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Convergence of the Finite Volume Method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation

We prove the convergence of the explicit-in-time Finite Volume method with monotone fluxes for the approximation of scalar first-order conservation laws with multiplicative, compactly supported noise.


Introduction
Stochastic first-order scalar conservation law. Let (Ω, F, P, (F t ), (β k (t))) be a stochastic basis and let T > 0. Consider the first-order scalar conservation law with stochastic forcing du(x, t) + div(A(u(x, t)))dt = Φ(x, u(x, t))dW (t), x ∈ T N , t ∈ (0, T ). (1.1) Equation (1.1) is periodic in the space variable: x ∈ T N where T N is the N -dimensional torus. The flux function A in (1.1) is supposed to be of class C 2 : A ∈ C 2 (R; R N ). We assume that A and its derivatives have at most polynomial growth. Without loss of generality, we will assume also that A(0) = 0. The right-hand side of (1.1) is a stochastic increment in infinite dimension. It is defined as follows (see [7] for the general theory): W is a cylindrical Wiener process, W = k≥1 β k e k , where the coefficients β k are independent Brownian processes and (e k ) k≥1 is a complete orthonormal system in a Hilbert space H. For each x ∈ T N , u ∈ R, Φ(x, u) ∈ L 2 (H, R) is defined by Φ(x, u)e k = g k (x, u) where g k (·, u) is a regular function on T N . Here, L 2 (H, K) denotes the set of Hilbert-Schmidt operator from the Hilbert space H to an other Hilbert space K. Since K = R in our case, this set is isomorphic to H, thus we may also define Φ(x, u) = k≥1 g k (x, u)e k , the action of Φ(x, u) on e ∈ H being given by Φ(x, u), e H . We assume g k ∈ C(T N × R), with the bounds where x, y ∈ T N , u, v ∈ R, and h is a continuous non-decreasing function on R + such that h(0) = 0. We assume also 0 ≤ h(z) ≤ 1 for all z ∈ R + .
Notation: in what follows, we will use the convention of summation over repeated indices k. For example, we write W = β k e k .
Compactly supported multiplicative noise. In this paper, we study the numerical approximation of (1.1): our aim is to prove the convergence of the Finite Volume method with monotone fluxes, see Theorem 7.4. Our analysis will be restricted to the case of multiplicative noise with compact support. Indeed, from Section 3 to Section 7, we will work under the following hypothesis: there exists a, b ∈ R, a < b, such that g k (x, u) = 0 for all u outside the compact [a, b], for all x ∈ T N , k ≥ 1. For simplicity, we will take a = −1, b = 1. We will assume therefore that for all u ∈ R, |u| ≥ 1 ⇒ g k (x, u) = 0, (1.4) for all x ∈ T N , k ≥ 1, and consider initial data with values in [−1, 1]. The solution of the continuous equation (1.1) then takes values in [−1, 1] almost-surely (see [9,Theorem 22]).
There is no loss in generality in considering that A is globally Lipschitz continuous then: Lip(A) := sup ξ∈R |A ′ (ξ)| < +∞. (1.5) In that framework, we will build a stable and convergent approximation to (1.1) by an explicit-in-time Finite Volume method. Under (1.4), it is also natural to assume which is of course stronger than (1.2). We may also perform the analysis of convergence of the Finite Volume method under (1.2) instead of (1.6), but this puts exponential factors in various estimates, whereas these factors are close to 1 in the real implementation of the scheme.
Kinetic formulation. To prove the convergence of the Finite Volume method with monotone fluxes, we will use the companion paper [9] and a kinetic formulation of the Finite Volume scheme. The subject of [9] is the convergence of approximations to (1.1) in the context of the kinetic formulation of scalar conservation laws. Such kinetic formulations have been developed in [26,27,28,30,31]. In [28], a kinetic formulation of Finite Volume E-schemes is given (and applied in particular to obtain sharp CFL criteria). For Finite Volume schemes with monotone fluxes, the kinetic formulation is simpler, we give it explicitly in Proposition 4.1. Based on the kinetic formulation and an energy estimate, we derive some a priori bounds on the numerical approximation (theses are "weak BV estimates" in the terminology of [11,Lemma 25.2]), see Section 5. These estimates are used in the proof of consistency of the scheme when we show that it gives rise to an approximate solution to (1.1) in the sense of Definition 2.6. Our final result, cf. Theorem 7.4, should be compared to [3,Theorem 2]. This latter gives the convergence of the Finite Volume method with monotone fluxes in a very similar context, under the slightly stronger hypothesis that the ratio of the time step ∆t with the spatial characteristic size h of the mesh tends to 0 when h tends to 0.
Plan of the paper. The plan of the paper is the following one. In the preliminary section 2, we give a brief summary of the notion of solution and approximate solution to (1.1) developed in [9]. In Section 3 is described the kind of approximation to (1.1) by the Finite Volume method which we consider here. In Section 4 we establish the kinetic formulation of the scheme. This numerical kinetic formulation is analysed as follows: energy estimates are derived in Section 5, then we show in Section 6 that this gives rise to an approximate generalized solution in the sense of Definition 2.6. We show some additional estimates and then conclude to the convergence of the scheme in Section 7. This result is stated in Theorem 7.4.

Generalized solutions, approximate solutions
The object of this section is to recall several results concerning the solutions to the Cauchy Problem associated to (1.1) and their approximations. We give the main statements, without much explanations or comments; those latter can be found in [9]: we give the precise references when needed.

Solutions
Definition 2.1 (Random measure). Let X be a topological space. If m is a map from Ω into the set of non-negative finite Borel measures on X such that, for each continuous and bounded function φ on X, m, φ is a random variable, then we say that m is a random measure on X.
A random measure m is said to have a finite first moment if T ] is said to be a solution to (1.1) with initial datum u 0 if u and f := 1 u>ξ have the following properties: 4. there exists a random measure m with first moment (2.1), such that for all ϕ ∈ a.s., where f 0 (x, ξ) = 1 u 0 (x)>ξ , G 2 := ∞ k=1 |g k | 2 and a(ξ) := A ′ (ξ).
In item 1, the index P in u ∈ L 1 P (T N × [0, T ] × Ω) means that u is predictable. See [9, Section 2.1.1]. The function denoted f := 1 u>ξ is given more precisely by This is the characteristic function of the subgraph of u. To study the stability of solutions, or the convergence of approximate solutions (there are two similar problems), we have to consider the stability of this property, the fact of being the " characteristic function of the subgraph of a function". If (u n ), is a sequence of functions, say on a finite measure space X, p ∈ (1, ∞) and (u n ) is bounded in L p (X), then there is a subsequence still denoted (u n ) which converges to a function u in L p (X)-weak. Up to a subsequence, the sequence of equilibrium functions f n := 1 un>ξ is converging to a function f in L ∞ (X × R)-weak star. The limit f is equal to f := 1 u>ξ if, and only if, (u n ) is converging strongly, see [9,Lemma 2.6]. When strong convergence remains a priori unknown, the limit f still keeps some structural properties. This is a kinetic function in the sense of Definition 2.4 below, [9, Corollary 2.5]. Our notion of generalized solution is based on this notion.

Generalized solutions
Definition 2.3 (Young measure). Let (X, A, λ) be a finite measure space. Let P 1 (R) denote the set of probability measures on R. We say that a map ν : X → P 1 (R) is a Young measure on X if, for all φ ∈ C b (R), the map z → ν z (φ) from X to R is measurable. We say that a Young measure ν vanishes at infinity if, for every p ≥ 1, Definition 2.4 (Kinetic function). Let (X, A, λ) be a finite measure space. A measurable function f : X × R → [0, 1] is said to be a kinetic function if there exists a Young measure ν on X that vanishes at infinity such that, for λ-a.e. z ∈ X, for all ξ ∈ R, We say that f is an equilibrium if there exists a measurable function u : X → R such that f (z, ξ) = f(z, ξ) = 1 u(z)>ξ a.e., or, equivalently, ν z = δ ξ=u(z) for a.e. z ∈ X.
have the following properties: 1. for all t ∈ [0, T ], almost-surely, f (t) is a kinetic function, and, for all 4. there exists a random measure m with first moment (2.1), such that for all ϕ ∈ C 1 c (T N × R), for all t ∈ [0, T ], almost-surely, The following statement is Theorem 3.2. in [9].
Then we have the following results: 1. there is at most one solution with initial datum u 0 to (1.1).
3. if u 1 , u 2 are two solutions to (1.1) associated to the initial data u 1,0 , u 2,0 ∈ L ∞ (T N ) respectively, then This implies the L 1 -contraction property, and comparison principle for solutions.

Approximate solutions
In [9], we prove the convergence of solutions to different problems, which are approximations of (1.1). The solutions of these approximate problems give rise to approximate solutions and, more precisely, of approximate generalized solutions, according to the following definition (see Definition 4.1 and Section 5 in [9]). |ε n (t, ϕ)| = 0 in probability, (2.8) and there exists some random measures m n with first moment (2.1), such that, for all n, for all ϕ ∈ C d c (T N × R), for all t ∈ [0, T ], almost-surely, Then we say that (f n ) is a sequence of approximate generalized solutions to (1.1) with initial datum f n 0 .
Consider a sequence (f n ) of approximate solutions to (1.1) satisfying the following (minimal) bounds.
1. There exists C p ≥ 0 independent on n such that ν n := −∂ ξ f n satisfies E sup We give in [9] the proof of the following convergence result, see Theorem 40 in [9].
Theorem 2.2 (Convergence). Suppose that there exists a sequence of approximate generalized solutions (f n ) to (1.1) with initial datum f n 0 satisfying (2.10), (2.11) and the tightness condition (2.12) and such that (f n 0 ) converges to the equilibrium function Then, for all p ∈ [1, ∞[, (u n ) is converging to u with the following two different modes of convergence: u n → u in L p (T N × (0, T ) × Ω) and almost surely, for all t ∈ [0, T ], u n (t) → u(t) in L p (T N ).
In the next section, we define the numerical approximation to (1.1) by the Finite Volume method. To prove the convergence of the method, we will show that the hypotheses of Theorem 2.2 are satisfied. The most difficult part in this programme is to prove that the numerical approximations generates a sequence of approximate generalized solutions, see Section 6.

The finite volume scheme
Mesh A mesh of T N is a family T /Z N of disjoint connected open subsets K ∈ (0, 1) N which form a partition of (0, 1) N up to a negligible set. We denote by T the mesh {K + l; l ∈ Z N , K ∈ T /Z N } deduced on R N . For all distinct K, L ∈ T , we assume that K ∩ L is contained in an hyperplane; the interface between K and L is denoted K|L := K ∩ L. The set of neighbours of K is We use also the notation ∂K = L∈N (K)

K|L.
In general, there should be no confusion between ∂K and the topological boundary K \ K. • Monotony: • Lipschitz regularity: there exists L A ≥ 0 such that • Consistency: for all v ∈ R, where n K,L is the outward unit normal to K on K|L.
• Conservative symmetry: The conservative symmetry property ensures that the numerical flux Q n K→L defined below in (3.6) satisfies Q n K→L = −Q n L→K for all K, L. Let t n < t n+1 be two given discrete time. Let ∆t n = t n+1 − t n . Knowing v n K , an approximation of the value of the solution u to (1.1) in the cell K at time t n , we compute v n+1 K , the approximation to the value of u in K at the next time step t n+1 , by the formula We have also defined Then, the (X n+1 k ) k≥1,n∈N are i.i.d. random variables with normalized centred normal law N (0, 1). Besides, for each n ≥ 1, the sequence (X n+1 k ) k≥1 is independent on F n , the sigma-algebra generated by {X m+1 k ; k ≥ 1, m < n}. The numerical functions g k,K are defined by the average Then, in virtue of (1.6) we have where v ∈ R, K ∈ T . We deduce (3.9) from (1.6) and Jensen's Inequality. Similarly, we deduce from (1.3) and Jensen's Inequality that for all y ∈ T N . In particular (switching from the variable y to the variable x), and assuming diam(K) ≤ h (this is the hypothesis (5.5), which we will make later), we have the following consistency estimate for all x ∈ K, which will be used later (see (6.47) for example).

The kinetic formulation of the finite volume scheme
The kinetic formulation of the Finite Volume method has been introduced by Makridakis and Perthame in [28]. The principle is the following one. For linear transport equations, which corresponds to a linear flux function A(u) = au, a ∈ R N , the upwind numerical flux A K→L in (3.6) is given by with the usual notation v + = max(v, 0), w − = (−w) + . The discrete approximation of the transport equation by the Finite Volume method is therefore Recall that G K is defined by (3.9). A kinetic formulation of (3.4) consistent with (4.3) would be where, for K ∈ T , n ∈ N, ξ ∈ R, m n K (ξ) ≥ 0, and where This is not exactly the kinetic formulation that we will consider. See (6.9) for a correct version of (4.4). We will mainly work with a kinetic formulation (see (4.12)), obtained thanks to the following splitting method. For K ∈ T and n ∈ N, let us define v n+1/2 K as the solution to Then v n+1/2 K is the state reached after a step of deterministic evolution, by the discrete approximation of the equation u t + div(A(u)) = 0. To this step corresponds the kinetic formulation In (4.7), a n K→L (ξ) is a function a n K→L (ξ) = a K→L (ξ, v n K , v n L ), (4.9) where (ξ, v, w) → a K→L (ξ, v, w) satisfies the following consistency conditions: for all ξ, v, w ∈ R, where a * K→L is defined by (4.2). Before we prove the existence of the kinetic formulation (4.7)-(4.8)-(4.10)-(4.11), see Proposition 4.1, let us first deduce from (4.7) the kinetic formulation of the whole scheme (3.4). This is the equation We may try to develop the term f n+1 K (ξ) − f n+1/2 K (ξ) (this is done in (6.13) to obtain (6.9)), but (4.12) will be sufficient for the moment. It will be sufficient in particular to obtain the so-called energy estimates of Section 5.
Remark 4.1 (Support of m n K ). By (4.14), the definition (4.13) of a K→L and the equation Proof of Proposition 4.1. We check at once (4.7) and (4.10), (4.11), (4.16). To show that m n K (ξ) ≥ 0, let us introduce A simple computation gives the formula By comparison with the identity (v − ξ) + = v − v ∧ ξ, the quantities Φ n K→L (ξ) appears, in virtue of (4.19), as the numerical entropy fluxes associated to the entropy η(v) : It is a classical fact that, under the CFL condition (4.15), the deterministic Finite Volume scheme (4.6) has the following monotony property: v

Energy estimates
The Finite Volume scheme (3.4) may be compared to the stochastic parabolic equation For (5.1), we have the energy estimate 2)). In the following Proposition 5.1, we obtain an analogous result for the Finite Volume scheme (3.4). To state Proposition 5.1, we need first some notations.

Notations
For a fixed final time T > 0, we denote by d T the set of admissible space-step and timesteps, defined as follows: if h > 0 and (∆t) = (∆t 0 , . . . , ∆t N T −1 ), N T ∈ N * , then we say For a given mesh parameter δ = (h, (∆t)) ∈ d T , we assume that a mesh T is given, with the following properties: is the diameter of K and α N is a given positive absolute constant depending on the dimension N only. Note the following consequence of (5.6)-(5.7): We will also need the intermediary discrete function defined for n ∈ N. Let us define the conjugate functionf = 1 − f . We introduce the following conjugate quantities: We recognize in (5.12) a numerical flux associated to the entropy From the explicit formula (4.13), we obtain the identitȳ Note that, for a K→L defined by (4.13), we have (using the fact that Formula (5.13) gives the estimate which is not optimal as (5.14) may be, since it has an additional factor 2. Consequently, we will use a slightly different formulation for Φ K→L : Now forb K→L , we have an estimate similar to (5.14):

Energy estimate and controls by the dissipation
Proposition 5.1 (Energy estimate for the Finite Volume Scheme). Let u 0 ∈ L ∞ (T), T > 0 and δ ∈ d T . Let (v δ (t)) be the numerical unknown defined by (3.4)-(3.5)-(5.9). Set Then, under the CFL condition (4.15), we have the energy estimate In the following proposition we derive various estimates, where the right-hand side is controlled by the dissipation term E(T ) introduced in (5.20).
where θ ∈ (0, 1), we have the following control: 23) and where θ ∈ (0, 1), (and whereb n K→L is defined by (5.18)) we have the following control: 26) and Eventually, as a corollary to Proposition 5.2, we obtain the following estimates.

Proof of Proposition 5.1, Proposition 5.2, Corollary 5.3
Proof of Proposition 5.1. We multiply first (4.7) by ξ and sum the result over K ∈ T /Z N and ξ ∈ R to get the following balance equation Adding (5.32) to (5.37) gives (5.21).
Remark 5.1. Note that (5.36) also gives for all 0 ≤ n ≤ N T .
Proof of Proposition 5.2. We begin with the proof of the estimates (5.23) and (5.24). Multiplying Equation (4.7) byf n K := 1 − f n K , we obtain Next, we multiply (5.39) by (ξ − v n K ) and sum the result over ξ, K. We use the first identity (once again, we use the fact that m n K is compactly supported to do the integration by parts in (5.40), cf. Remark 4.1) and the second identity We transform the right-hand side of (5. Then we can relabel the indices in (5.42) and use the conservative symmetry relation (consequence of (5.35)) to see that Note that the integrand (f n L (ξ)−f n K (ξ))Φ n K→L (ξ) is non-negative thanks to the monotony properties of A K→L and (4.19). At this stage, in order to deduce (5.23) from (5.44), we have to prove that, under the CFL condition (5.22), a fraction of the right-hand side of (5.44) controls the term , (see the estimate (5.49) below). To this end, we integrate Equation (5.39) over ξ ∈ R. This gives by (4.18) (note that it is also the discrete entropy inequality (4.20) with ξ = v n K ). Taking the square, using the Cauchy-Schwarz Inequality and summing over K ∈ T /Z N , In that case, it can be decomposed as Using (5.44) then, we deduce the two estimates (5.23)-(5.24).
To prove the estimates (5.26) and (5.27), we proceed similarly: we start from the following equation onf n K , which is equivalent to (4.7) : which is the analogue to (5.39). In a first step, we multiply (5.51) by (v n K − ξ) and sum the result over ξ ∈ R, K ∈ T /Z N . This gives (compare to (5.41)-(5.44)) for all ξ ∈ R. This estimate will be used in the proof of Lemma 6.4 below.
Proof of Corollary 5.3. Assume that (5.28) is satisfied. It is clear, in virtue of the estimate (5.8) and the bound (5.14) and (5.19) on a n K→L andb n K→L , that (5.28) implies the CFL conditions (5.22) and (5.25). Besides, due to (3.9), we have the bound This gives which, inserted in the energy estimate (5.21), shows that  6 Approximate kinetic equation 6

.1 Discrete unknown
In this section we will show that, when δ → 0, some discrete kinetic unknowns f δ associated to the scheme (3.4) are approximate kinetic solutions. There may be several way to define f δ : it depends for example on the manner in which the discrete data f n K are assembled by interpolation. One of the constraints due to our definition of approximate kinetic solution is the formulation "at fixed t" (2.9) (in opposition to a formulation which would be weak in time). To establish such a formulation in our context, a minimal amount of regularity of the function t → f δ (t) is required (in particular, for all ϕ, f δ (t), ϕ should be a càdlàg process). For t ∈ [t n , t n+1 ), we will therefore consider the function f δ (t) defined as the interpolation We also denote by f δ the piecewise constant function We check first that f δ and f δ are close to each other.
Proof of Lemma 6.1. Since for t ∈ [t n , t n+1 ) and since the factor t−tn ∆tn is less than 1, the quantity we want to estimate is bounded by the following L 2 -norm: By definition of v ♯ δ (t) and independence and (3.9), we obtain .
Remark 6.1. Note for a future use (cf. (6.45)) that we have just proved the estimate To f δ we will associate the Young measure We also denote by m δ the discrete random measure given by  Let (v δ (t)) be the numerical unknown defined by (3.4)-(3.5)-(5.9) and let f δ , ν δ , m δ be defined by (6.1), (6.7), (6.8) respectively. Then f δ satisfies the following discrete kinetic formulation: for all t ∈ [t n , t n+1 ], x ∈ K, for all ψ ∈ C 1 c (R), In (6.9), f δ (x, t), ψ stands for the product Proof of Proposition 6.2. Let Ψ be a primitive for ψ and let x ∈ K, t ∈ [t n , t n+1 ).
By definition of f δ , see Equation (6.1), we have which we decompose as the sum of two terms: and We use first the deterministic kinetic formulation (4.7), which we multiply by ψ(ξ). By integration over ξ ∈ R, it gives By Itō's Formula on the other hand (cf. (6.2)), the term (6.10) is equal to Summing (6.12) and (6.13), we obtain (6.9).
We will prove now that the Finite Volume scheme (3.4) is consistent with (1.1). Indeed, we will show, thanks to the estimates obtained in Section 5, that an approximate kinetic equation for f δ in the sense of (2.9) can be deduced from the discrete kinetic formulation (6.9). ) and let f δ , ν δ , m δ be defined by (6.1), (6.7), (6.8) respectively. If (δ m ) is a sequence in d T which tends to zero according to (5.4), then (f δm ) is a sequence of approximate solutions to (1.1) of order d = 2. Besides, (f δm (0)) converges to the equilibrium function Proof of Proposition 6.3. The last assertion is clear: (f δm (0)) converges to the equilibrium function on T N . We will show that, for all t ∈ [0, T ], for all ϕ ∈ C ∞ c (T N × R), (6.16) where the error term ε δ (t, ϕ) satisfies for all ϕ ∈ C 2 c (T N × R). Note that the convergence in probability (2.8) follows from (6.17). Given ϕ ∈ C 2 c (T N × R), we introduce the averages over the cells K ∈ T To prove (6.16), we apply the discrete kinetic equation (6.9) to ξ → ϕ(x, ξ) for a fixed x ∈ K. Then we sum the result over x ∈ T N . By the telescopic formula where the measure µ δ x,s,t on R × R is defined by (6.22) and the discrete coefficient g k,δ (x, ξ) is equal to g k,K (ξ) (cf. (3.8)) when x ∈ K (similarly, G δ (x, ξ) := G K (ξ), x ∈ K). Note that µ δ x,s,t is simply the Dirac mass at (v δ (x, s), v ♯ δ (x, s)), except when t l ≤ s ≤ t (where l is the index such that t l ≤ t < t l+1 ), in which case it is the same Dirac mass, with an additional multiplicative factor t−t l ∆t l . The term (6.19) is a discrete space derivative: we will show that it is an approximation of the term (6.14). The two terms (6.20) and (6.21) are close to (6.15) and (6.16) respectively. We analyse those terms separately (see Section 6.2, Section 6.3). The conclusion of the proof of Proposition 6.3 is given in Section 6.4.

Space consistency
and, for all compact Proof of Lemma 6.4. To begin with, we replace f δ by f δ in the left-hand side of (6.23). This accounts for the first error term Thanks to Lemma 6.1, we have the estimate (6.24) for ε δ space,0 (t, ϕ). Then, we use the following development: Since f δ (s) has a constant value f n K in K × [t n , t n+1 ), we obtain, thanks to the Stokes formula, where a * K→L (ξ) is defined by (4.2) and ϕ K|L is the mean-value of ϕ over K|L: We add a corrective term to (6.27) to obtain Equation (6.28) follows indeed from (6.27) by the anti-symmetry property (5.35) of a K→L . Note that Equation (6.28) is more natural than Equation (6.27) (when one has in mind the decomposition of a volume integral over each cells K), by use of the correspondence By (5.35), the discrete convective term in (6.19) is To estimate how close is the right-hand side of (6.28) to (6.29), we have to compare a n K→L (ξ) and a * by the consistency hypothesis (4.11) and the support condition (4.16). Using an integration by parts and (4.17), we obtain We deduce that (6.23) is satisfied with an error term a n K→L (ξ)ϕ K (ξ)dξ, (6.33) which is bounded as follows: for all ξ ∈ R. If ϕ is compactly supported in T N × Λ, we obtain thus the bound where B space is equal to We seek for a bound of order h −1/2 on B space . For notational convenience we will estimate only the first part since the bound on the second part in B space will be similar. By the Cauchy Schwarz inequality, we have We use the estimate (5.53), which gives thanks to (5.14). We also use (5.8), and get With (5.29) and (6.35), we conclude to (6.25).

Convergence
To apply Theorem 2.2 on the basis of Proposition 6.3, we need to establish some additional estimates on the numerical Young measure ν δ and on the numerical random measure m δ . This is done in Section 7.1. We conclude to the convergence of the Finite Volume method in Section 7, Theorem 7.4.
Proof of Lemma 7.1. It is sufficient to do the proof for p ∈ 2N * since 1 + |ξ| p ≤ 2(1 + |ξ| q ) for all ξ ∈ R if q ≥ p. Note that for t ∈ [t n , t n+1 ). Recall also that v ♯ δ is defined by (6.2). Let We multiply Equation (4.7) by ϕ p (ξ) and sum the result over K, ξ. We obtain then, thanks to (5.35), In particular, we have the Let us now estimate the increase of L p -norm due to the stochastic evolution. By Itō's Formula and (6.2), we have

Using (7.3) and induction, we obtain
where (M N ) is the martingale and Note that the argument v ♯ δ (t) p−2 , Γ n L 2 (T N ) in B N is non-negative since Γ n ≥ 0 and p − 2 ∈ 2N. Consequently, E sup 0≤n≤N T B n = EB N T and we deduce the following bound We have used (3.9) to obtain (7.6).
is, conditionally to F n , a Gaussian random variable with variance, for x ∈ K, (3.9). In particular, we have the bound where C(p) is a constant depending on p. It follows, using (7.3), that we have the estimate where C(p, D 0 ) is a constant depending on p and D 0 . In particular, we have By (7.6), we conclude that for possibly a different constant C(p, D 0 ). Let us now turn to the estimate of the quantity E sup 0≤n≤N T |M n |. The martingale (M N ) can be rewritten as a stochastic integral (with an integrand which is a simple function). Consequently, the quadratic variation of M N T is by (3.9). Using (7.7) (with 2p instead of p) gives thus By Burkholder -Davis -Gundy's Inequality, there exists a constant C BDG such that By Jensen's Inequality and the estimate (7.9), we obtain We can conclude now. Since EM N T = 0, taking expectation in (7.5) (where we replace By (7.8), this gives By iteration on p ∈ 2N * , we deduce, for every such p, that where the constant C p depends on p, D 0 , T and u 0 L ∞ (T N ) . Denote generally by C p any such constant, possibly different from line to line, depending only on p, D 0 , T and u 0 L ∞ (T N ) . By (7.11) with 2p − 2 instead of p, we have E sup 0≤n≤N T |M n | ≤ C p . Then we use (7.8) with p − 2 instead of p to obtain E sup 0≤n≤N T B n ≤ C p . By (7.5), we deduce which concludes the proof of the lemma.
where C p is a constant depending on D 0 , p, T and u 0 L ∞ (T N ) only.
Remark 7.1. If the X n+1 k are merely i.i.d. random variables with normalized centred normal law N (0, 1), then (v δ ) is converging to u in L p (T N × (0, T )) in law when δ → 0. Indeed, the identity (3.7) is only satisfied in law now, hence v δ has the same law as the functionṽ δ defined by (3.4)-(3.5)-(3.6), with X n+1 k replaced by the right-hand side of (3.7). We apply the conclusion of Theorem 7.4 toṽ δ . As a corollary, we obtain the convergence in law of (ṽ δ ) to u in L p (T N × (0, T )). A slightly different manner of expressing the same thing is to notice that, when the discrete increments (X n+1 k ) are some given normal law N (0, 1), then we can construct a set of Brownian motionsβ k (t) such that X n+1 k =β k (t n+1 ) −β k (t n ) (∆t n ) 1/2 . , . . . , G k−1 = X n+1 k−1 , G k = X n+1 k+1 , G k+1 = X n+1 k+2 , . . . and we setβ where the H p 's are the Haar basis of L 2 (0, 1). Then (7.15) follows from the fact that Remark 7.2. Theorem 7.3 has already been proved in [8] (see also Section 5 in [9]) under less restrictive hypotheses (having a compactly supported noise is unnecessary).
We give the statement together with Theorem 7.4, however, to emphasize the fact that the convergence of the Finite Volume method, in the framework which we use, give both the existence-uniqueness of the solution to the limit continuous problem, and the convergence of the numerical method to this solution. It is not necessary to provide the existence of the solution u to (1.1) by an external means.
Proof of Theorem 7.3 and Theorem 7.4. Let us first prove the theorem 7.4. We take the existence of u, solution to (1.1) with initial datum u 0 for granted. By Proposition 6.3, Lemma 7.1 and Lemma 7.2, we may apply Theorem 2.2 to f δ : we obtain (7.14) with z δ instead of v δ , where for t ∈ [t n , t n+1 ]. By (6.6), we have the estimate on the difference between z δ and v δ . This gives (7.14) for p ≤ 2. If p > 2, we use the inequality L 2(p−1) (T N ×(0,T )) , (7.16) where η is a positive parameter. Thanks to the uniform bounds (2.2)-(7.12), we can choose η independent on δ to have the second term in (7.16) smaller than an arbitrary threshold. By the convergence result for p = 2 the first term in (7.16) is then also small for δ close to 0. This concludes the proof of theorem 7.4. To prove 7.3, we just need to construct an approximation scheme satisfying (3.1), (5.6), (5.7) and (5.28) and to compute v δ by (3.4)-(3.5)-(3.6)-(3.7). We can use a cartesian grid for this purpose: let h m = 1 m , where m ∈ N * . Let T /Z N be the set of open hypercubes of length h m obtained by translates of the original hypercube (0, h m ) N by vectors h m x, x having components in {0, l . . . , m − 1}. Then (5.6), (5.7) are satisfied with α N = 2 −N since an hypercube has 2 N sides. We can choose the Godunov numerical fluxes, defined as follows: These fluxes A K→L are monotone (non-increasing in the first variable, non-increasing in the second variable). They satisfy the hypotheses of regularity, consistency (3.1) and (3.2) respectively with L A = Lip(A). The conservative symmetry property (3.3) is also satisfied. At last, to ensure (5.28), we just need to take a uniform time step ∆t like ∆t = 1 2 We have then (5.28) with θ = 1 2 . At this stage, Proposition 6.3 provides a sequence of approximate solutions (f m ). Thanks to the uniform bounds established in Section 7.1, we can apply Theorem 2.2: this gives the existence of a unique solution u to (1.1) with initial datum u 0 . By Corollary 3.3 in [9], we have u ∈ C([0, T ]; L p (T N )) almost-surely.