 ) converges weakly to 𝑔 𝑢 in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))) as Δ𝑡 𝑗 tends to zero. For later use, we define

BΔ𝑡 𝑗 = 𝑁 ∑ 𝑛=1 [ 𝐵 𝑛 -𝐵 𝑛-1 Δ𝑡 𝑗 [𝑡 -(𝑛 -1)Δ𝑡 𝑗 ] + 𝐵 𝑛-1 ] ⋅ 𝟏 [(𝑛-1)Δ𝑡 𝑗 ,𝑛Δ𝑡 𝑗 ) (43)
where

𝐵 𝑛 = 𝑛-1 ∑ 𝑘=0 (𝑊 𝑘+1 -𝑊 𝑘 )𝑔(𝑢 𝑘 ) = 𝑛Δ𝑡 𝑗 ∫ 0 𝑔(𝑢 Δ𝑡 𝑗 (⋅ -Δ𝑡 𝑗 ))𝑑𝑊 , ( 44
)
in which we set 𝑢 Δ𝑡 𝑗 (𝑠) = 𝑢 Δ𝑡 𝑗 (0) = 𝑢 0 , for all 𝑠 < 0. One can check that { BΔ𝑡 𝑗 } is uniformly bounded in 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω × 𝕋 𝑑 )).

Next, we identify the weak limit of ∫ 𝑡 0 𝑔(𝑢 Δ𝑡 𝑗 (𝑠 -Δ𝑡 𝑗 ))𝑑𝑊 (𝑠) when Δ𝑡 𝑗 → 0 in 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 )) for all 𝑡 ∈ [0, 𝑇 ]. Lemma 8. We have the following limit property

𝑡 ∫ 0 𝑔(𝑢 Δ𝑡 𝑗 (⋅ -Δ𝑡 𝑗 ))𝑑𝑊 (𝑥, 𝑡) → 𝑡 ∫ 0 𝑔 𝑢 𝑑𝑊 (𝑥, 𝑡) (45)
as Δ𝑡 𝑗 , weakly in 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 )) for all 𝑡 ∈ [0, 𝑇 ].

Proof. We set 𝑢 Δ𝑡 𝑗 (𝑠) = 𝑢 0 , for all 𝑠 < 0.

We define the operator  as

 ( ) = 𝑡 ∫ 0  𝑑𝑊 (𝑥, 𝑠) (46)
where 𝑊 (𝑥, 𝑡) is the 𝑄-Brownian motion defined by (3) satisfying the condition (4). Using the same idea as in the proof of (45) in [c.f. 7 , Lemma 5.4], we conclude that along a subsequence

 (𝑔(𝑢 Δ𝑡 𝑗 (⋅ -Δ𝑡 𝑗 ))) ⇀  ( 𝑔 𝑢 )
weakly in 𝐿 2 (Ω × 𝕋 𝑑 ),as Δ𝑡 𝑗 → 0, or in other words, in view of the definition (46) of  ,

𝑡 ∫ 0 𝑔(𝑢 Δ𝑡 𝑗 (⋅ -Δ𝑡 𝑗 ))𝑑𝑊 (𝑥, 𝑡) → 𝑡 ∫ 0 𝑔 𝑢 𝑑𝑊 (𝑥, 𝑡)
weakly in 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 )) for all 𝑡 ∈ [0, 𝑇 ].

Lemma 9.

For almost all 𝑡 ∈ (0, 𝑇 ), P-a.s., there holds,

𝑢(𝑡) = 𝑢 𝜖 0 + 𝜖 𝑡 ∫ 0 Δ𝑢𝑑𝑠 - 𝑡 ∫ 0 div(𝐯𝑓 𝑢 )𝑑𝑠 + 𝑡 ∫ 0 𝑔 𝑢 𝑑𝑊 (𝑠), in ( 𝐻 1 (𝕋 𝑑 ) ) ′
, where 𝑢 is the weak limit of 𝑢 Δ𝑡 𝑗 in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))) and its weak- * limit in 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))).

Proof. For all 𝑡 ∈ [0, 𝑇 ], there exists 𝑛 such that 𝑡 ∈ [𝑛Δ𝑡 𝑗 , (𝑛 + 1)Δ𝑡 𝑗 ). We return to (29) and suppose that the test function 𝑣 ∈ 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 )) and is  𝑡 -adapted, and sum 𝑘 from 0 to 𝑛 to deduce that P-a.s.

𝑛 ∑ 𝑘=0 ∫ 𝕋 𝑑 (𝑢 𝜖 𝑘+1 -𝑢 𝜖 𝑘 )𝑣𝑑𝑥 + Δ𝑡 𝑗 𝑛 ∑ 𝑘=0 ∫ 𝕋 𝑑 ( 𝜖∇𝑢 𝜖 𝑘+1 ⋅ ∇𝑣 -𝑓 (𝑢 𝜖 𝑘+1 )𝐯 ⋅ ∇𝑣 ) 𝑑𝑥 = 𝑛 ∑ 𝑘=0 ∫ 𝕋 𝑑 [ 𝑣𝑔(𝑢 𝜖 𝑘 )(𝑊 𝑘+1 -𝑊 𝑘 ) ] 𝑑𝑥, which yields ∫ 𝕋 𝑑 (𝑢 𝜖 𝑛+1 𝑣 -𝑢 𝜖 0 𝑣)𝑑𝑥 + Δ𝑡 𝑗 𝑛 ∑ 𝑘=0 ∫ 𝕋 𝑑 ( 𝜖∇𝑢 𝜖 𝑘+1 ⋅ ∇𝑣 -𝑓 (𝑢 𝜖 𝑘+1 )𝐯 ⋅ ∇𝑣 ) 𝑑𝑥 = Δ𝑡 𝑗 𝑛 ∑ 𝑘=0 ∫ 𝕋 𝑑 [ 𝑣𝑔(𝑢 𝜖 𝑘 ) 𝑊 𝑘+1 -𝑊 𝑘 Δ𝑡 𝑗 ] 𝑑𝑥
and by definition (30) it is equivalent to

E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑢 Δ𝑡 𝑗 (𝑡)𝑣 -𝑢 𝜖 0 𝑣)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ⎛ ⎜ ⎜ ⎝ 𝑡 ∫ 0 𝜖∇𝑢 Δ𝑡 𝑗 (𝑠) -𝑓 (𝑢 Δ𝑡 𝑗 (𝑠))𝐯𝑑𝑠 ⎞ ⎟ ⎟ ⎠ ⋅ ∇𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎦ -E ⎡ ⎢ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ⎛ ⎜ ⎜ ⎜ ⎝ 𝑡 ∫ 𝑛Δ𝑡 𝑗 𝜖∇𝑢 Δ𝑡 𝑗 (𝑠) -𝑓 (𝑢 Δ𝑡 𝑗 (𝑠))𝐯𝑑𝑠 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ ∇𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ = E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ⎛ ⎜ ⎜ ⎝ 𝑡 ∫ 0 𝑔(𝑢 Δ𝑡 𝑗 (𝑠 -Δ𝑡 𝑗 ))𝑑𝑊 (𝑠) ⎞ ⎟ ⎟ ⎠ 𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎦ -E ⎡ ⎢ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ⎛ ⎜ ⎜ ⎜ ⎝ 𝑡 ∫ 𝑛Δ𝑡 𝑗 𝑔(𝑢 Δ𝑡 𝑗 (𝑠 -Δ𝑡 𝑗 ))𝑑𝑊 (𝑠) ⎞ ⎟ ⎟ ⎟ ⎠ 𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ . ( 47
)
Next we estimate the terms involving the time integral on [𝑛Δ𝑡 𝑗 , 𝑡).

| | | | | | | | E ⎡ ⎢ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑡 ∫ 𝑛Δ𝑡 𝑗 𝜖∇𝑢 Δ𝑡 𝑗 (𝑠) ⋅ ∇𝑣𝑑𝑠𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ | | | | | | | | ≤ E ⎡ ⎢ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑡 ∫ 𝑛Δ𝑡 𝑗 | | | 𝜖∇𝑢 Δ𝑡 𝑗 (𝑠) ⋅ ∇𝑣 | | | 𝑑𝑠𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ ≤ ⎛ ⎜ ⎜ ⎜ ⎝ E ⎡ ⎢ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑡 ∫ 𝑛Δ𝑡 𝑗 | | | ∇𝑣 | | | 2 𝑑𝑠𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎟ ⎠ 1∕2 ⎛ ⎜ ⎜ ⎜ ⎝ E ⎡ ⎢ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑡 ∫ 𝑛Δ𝑡 𝑗 | | | 𝜖∇𝑢 Δ𝑡 𝑗 (𝑠) | | | 2 𝑑𝑠𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎟ ⎠ 1∕2 ≤𝜖 √ Δ𝑡 𝑗 ‖𝑣‖ 𝐿 2 (Ω;𝐻 1 (𝕋 𝑑 )) ‖𝑢 Δ𝑡 𝑗 (𝑠)‖ 𝐿 2 (0,𝑇 ;𝐿 2 (Ω;𝐻 1 (𝕋 𝑑 ))) .
We deduce from a similar computation that

| | | | | | | | E ⎡ ⎢ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑡 ∫ 𝑛Δ𝑡 𝑗 𝑓 (𝑢 Δ𝑡 𝑗 (𝑠))𝐯𝑑𝑠 ⋅ ∇𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ | ⎡ ⎢ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ⎛ ⎜ ⎜ ⎜ ⎝ 𝑡 ∫ 𝑛Δ𝑡 𝑗 𝑔(𝑢 Δ𝑡 𝑗 (𝑠 -Δ𝑡 𝑗 ))𝑑𝑊 (𝑠) ⎞ ⎟ ⎟ ⎟ ⎠ 𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ | | | | | | | | ≤ ⎛ ⎜ ⎜ ⎝ E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑣 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠ 1∕2 ⎛ ⎜ ⎜ ⎜ ⎝ E ⎡ ⎢ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ⎛ ⎜ ⎜ ⎜ ⎝ 𝑡 ∫ 𝑛Δ𝑡 𝑗 𝑔(𝑢 Δ𝑡 𝑗 (𝑠 -Δ𝑡 𝑗 ))𝑑𝑊 (𝑠) ⎞ ⎟ ⎟ ⎟ ⎠ 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎟ ⎠ 1∕2 =‖𝑣‖ 𝐿 2 (Ω;𝐿 2 (𝕋 𝑑 )) 𝑀 𝑔 ⎛ ⎜ ⎜ ⎝ E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑊 (𝑡) -𝑊 𝑛 ) 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠ 1∕2 ≤Δ𝑡 𝑗 Λ 1 𝑀 𝑔 ‖𝑣‖ 𝐿 2 (Ω;𝐿 2 (𝕋 𝑑 ))
We recall that as Δ𝑡 𝑗 → 0, and also using that 𝑢 Δ𝑡 𝑗 is bounded in 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))), we deduce that for a.e. 𝑡 > 0, there exists a subsequence of 𝑢 Δ𝑡 𝑗 (𝑡) which we denote again by 𝑢 Δ𝑡 𝑗 (𝑡) which converges to 𝑢(𝑡) weakly in 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 )). We let Δ𝑡 𝑗 → 0 in (47) to deduce that

𝑢 Δ𝑡 𝑗 ⇀ 𝑢 in 𝐿 2 (
E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑢(𝑡)𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎦ -E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑢 𝜖 0 𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + 𝜖 E ⎡ ⎢ ⎢ ⎣ 𝑡 ∫ 0 ∫ 𝕋 𝑑 ∇𝑢 ⋅ ∇𝑣 𝑑𝑥𝑑𝑠 ⎤ ⎥ ⎥ ⎦ -E ⎡ ⎢ ⎢ ⎣ 𝑡 ∫ 0 ∫ 𝕋 𝑑 𝑓 𝑢 𝐯 ⋅ ∇𝑣 𝑑𝑥𝑑𝑠 ⎤ ⎥ ⎥ ⎦ = E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑡 ∫ 0 𝑔 𝑢 𝑑𝑊 (𝑠)𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎦
for all 𝑣 ∈ 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 )) and all 𝑡 ∈ [0, 𝑇 ], so that

E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ⎛ ⎜ ⎜ ⎝ 𝑢(𝑡) - 𝑡 ∫ 0 𝑔 𝑢 𝑑𝑊 (𝑠) ⎞ ⎟ ⎟ ⎠ 𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎦ -E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑢 𝜖 0 𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + 𝜖 E ⎡ ⎢ ⎢ ⎣ 𝑡 ∫ 0 ∫ 𝕋 𝑑 ∇𝑢 ⋅ ∇𝑣 𝑑𝑥𝑑𝑠 ⎤ ⎥ ⎥ ⎦ -E ⎡ ⎢ ⎢ ⎣ 𝑡 ∫ 0 ∫ 𝕋 𝑑 𝑓 𝑢 𝐯 ⋅ ∇𝑣 𝑑𝑥𝑑𝑠 ⎤ ⎥ ⎥ ⎦ = 0 which is equivalent to E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 ⎛ ⎜ ⎜ ⎝ 𝑢(𝑡) - 𝑡 ∫ 0 𝑔 𝑢 𝑑𝑊 (𝑠) ⎞ ⎟ ⎟ ⎠ 𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎦ -E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑢 𝜖 0 𝑣𝑑𝑥 ⎤ ⎥ ⎥ ⎦ + E ⎡ ⎢ ⎢ ⎣ 𝑡 ∫ 0 ∫ 𝕋 𝑑 ⟨ -𝜖Δ𝑢 + div(𝑓 𝑢 𝐯), 𝑣 ⟩ (𝐻 1 ) ′ ,𝐻 1 𝑑𝑥𝑑𝑠 ⎤ ⎥ ⎥ ⎦ = 0.
Thus for almost all 𝑡 ∈ (0, 𝑇 ), P-a.s., there holds, [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] . In view of (33), we deduce that

𝑢(𝑡) = 𝑢 𝜖 0 (⋅) + 𝜖 𝑡 ∫ 0 Δ𝑢𝑑𝑠 - 𝑡 ∫ 0 div(𝐯𝑓 𝑢 )𝑑𝑠 + 𝑡 ∫ 0 𝑔 𝑢 𝑑𝑊 (𝑠), (48) in
) ∫ ℝ 𝑑 ∇[𝑢 𝑛+1 -𝑢 𝑛 ] ⋅ ∇ℎ(𝑢 𝑛 ) in c.f.
E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 )∇(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 ) ⋅ ∇𝑔(𝑢 𝜖 𝑛 )𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ≤ E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 ) 2 |∇𝑔(𝑢 𝜖 𝑛 )| 2 𝑑𝑥 + 1 4 ∫ 𝕋 𝑑 |∇(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 )| 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ≤ ∫ 𝕋 𝑑 E [ (𝑊 𝑛+1 -𝑊 𝑛 ) 2 ] E [ |∇𝑔(𝑢 𝜖 𝑛 )| 2 ] 𝑑𝑥 + 1 4 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 |∇(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 )| 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ≤Λ 1 Δ𝑡 𝑗 ∫ 𝕋 𝑑 E [ |∇𝑔(𝑢 𝜖 𝑛 )| 2 ] 𝑑𝑥 + 1 4 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 |∇(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 )| 2 𝑑𝑥 ⎤ ⎥ ⎥ ⎦ .
One can obtain an estimate for the term

E [ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 )∇(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 ) ⋅ ∇𝑔(𝑢 𝜖 𝑛 )𝑑𝑥
] in a similar fashion by using (5). We deduce that

E [ ‖𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 -(𝑊 𝑛+1 -𝑊 𝑛 )𝑔(𝑢 𝜖 𝑛 )‖ 2 𝐿 2 (𝕋 𝑑 ) ] + Δ𝑡 𝑗 𝜖 E [ ‖∇𝑢 𝜖 𝑛+1 ‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑 -‖∇𝑢 𝜖 𝑛 ‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑 + 1 2 ‖∇(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 )‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑 ] ≤Λ 1 (Δ𝑡 𝑗 ) 2 𝜖 E [ ‖∇𝑔(𝑢 𝜖 𝑛 )‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑 ] + Λ 2 (Δ𝑡 𝑗 ) 2 𝜖 E [ ‖𝑔(𝑢 𝜖 𝑛 )‖ 2 𝐿 2 (𝕋 𝑑 ) ] + 𝐶(𝐯, 𝐶 𝑓 )(Δ𝑡 𝑗 ) 2 E [ ‖∇𝑢 𝜖 𝑛+1 ‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑
] .

We sum the above inequality on the variable 𝑛 from 0 to 𝑁 -1 and apply Lemma 7 to deduce that

𝑁-1 ∑ 𝑛=0 Δ𝑡 𝑗 E [ ‖ 𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 -(𝑊 𝑛+1 -𝑊 𝑛 )𝑔(𝑢 𝜖 𝑛 ) Δ𝑡 𝑗 ‖ 2 𝐿 2 (𝕋 𝑑 ) ] + 𝜖 E [ ‖∇𝑢 𝜖 𝑁 ‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑 ] + 𝜖 2 𝑁-1 ∑ 𝑛=0 E [ ‖∇(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 )‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑 ‖ ] ≤𝐶(𝐯, 𝐶 𝑓 , 𝐶 𝑔 , 𝜖, Λ 1 )Δ𝑡 𝑗 𝑁 ∑ 𝑛=1 E [ ‖∇𝑢 𝜖 𝑛 ‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑 ] + 𝜖 E [ ‖∇𝑢 𝜖 0 ‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑 ] + Λ 2 Δ𝑡 𝑗 𝜖 𝑁 ∑ 𝑛=0 E [ ‖𝑔(𝑢 𝜖 𝑛 )‖ 2 𝐿 2 (𝕋 𝑑 ) ] ≤ 𝐶, (49)
where 𝐶 depends on 𝐯, 𝐶 𝑓 , 𝐶 𝑔 , Λ 1 , Λ 2 , 𝑢 0 and 𝜖. We take the difference of (31) and (43) to deduce that

ũΔ𝑡 𝑗 -BΔ𝑡 𝑗 = 𝑁 ∑ 𝑛=1 [ 𝑢 𝑛 -𝑢 𝑛-1 -(𝐵 𝑛 -𝐵 𝑛-1 ) Δ𝑡 𝑗 (𝑡 -(𝑛 -1)Δ𝑡 𝑗 ) + (𝑢 𝑛-1 -𝐵 𝑛-1 ) ] 𝟏 [(𝑛-1)Δ𝑡 𝑗 ,𝑛Δ𝑡 𝑗 ) ,
which we differentiate with respect to 𝑡 to obtain

𝜕 𝑡 [ ũΔ𝑡 𝑗 -BΔ𝑡 𝑗 ] = 𝑁 ∑ 𝑛=1 [ 𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 -(𝐵 𝑛 -𝐵 𝑛-1 ) Δ𝑡 𝑗 ] 𝟏 [(𝑛-1)Δ𝑡 𝑗 ,𝑛Δ𝑡 𝑗 ) = 𝑁 ∑ 𝑛=1 [ 𝑢 𝜖 𝑛 -𝑢 𝜖 𝑛-1 -(𝑊 𝑛 -𝑊 𝑛-1 )𝑔(𝑢 𝜖 𝑛-1 ) Δ𝑡 𝑗 ] 𝟏 [(𝑛-1)Δ𝑡 𝑗 ,𝑛Δ𝑡 𝑗 )
where we have also used the definition of 𝐵 𝑛 in (44). The result of Lemma 10 then follows from (49).

Lemma

] ≤ 4 E [ (𝑤 𝑀 (𝑥, 𝑇 ) -𝑤 𝑁 (𝑥, 𝑇 )) 2 ] = 4 𝑀 ∑ 𝑗=𝑁+1 𝜆 𝑗 𝑒 2 𝑗 (𝑥) 𝑇 ∫ 0 E [ 𝑔 2 𝑢 ] 𝑑𝑠. ( 50
)
Recalling the assumption that

∞ ∑ 𝑗=1 𝜆 𝑗 ‖𝑒 𝑗 ‖ 2 𝐿 ∞ (𝕋 𝑑 ) ≤ Λ 1 and using the fact that 𝑔 𝑢 ∈ 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))), we deduce that ∞ ∑ 𝑗=1 𝜆 𝑗 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑇 ∫ 0 𝑒 2 𝑗 𝑔 2 𝑢 𝑑𝑠𝑑𝑥 ⎤ ⎥ ⎥ ⎦ = E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑇 ∫ 0 ∞ ∑ 𝑗=1 𝜆 𝑗 𝑒 2 𝑗 𝑔 2 𝑢 𝑑𝑠𝑑𝑥 ⎤ ⎥ ⎥ ⎦ ≤ Λ 1 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑇 ∫ 0 𝑔 2 𝑢 𝑑𝑠𝑑𝑥 ⎤ ⎥ ⎥ ⎦ < ∞,
so that in view of (50)

E [ sup 0≤𝑡≤𝑇 ‖𝑤 𝑀 (𝑡) -𝑤 𝑁 (𝑡)‖ 2 𝐿 2 (𝕋 𝑑 ) ] ≤ ∫ 𝕋 𝑑 𝑑𝑥 E [ sup 0≤𝑡≤𝑇 (𝑤 𝑀 (𝑡) -𝑤 𝑁 (𝑡)) 2 ] ≤ 4 𝑀 ∑ 𝑗=𝑁+1 𝜆 𝑗 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑇 ∫ 0 𝑒 2 𝑗 𝑔 𝑢 𝑑𝑠𝑑𝑥 ⎤ ⎥ ⎥ ⎦ → 0,
∫ 0 𝑒 -𝑐𝑡 E [ ‖𝑢 Δ𝑡 𝑗 (𝑡)‖ 2 ] 𝑑𝑡 ≤𝑇 ‖𝑢 0 ‖ 2 + 2 𝑇 ∫ 0 𝑡 ∫ 0 𝑒 -𝑐𝑠 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑓 𝑢 𝐯 ⋅ ∇𝑢𝑑𝑥 ⎤ ⎥ ⎥ ⎦ 𝑑𝑠𝑑𝑡 + 2 𝑇 ∫ 0 𝑡 ∫ 0 𝑒 -𝑐𝑠 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑔 𝑢 𝑔(𝑢)𝑄(𝑥, 𝑥)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ 𝑑𝑠𝑑𝑡 (51) - 𝑇 ∫ 0 𝑡 ∫ 0 𝑒 -𝑐𝑠 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑔 2 (𝑢)𝑄(𝑥, 𝑥)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ 𝑑𝑠𝑑𝑡 -𝑐 𝑇 ∫ 0 𝑡 ∫ 0 𝑒 -𝑐𝑠 E [ ‖𝑢‖ 2 ] 𝑑𝑠𝑑𝑡 -2𝜖 𝑇 ∫ 0 𝑡 ∫ 0 𝑒 -𝑐𝑠 E [ ‖∇𝑢‖ 2 ] 𝑑𝑠𝑑𝑡.
One can prove this inequality by using similar ideas as in [START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] and the fact that |𝑄(𝑥, 𝑥)| ≤ Λ 1 . Next, we use this result to prove Lemma 12.

Proof. We apply the Itô's formula[cf. 10 , Theorem 4.2.5] and also using the equality (4.2.28) in [START_REF] Prévot | A Concise Course on Stochastic Partial Differential Equations[END_REF] , to deduce that

𝑒 -𝑐𝑡 E [ ‖𝑢(𝑡)‖ 2 ] + 2𝜖 𝑡 ∫ 0 𝑒 -𝑐𝑠 E [ ‖∇𝑢‖ 2 ] 𝑑𝑠 -2 𝑡 ∫ 0 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑒 -𝑐𝑠 𝐯𝑓 𝑢 ⋅ ∇𝑢 ⎤ ⎥ ⎥ ⎦ 𝑑𝑥𝑑𝑠 = E [ ‖𝑢 0 ‖ 2 ] -𝑐 𝑡 ∫ 0 𝑒 -𝑐𝑠 E [ ‖𝑢(𝑠)‖ 2 ] 𝑑𝑠 + 𝑡 ∫ 0 𝑒 -𝑐𝑠 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑔 2 𝑢 𝑄(𝑥, 𝑥)𝑑𝑥 ⎤ ⎥ ⎥ ⎦ 𝑑𝑠. ( 52
)
We integrate (52) on time interval [0, 𝑇 ) which yields In what follows, we prove that 𝑢 Δ𝑡 𝑗 converges to 𝑢 strongly in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω × 𝕋 𝑑 )). We have that ‖𝑒 -𝑐𝑡∕2 (𝑢 Δ𝑡 𝑗 -𝑢)‖ ‖𝑒 -𝑐𝑡∕2 (𝑢 Δ𝑡 𝑗 -𝑢)‖ 2 𝐿 2 (0,𝑇 ;𝐿 2 (Ω×𝕋 𝑑 )) = 0. Finally we prove that 𝑓 𝑢 = 𝑓 (𝑢) and 𝑔 𝑢 = 𝑔(𝑢). Since 𝑓 is Lipschitz continuous we have ‖𝑓 (𝑢) -𝑓 (𝑢 Δ𝑡 𝑗 )‖ 𝐿 2 (0,𝑇 ;𝐿 2 (Ω×𝕋 𝑑 )) ≤ 𝐶 𝑓 ‖𝑢 -𝑢 Δ𝑡 𝑗 ‖ 𝐿 2 (0,𝑇 ;𝐿 2 (Ω×𝕋 𝑑 )) .

𝑇 ‖𝑢 0 ‖ 2 + 2 𝑇 ∫ 0 𝑡 ∫ 0 𝑒 -𝑐𝑠 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑 𝑓 𝑢 𝐯 ⋅ ∇𝑢𝑑𝑥 ⎤ ⎥ ⎥ ⎦ 𝑑𝑠𝑑𝑡 -2𝜖 𝑇 ∫ 0 𝑡 ∫ 0 𝑒 -𝑐𝑠 E [ ‖∇𝑢‖ 2 ] 𝑑𝑠𝑑𝑡 -𝑐 𝑇 ∫ 0 𝑡 ∫ 0 𝑒 -𝑐𝑠 E [ ‖𝑢‖ 2 ] 𝑑𝑠𝑑𝑡 = 𝑇 ∫ 0 𝑒 -𝑐𝑡 E [ ‖𝑢(𝑡)‖ 2 ] 𝑑𝑡 - 𝑇 ∫ 0 𝑡 ∫ 0 𝑒 -𝑐𝑠 E ⎡ ⎢ ⎢ ⎣ ∫ 𝕋 𝑑
Using the fact that ‖𝑢 -𝑢 Δ𝑡 𝑗 ‖ 𝐿 2 (0,𝑇 ;𝐿 2 (Ω×𝕋 𝑑 )) → 0, we deduce that ‖𝑓 (𝑢) -𝑓 (𝑢 Δ𝑡 𝑗 )‖ 𝐿 2 (0,𝑇 ;𝐿 2 (Ω×𝕋 𝑑 )) → 0. Hence, 𝑓 (𝑢 Δ𝑡 𝑗 ) → 𝑓 (𝑢) strongly in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω × 𝕋 𝑑 )) as Δ𝑡 𝑗 → 0, which in turn implies that 𝑓 𝑢 = 𝑓 (𝑢). A similar proof implies that 𝑔 𝑢 = 𝑔(𝑢).

in 𝐿 2 (𝕋 𝑑 ), so that 𝑢 is a strong solution in the sense of Definition 4, which can be denoted by 𝑢 𝜖 .

Lemma 14. 𝜖Δ𝑢 𝜖 -div(𝐯𝑓 (𝑢 𝜖 )) belongs to 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))).

Proof it follows that 𝜖Δ𝑢 𝜖 -div(𝐯𝑓 (𝑢 𝜖 )) ∈ 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))). This result is used in the proof of Lemma 4.

The uniqueness of the strong solution 𝑢 𝜖

Finally, we prove the uniqueness of the strong solution of Problem ( 𝜖 ). Let 𝑢 𝜖 1 and 𝑢 𝜖 2 be two strong solutions of Problem ( 𝜖 ). Applying Itô's formula to the difference of the equations for 𝑢 𝜖 1 and 𝑢 𝜖 2 , we deduce that

‖𝑢

  𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝑔(û(𝑥, 𝑡, 𝛼))𝑔(𝑢 𝜖 (𝑦, 𝑡))𝑑𝛼 ⎤ ⎥ ⎥ ⎦ 𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑄(𝑥, 𝑦)𝑑𝑦𝑑𝑥𝑑𝑡.

(

  𝑔 𝑢 -𝑔(𝑢)) 2 𝑄(𝑥, 𝑥)𝑑𝑥

1 (

 1 Thus for almost all 𝑡 ∈ (0, 𝑇 ), P-a.s., there holds, 𝑢(⋅, 𝑡) = 𝑢 𝜖 0 𝕋 𝑑 ) ) ′ . Next we prove that this equality also holds in 𝐿 2 (𝕋 𝑑 ). SinceũΔ𝑡 𝑗 -BΔ𝑡 𝑗 ⇀ 𝑢(⋅, 𝑡) -𝑡 ∫ 0 𝑔(𝑢)𝑑𝑊 (𝑠)is in 𝐿 2 (0, 𝑇 ; 𝐿 2 (𝕋 𝑑 )), we have that 𝑢(⋅, 𝑡)-∫ 𝑡 0 𝑔(𝑢)𝑑𝑊 (𝑠) ∈ 𝐿 2 (0, 𝑇 ; 𝐿 2 (𝕋 𝑑 )) P-a.s.. In view of Lemma 11 and the identification 𝑔 𝑢 = 𝑔(𝑢), we deduce that ∫ 𝑡 0 𝑔(𝑢)𝑑𝑊 (𝑠) ∈ 𝐶([0, 𝑇 ]; 𝐿 2 (𝕋 𝑑 )) and thus 𝑢 ∈ 𝐿 2 (0, 𝑇 ; 𝐿 2 (𝕋 𝑑 )). This completes the proof that for almost all 𝑡 ∈ (0, 𝑇 ), P-a.s., there holds, 𝑢(⋅, 𝑡) = 𝑢 𝜖 0

=

  𝜖Δ𝑢 𝜖 -div(𝐯𝑓 (𝑢 𝜖 )),

  𝑥, 𝑡, 𝛼))𝑒 𝑗 (𝑥) -𝑔(𝑢 𝜖 (𝑦, 𝑡))𝑒 𝑗 (𝑦) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑑𝛼𝑑𝑦𝑑𝑥𝑑𝑡 𝑡, 𝛼)) -𝑔(𝑢 𝜖 (𝑦, 𝑡)))𝑒 𝑗 (𝑥) + 𝑔(𝑢 𝜖 (𝑦, 𝑡))(𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦))

	Proof. We remark that
		∞ ∑	𝜆 𝑗	[ 𝑔(û(𝑥, 𝑡, 𝛼))𝑒 𝑗 (𝑥) -𝑔(𝑢 𝜖 (𝑦, 𝑡))𝑒 𝑗 (𝑦)	] 2
		𝑗=1
	=	∞ ∑	𝜆 𝑗	[ (𝑔(û(𝑥, ] 2
		𝑗=1
	≤ 2𝑄(𝑥, 𝑥)	[ 𝑔(û(𝑥, 𝑡, 𝛼)) -𝑔(𝑢 𝜖 (𝑦, 𝑡))
										⎤ ⎥ ⎦ ⎥	.
	Next, we prove the following result
	Lemma 5. There holds				lim 𝜖→0	lim 𝛿→0	lim 𝑚→∞	lim 𝑙→∞	lim 𝑛→∞	[ 𝐼 6 + 𝐽 6 + 𝐼 7	]	= 0.

  ∶= 𝑓 𝑗 (𝑧), we deduce that

		lim 𝑚→∞ ∫ 𝕋 𝑑		∫ 0	𝑇	𝕋 𝑑 ∫	𝜌 𝑚 (𝑥 -𝑦)	𝑗=1 𝐾 ∑	𝜆 𝑗	[ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦)	] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑡
	=	𝐾 ∑ 𝑗=1	lim 𝑚→∞	0 ∫	𝑇	𝕋 𝑑 ∫	𝕋 𝑑 ∫	𝜆 𝑗 𝜌 𝑚 (𝑧)𝑓 𝑗 (𝑧)𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑧𝑑𝑦𝑑𝑡
	=	𝐾 ∑ 𝑗=1	lim 𝑚→∞	0 ∫	𝑇	𝕋 𝑑 ∫	𝕋 𝑑 ∫	𝜆 𝑗 𝑚 𝑑 𝑘 𝜌 𝑒	-1 1-|𝑚𝑧+1| 𝜒 {𝑧∈𝕋 𝑑 ,|𝑚𝑧+1|<1} 𝑓 𝑗 (𝑧)𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑧𝑑𝑦𝑑𝑡	(26)
	=	𝐾 ∑ 𝑗=1	lim 𝑚→∞	0 ∫	𝑇	𝕋 𝑑 ∫	𝕋 𝑑 ∫	𝜆 𝑗 𝑚 𝑑 𝑘 𝜌 𝑒	-1 1-|𝜏| 𝜒 {𝜏∈𝕋 𝑑 , |𝜏|<1} 𝑓 𝑗 (	𝜏 -1 𝑚	)𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)	𝑑𝜏 𝑚 𝑑 𝑑𝑦𝑑𝑡
	=	𝐾 ∑ 𝑗=1	lim	0 ∫	𝑇	𝕋 𝑑 ∫	𝕋 𝑑 ∫	𝜆 𝑗 𝑘 𝜌 𝑒 -1 1-|𝜏| 𝜒 {𝜏∈𝕋 𝑑 , |𝜏|<1} 𝑓 𝑗 (	𝜏 -1 𝑚	)𝜂 ′′
	= 0.
	As for the second term, we use (24) to obtain,
					𝑇
		lim 𝑚→∞	∫		∫	∫	𝜌 𝑚 (𝑥 -𝑦)
				0		𝕋 𝑑	𝕋 𝑑
								𝑇
	𝑚→∞ 𝜌 In view of (25) and (26), ∫ 0 ∫ 𝕋 𝑑 ∫ 𝕋 𝑑

𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝜏𝑑𝑦𝑑𝑡 ∞ ∑ 𝑗=𝐾+1 𝜆 𝑗 [ 𝑒 𝑗 (𝑥) -𝑒 𝑗 (𝑦) ] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑡 ≤ 𝜖 1 lim 𝑚 (𝑥 -𝑦)𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜑(𝑦, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑡 ≤ 𝜖 1 𝑇 |𝕋 𝑑 |‖𝜂 ′′ 𝛿 ‖ 𝐿 ∞ ((0,𝑇 )×𝕋 𝑑 ) ‖𝜑‖ 𝐿 ∞ ((0,𝑇 )×𝕋 𝑑 ) .

  𝐽 𝑖 } + 𝐽 8 by , we obtain (12), which completes the proof of the Kato inequality. □

	𝑥, 𝑡, 𝛼))𝑒 𝑗 (𝑥) -𝑔(𝑢 𝜖 (𝑦, 𝑡))𝑒 𝑗 (𝑦)	] 2 𝜂 ′′ 𝛿 (û(𝑥, 𝑡, 𝛼) -𝑢 𝜖 (𝑦, 𝑡))𝜌 𝑚 (𝑥 -𝑦)𝜑(𝑦, 𝑡)𝑑𝛼𝑑𝑦𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎦ ⎥	;
	then in view of (23), we deduce that	lim 𝜖→0	lim 𝛿→0	lim 𝑚→∞	lim 𝑙→∞	lim 𝑛→∞	[ 𝐼 6 + 𝐽 6 + 𝐼 7	]	= 0.

which complete the proof of Lemma 5.

Passing to the limits 𝑛, 𝑙 → ∞, 𝛿, 𝜖 → 0 and 𝑚 → ∞ in 7 ∑ 𝑖=1 {𝐼 𝑖 +

  0, 𝑇 ; 𝐿 2 (Ω; 𝐻 1 (𝕋 𝑑 ))); 𝑓 (𝑢 Δ𝑡 𝑗 ) ⇀ 𝑓 𝑢 in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 )));𝑔 𝑢 𝑑𝑊 (𝑠) in 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 )) for all 𝑡 ∈ [0, 𝑇 ],

	𝑡		𝑡
	∫	𝑔(𝑢 Δ𝑡 𝑗 (𝑠 -Δ𝑡 𝑗 ))𝑑𝑊 (𝑠) ⇀	∫
	0		0

  ( 𝐻 1 (𝕋 𝑑 )) ′ . In the following we will identify 𝑓 𝑢 and 𝑔 𝑢 . 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))).

	Lemma 10. Suppose that the hypothesis (5) holds for some positive constant Λ 2 . Then 𝜕 𝑡	[ ũΔ𝑡 𝑗 -BΔ𝑡 𝑗	]	is bounded in
	𝐿 2 (0, Proof. The proof is similar to that of Bauzet-Vallet-Wittbold [c.f. 2 , proof of Lemma A.3, p.697 ]. We just consider the new terms E [ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 )∇(𝑢 𝜖 𝑛+1 -𝑢 𝜖 𝑛 ) ⋅ ∇𝑔(𝑢 𝜖 𝑛 )𝑑𝑥 ] and E [ ∫ 𝕋 𝑑 (𝑊 𝑛+1 -𝑊 𝑛 )∇(𝑢 𝜖 𝑛 ) ⋅ ∇𝑔(𝑢 𝜖 𝑛 )𝑑𝑥 ] which should be estimated 𝑛+1 -𝑢 𝜖 instead of 𝐸(𝑤 𝑛+1 -𝑤 𝑛

  𝜕 𝑡 [ũ Δ𝑡 𝑗 -BΔ𝑡 𝑗 ] ∈ 𝐿 2 (0, 𝑇 ; 𝐿 2 (𝕋 𝑑 )) ⊂ 𝐿[START_REF] Bauzet | The Cauchy problem for a conservation law with a multiplicative stochastic perturbation[END_REF] Using the fact that the embedding from 𝐿 2 (𝕋 𝑑 ) into ( 𝐻 1 (𝕋 𝑑 ) ) ′ is compact, and applying the Theorem of Aubin-Lions-Simon [cf. 1 , Theorem II.5.16], we deduce that the sequence 𝑇 ]; (𝐻 1 (𝕋 𝑑 )) ′ ) as Δ𝑡 𝑗 → 0. If

										( 0, 𝑇 ;	( 𝐻 1 (𝕋 𝑑 ) ) ′ )	.
								{ ũΔ𝑡 𝑗 -BΔ𝑡 𝑗	}	is relatively compact in 𝐶	( [0, 𝑇 ]; (𝐻 1 (𝕋 𝑑 )) ′ ) . Therefore
	ũΔ𝑡 𝑗 -BΔ𝑡 𝑗 → 𝑢 -	∫	𝑡	𝑔 𝑢 𝑑𝑊 (𝑠) strongly in 𝐶	( [0, 𝑡 ∫	𝑔 𝑢 𝑑𝑊 (𝑠) is continuous in time, then 𝑢(𝑡) is
	0 continuous in time. We define						0
				𝑤(𝑥, 𝑡) =	∫ 0	𝑡	𝑔 𝑢 𝑑𝑊 (𝑥, 𝑠) =	∞ ∑ 𝑗=1	√ 𝜆 𝑗 𝑒 𝑗 (𝑥)	0 ∫	𝑡	𝑔 𝑢 𝑑𝛽 𝑗 (𝑠),
				𝑤 𝑁 (𝑥, 𝑡) =	𝑁 ∑ 𝑗=1	√ 𝜆 𝑗 𝑒 𝑗 (𝑥)	0 ∫	𝑡	𝑔 𝑢 𝑑𝛽 𝑗 (𝑠)

11. P-a.s., 𝑡 ∫ 0 𝑔 𝑢 𝑑𝑊 (𝑠) is continuous in time and 𝑢 defined as (48) is in 𝐶 ( [0, 𝑇 ]; (𝐻 1 (𝕋 𝑑 )) ′ ) . Proof. We have proved that, for almost every 𝜔 ∈ Ω, ũΔ𝑡 𝑗 -BΔ𝑡 𝑗 ∈ 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (𝕋 𝑑 )); for all 𝑁 ∈ ℕ. Thus for all 1 ≤ 𝑁 < 𝑀, 𝑀, 𝑁 ∈ ℕ , by Doob's inequality and Itô isometry, E [ sup 0≤𝑡≤𝑇 (𝑤 𝑀 (𝑥, 𝑡) -𝑤 𝑁 (𝑥, 𝑡)) 2

  Next we identify 𝑓 𝑢 with 𝑓 (𝑢), 𝑔 𝑢 with 𝑔(𝑢) and 𝑢 with 𝑢 𝜖 . We have that 𝑢 Δ𝑡 𝑗 → 𝑢 strongly in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω × 𝕋 𝑑 )) as Δ𝑡 𝑗 → 0. In particular, this implies that that 𝑓 𝑢 = 𝑓 (𝑢) and 𝑔 𝑢 = 𝑔(𝑢). And 𝑢 is a strong solution of Problem ( 𝜖 ), we will denote it by 𝑢 𝜖 . Before proving the Lemma 12, we show a preliminary estimate.

	Lemma 12. Lemma 13. There holds
	𝑇
	lim sup
	Δ𝑡 𝑗 →0

as 𝑁 < 𝑀 and 𝑁 → ∞. This implies with the help of Borel-Cantelli's lemma that {𝑤 𝑁 (𝑡)} has a Cauchy subsequence {𝑤 𝑁 ′ (𝑡)} in 𝐶([0, 𝑇 ], 𝐿 2 (𝕋 𝑑 )), P-a.s.. Therefore since {𝑤 𝑁 ′ } converges to 𝑤 in 𝐶([0, 𝑇 ], 𝐿 2 (𝕋 𝑑 )), it follows that 𝑤 ∈ 𝐶([0, 𝑇 ], 𝐿 2 (𝕋 𝑑 )).

As P-a.s., 𝑢 -𝑡 ∫ 0 𝑔 𝑢 𝑑𝑊 (𝑠) is in 𝐶([0, 𝑇 ], 𝐻 1 (𝕋 𝑑 ) ′ ), we deduce that P-a.s., 𝑢 ∈ 𝐶([0, 𝑇 ], 𝐻 1 (𝕋 𝑑 ) ′ ).

  2 𝐿 2 (0,𝑇 ;𝐿 2 (Ω×𝕋 𝑑 )) = E ‖𝑒 -𝑐𝑡∕2 (𝑢 Δ𝑡 𝑗 -𝑢)‖ 2 𝐿 2 (0,𝑇 ;𝐿 2 (Ω×𝕋 𝑑 )) ≤ 0, so that also lim Δ𝑡 𝑗 →0 ‖𝑢 Δ𝑡 𝑗 -𝑢‖ 2 𝐿 2 (0,𝑇 ;𝐿 2 (Ω×𝕋 𝑑 )) ≤ 𝑒 𝑐𝑇 lim

		⎡ ⎢ ⎢ ⎣ ∫ 0	𝑇	𝑒 -𝑐𝑡	∫				⎤ ⎥ ⎥ ⎦
	= E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑇	∫ 𝕋 𝑑	(𝑒 -𝑐𝑡∕2 𝑢 Δ𝑡 𝑗 ) 2 𝑑𝑥𝑑𝑡	⎤ ⎥ ⎥ ⎦	-2 E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑇	𝑒 -𝑐𝑡	∫ 𝕋 𝑑	𝑢𝑢 Δ𝑡 𝑗 𝑑𝑥𝑑𝑡	⎤ ⎥ ⎥ ⎦	+ E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑇	∫ 𝕋 𝑑	𝑒 -𝑐𝑡 𝑢 2 𝑑𝑥𝑑𝑡	⎤ ⎥ ⎦ ⎥	.
	Passing to the limit and using (54), we deduce that
		lim Δ𝑡 𝑗 →0
						Δ𝑡 𝑗 →0

𝕋 𝑑 (𝑢 Δ𝑡 𝑗 -𝑢)(𝑢 Δ𝑡 𝑗 -𝑢)𝑑𝑥𝑑𝑡

  . According to Lemma 10 and the facts that ũΔ𝑡 𝑗 → 𝑢 𝜖 weakly in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))) and BΔ𝑡 𝑗 → ∫ 𝑡 𝑔(𝑢 𝜖 )𝑑𝑊 (𝑠) in 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 )) for all 𝑡, we obtain that 𝜕 𝑡 𝑇 ; 𝐿 2 (Ω; 𝐿 2 (𝕋 𝑑 ))). Since

	[ ũΔ𝑡 𝑗 -BΔ𝑡 𝑗	]	converges up to a subsequence to 𝜕 𝑡	[ 𝑢 𝜖 -∫ 0 𝑔(𝑢 𝜖 )𝑑𝑊 (𝑠) 𝑡	]	in
	𝐿 2 (0,

  𝑢 𝜖 2 ⟩ 𝐿 2 (𝕋 𝑑 ) 𝑑𝑊 (⋅, 𝑠) + 𝜆 𝑗 𝑒 𝑗 , 𝑒 𝑗 ⟩ 𝐿 2 (𝕋 𝑑 ) 𝑑𝑠.which combined with Gronwall's lemma yields the uniqueness result of 𝑢 𝜖 .

	so that	E	[ ‖𝑢 𝜖 1 (𝑡) -𝑢 𝜖 2 (𝑡)‖ 2 𝐿 2 (𝕋 𝑑 )	]	+ 2𝜖 E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑡	‖∇(𝑢 𝜖 1 -𝑢 𝜖 2 )‖ 2 𝐿 2 (𝕋 𝑑 )	⎤ ⎥ ⎥ ⎦
			≤ 2𝑉 E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑡	∫ 𝕋 𝑑	|∇(𝑢 𝜖 1 -𝑢 𝜖 2 )||𝑓 (𝑢 𝜖 1 ) -𝑓 (𝑢 𝜖 2 )|𝑑𝑥𝑑𝑡 ⎤ ⎥ ⎥ ⎦	+ Λ 1 𝐶 𝑔 E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑡	‖𝑢 𝜖 1 -𝑢 𝜖 2 ‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑𝑡	⎤ ⎥ ⎦ ⎥	,
										𝑡				𝑡
			𝜖 1 (𝑡) -𝑢 𝜖 2 (𝑡)‖ 2 𝐿 2 (𝕋 𝑑 ) =2𝜖	∫		⟨Δ(𝑢 𝜖 1 -𝑢 𝜖 2 ), 𝑢 𝜖 1 -𝑢 𝜖 2 ⟩ 𝐿 2 (𝕋 𝑑 ) -2	∫	⟨div(𝐯𝑓 (𝑢 𝜖 1 )) -div(𝐯𝑓 (𝑢 𝜖 2 )), 𝑢 𝜖 1 -𝑢 𝜖 2 ⟩ 𝐿 2 (𝕋 𝑑 )
									0					0
	+ 2 2 )] 2 Next we take the expectation to obtain the estimate 𝑡 ∫ 0 ⟨𝑔(𝑢 𝜖 1 ) -𝑔(𝑢 𝜖 2 ), 𝑢 𝜖 1 -𝑡 ∫ 0 ∞ ∑ 𝑗=0 ⟨[𝑔(𝑢 𝜖 1 ) -𝑔(𝑢 𝜖
	E	[ ‖𝑢 𝜖 1 (𝑡) -𝑢 𝜖 2 (𝑡)‖ 2 𝐿 2 (𝕋 𝑑 )	]	≤ -2𝜖 E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑡	‖∇(𝑢 𝜖 1 -𝑢 𝜖 2 )‖ 2 𝐿 2 (𝕋 𝑑 )	⎤ ⎥ ⎥ ⎦	-2 E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑡	⟨div(𝐯𝑓 (𝑢 𝜖 1 )) -div(𝐯𝑓 (𝑢 𝜖 2 )), 𝑢 𝜖 1 -𝑢 𝜖 2 ⟩𝑑𝑠	⎤ ⎥ ⎥ ⎦
							+ 2 E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑡	⟨𝑔(𝑢 𝜖 1 ) -𝑔(𝑢 𝜖 2 ), 𝑢 𝜖 1 -𝑢 𝜖 2 ⟩𝑑𝑊 (𝑥, 𝑠) ⎤ ⎥ ⎥ ⎦	+	∞ ∑ 𝑗=0	𝜆 𝑗 ‖𝑒 𝑗 ‖ 2 𝐿 ∞ (𝕋 𝑑 ) E	⎡ ⎢ ⎢ ⎣ ∫ 0	𝑡	‖𝑔(𝑢 𝜖 1 ) -𝑔(𝑢 𝜖 2 )‖ 2 𝐿 2 (𝕋 𝑑 ) 𝑑𝑠 ⎤ ⎥ ⎦ ⎥	,

ACKNOWLEDGMENTS

The authors would like to thank Dr. Perla El Kettani for helpful discussions and the GDRI ReaDiNet as well as the research funding of MathAM-OIL, AIST c/o AIMR, Tohoku University for financial support.