Benoît Fabrèges

Hélène Hivert

Kevin Le Balc'h

François Delarue

Frédéric Lagoutière

Nicolas Vauchelet

Le double encadrement dont j'ai pu bénéficier auprès de Sébastien Boyaval et Julien Reygner a constamment été au-delà de ce que je pouvais en attendre. Je souhaite ici les remercier sincèrement pour l'énergie qu'ils ont consacré, la disponibilité qu'ils m'ont accordée, la patience dont ils ont fait preuve pendant toutes les étapes d'émergence, de développement et d'aboutissement des travaux contenus dans ce manuscrit. J'ajoute également que leur complémentarité a été très enrichissante. La culture scientifique étendue de Sébastien et sa mise en perspective constante des questions mathématiques avec les problèmes physiques m'ont incité à prendre un recul considérable par rapport à ce sujet de thèse. J'ai également beaucoup apprécié travailler avec Julien qui a quant à lui fait preuve d'une pédagogie remarquable par sa rigueur méticuleuse, son sens de la clarté élevé jusqu'à une certaine forme d'art, et ses intuitions fulgurantes.

Gabriel Lord et Julien Vovelle m'ont fait l'honneur d'être les rapporteurs de ce manuscrit. Je les remercie ici pour leur relecture minutieuse et leurs points de vue aiguisés sur ce travail. Je remercie également Caroline Bauzet, Alexandre Boritchev, Mireille Bossy et Frédéric Lagoutière pour avoir rapidement accepté de faire partie de mon jury et pour toutes leurs questions pertinentes. Mention spéciale à François Bouchut qui, dans des circonstances un peu particulières, a parfaitement géré le rôle de président du jury qui l'incombait.

Je tiens aussi à remercier tous les chercheurs qui, au cours de cette thèse, m'ont accordé des discussions dont je suis sorti à chaque fois plus éclairé : Alexandre Boritchev, Charles-Édouard Bréhier et Tony Lelièvre, Frédéric Cérou et Mathias Rousset, Julia Charrier, ainsi que Julien Vovelle.

Ma participation au CEMRACS 2017 m'a permis de travailler sur un projet dans lequel j'ai retrouvé un certain nombre de concepts et d'outils utilisés dans cette thèse mais ordonnés différemment ! Pendant cette période intense, je pense avoir beaucoup appris auprès de François Delarue, Benoît Fabrèges, Frédéric Lagoutière et Nicolas Vauchelet, sans oublier mes camarades de projet Hélène et Kévin.

Je souhaite maintenant exprimer toute ma gratitude à l'ensemble des membres du laboratoire Saint-Venant que j'ai côtoyés pendant ces trois ans, et en premier lieu à Nicole Goutal qui a été une directrice de laboratoire toujours très bienveillante. Je remercie Alex et Adrien pour m'avoir accueilli dans leur bureau de la meilleure des manières, et tous ceux avec qui j'ai pu partager l'ambiance paisible et isolée de l'enclave du deuxième étage : Thomas, Konstantin

Préambule

Cette thèse se consacre à une analyse théorique puis numérique d'une certaine classe d'équations aux dérivées partielles stochastiques (EDPS) : les lois de conservation scalaires avec viscosité et avec un forçage aléatoire de type additif et bruit blanc en temps. Un exemple typique est l'équation de Burgers stochastique, motivée par la théorie de la turbulence. On s'intéresse particulièrement au comportement en temps long des solutions de ces équations à travers une étude des mesures invariantes.

La partie théorique de la thèse constitue le chapitre 2. Dans ce chapitre, on prouve l'existence et l'unicité d'une solution au sens fort. Pour cela, des estimations sur les normes de Sobolev jusqu'à l'ordre 2 sont établies. Dans la seconde partie du chapitre 2, on montre que la solution de l'EDPS admet une unique mesure invariante.

On se propose dans le chapitre 3 d'approcher numériquement cette mesure invariante. À cette fin, on introduit un schéma numérique dont la discrétisation spatiale est de type Volumes Finis et dont la discrétisation temporelle est une méthode d'Euler à pas fractionnaire. Il est montré que ce type de schéma respecte certaines propriétés fondamentales de l'EDPS telles que la dissipation d'énergie et la contraction L 1 . Ces propriétés assurent l'existence et l'unicité d'une mesure invariante pour le schéma. À l'aide d'un certain nombre d'estimations de régularité, on montre ensuite que cette mesure invariante discrète converge, lorsque le pas de temps et le pas d'espace tendent vers zéro, vers l'unique mesure invariante pour l'EDPS au sens de la distance de Wasserstein d'ordre 2.

Dans le chapitre 4, des expériences numériques sont effectuées sur l'équation de Burgers pour illustrer cette convergence ainsi que des propriétés à petites échelles spatiales relatives à la turbulence.

Un appendice est consacré à l'étude de la stationnarité des schémas numériques introduits au chapitre 3 dans le cas où la viscosité est nulle et où le domaine spatial est de dimension quelconque.

Preamble

This thesis is devoted to the theoretical and numerical analysis of a certain class of stochastic partial differential equations (SPDEs), namely scalar conservation laws with viscosity and with a stochastic forcing which is an additive white noise in time. A particular case of interest is the stochastic Burgers equation, which is motivated by turbulence theory. We focus on the long time behaviour of the solutions of these equations through a study of the invariant measures.

The theoretical part of the thesis constitutes the second chapter. In this chapter, we prove the existence and uniqueness of a solution in a strong sense. To this end, estimates on Sobolev norms up to the second order are established. In the second part of Chapter 2, we show that the solution of the SPDE admits a unique invariant measure.

In the third chapter, we aim to approximate numerically this invariant measure. For this purpose, we introduce a numerical scheme whose spatial discretisation is of the finite volume type and whose temporal discretisation is a split-step backward Euler method. It is shown that this kind of scheme preserves some fundamental properties of the SPDE such as energy dissipation and L 1 -contraction. Those properties ensure the existence and uniqueness of an invariant measure for the numerical scheme. Thanks to a few regularity estimates, we show that this discrete invariant measure converges, as the space and time steps tend to zero, towards the unique invariant measure for the SPDE in the sense of the second order Wasserstein distance.

In Chapter 4, numerical experiments are performed on the Burgers equation in order to illustrate this convergence as well as some small-scale properties related to turbulence.

An appendix is devoted to the study of the stationarity of the numerical schemes introduced in Chapter 3, in the case where the viscosity coefficient is zero and the spatial domain is of arbitrary dimension.

Chapter 1

Introduction

Viscous scalar conservation laws

Conservation laws are among the most fundamental principles describing a physical process. They state that a particular measurable quantity of an isolated physical system does not evolve over time. This quantity may be a mass, an energy, a momentum, an electric charge, etc... When the physical system consists of a continuous material, this conservation principle is appropriately complemented with the statement that the conserved quantity is also locally conserved: the amount of the quantity in a given volume of space varies only by the amount flowing in or out of the volume through its boundaries. In more general words, the quantity moves in space as a continuous flow. This assumption, called the continuum hypothesis, constitutes the core of continuum mechanics [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF]. The physical system may also create and destroy some of the conserved quantity by itself. We say that it contains a source at the time and place where the conserved quantity is created, and a sink where it is destroyed. The conservation law still holds as long as the sources and the sinks compensate each other.

Let u(t, x) denote the density per unit volume of the conserved quantity at a time t and at a point x of the physical space. Similarly, let F(t, x) denote the flux of the conserved quantity, and S(t, x) the generation of the conserved quantity per unit volume per unit time (it corresponds to a source when S > 0 and a sink when S < 0). Let V be an arbitrary volume of space. The conservation law in the volume V has the expression

d dt V udx = - ∂V F(t, x) • ndx + V S(t, x)dx.
Indeed, the left-hand side expresses the variation per unit time of the amount of the conserved quantity inside V . As n denotes the outward unit normal to V , the first term of the right-hand side expresses the amount of the conserved quantity flowing into V through its boundary at time t, and the second term expresses the amount of the conserved quantity generated inside V at time t. Using Green's formula, we can rewrite this equation in the following way:

d dt V udx = - V div(F(t, x))dx + V S(t, x)dx.
Since this equation must hold for any volume of space V , the integrals may vanish and we shall write ∂ t u = -div(F(t, x)) + S(t, x).

(1.1)

In most physical applications, the function F depends on t and x through the density u. When it depends on u up to the first-order space differential, i.e. when F(t, x) = F(t, x, u(t, x), ∇u(t, x)), Chapter 1. Introduction then Equation (1.1) defines a partial differential equation that we may call a second-order scalar conservation law.

In the present work, we will restrict ourselves to second-order scalar conservation laws where the flux is homogeneous in time and space and takes the form: F (t, x, u(t, x), ∇u(t, x)) = A(u(t, x)) -ν∇u(t, x), where A is a continuous function and ν is a positive constant. Furthermore, the time interval considered will always be [0, +∞). Taking these assumptions into account, Equation (1.1) now writes

∂ t u = -div(A(u)) + ν∆u + S(t, x), t ≥ 0, x ∈ D, (1.2)
where D is the space domain. It could be either R d , a subdomain1 of R d , or even the d-dimensional torus T d , where T := R/Z. Observe that the expression of the flux of the conserved quantity has been split in two parts:

• The term -div(A(u)) accounts for the transport of the conserved quantity. Indeed, the equation ∂ t u = -div(A(u)) which by differentiation can also be written ∂ t u = -A (u) • ∇u, is a transport equation where the solution u is transported with a velocity A (u). As it relates to the nondiffusive part of the transport, we call A the flux function, and by extension, we call -div(A(u)) the flux term.

• The term ν∆u accounts for the spatial diffusion of the conserved quantity. Observe that when A ≡ 0 and S ≡ 0, Equation (1.2) is merely the heat equation. By analogy with the hydrodynamical context, the term ν∆u is called the viscous term and the constant ν is the viscosity coefficient.

In this regard, Equation (1.2) is often called a viscous scalar conservation law. A notable non-linear viscous conservation law is the viscous Burgers equation, namely Equation (1.2) in dimension d = 1 and with the flux function A(u) = u 2 /2. Particular physical motivations behind this equation will be discussed in Section 1. [START_REF] Andersson | Weak convergence for a spatial approximation of the nonlinear stochastic heat equation[END_REF].

In the recent years, a great interest has been brought to hyperbolic (or inviscid, or first-order ) scalar conservation laws, i.e. to equations of the type (1.2) with ν = 0: ∂ t u = -div(A(u)) + S(t, x), t ≥ 0, x ∈ D.

(1.3)

When A is non-linear, solutions of (1.3) usually develop shocks in finite time, even when the initial condition u 0 is smooth. In general, formulating Equation (1.3) in the weak sense is not sufficient to establish well-posedness as several weak solutions may co-exist after a shock has occured. The usual approach thus consists in establishing a formulation of (1.3) that contains a criterion discriminating the weak solution that has a proper physical meaning. In Kruzkov's seminal paper [START_REF] Stanislav | First order quasilinear equations in several independent variables[END_REF], such a formulation was introduced with the notion of entropic solution. Another notable formulation of (1.3) is the notion of kinetic solution introduced by Lions, Perthame and Tadmor [START_REF] Lions | A Kinetic Formulation of Multidimensional Scalar Conservation Laws[END_REF] as a generalisation of the entropic one.

To establish a proper notion of solution for Equation (1.3), one usually first considers the solution to Equation (1.2). Indeed, (1.2) is a second-order parabolic equation for which it is often possible (depending on the choice of the different parameters) to find a unique solution in the classical sense [START_REF] Ladyženskaja | Linear and quasi-linear equations of parabolic type[END_REF]. In this regard, (1.2) is often presented as the parabolic approximation of (1.3). For instance, the entropic (or the kinetic) solution of (1.3) is found by the so-called vanishing viscosity method : it is sought as the limit of the solutions of (1.2) as ν tends to 0. When the physical system described by the conservation law is subject to unpredictable external disturbances, it can be relevant to consider the source term S to be random. According to the nature of these disturbances, the term S may be chosen among a wide variety of random forcing terms. In this respect, white noises are almost always successful candidates. Indeed, they express the chaotic nature of the physical system at all scales of time and/or space and they offer mathematical convenience by the use of familiar probabilistic tools provided by Gaussian processes. The noise is said to be • white in time and in space when E[S(t, x)S(s, y)] = δ t=s δ x=y ;

• white in time and spatially correlated when E[S(t, x)S(s, y)] = δ t=s c(x, y), where c is a function on D 2 expressing the spatial correlations.

We will be interested in cases where the random source is written S(t, x) = Φ(t, x, u(t, x)) Ẇ (t), where (W (t)) t≥0 is a cylindrical Brownian motion, which formally writes

W (t) = k≥1 W k (t)e k , t ≥ 0,
where (e k ) k≥1 is a complete orthonormal basis of a Hilbert space H and (W k ) k≥1 is a family of independent real Brownian motions. The mapping Φ is the noise operator. For any (t, x, u) ∈ [0, +∞)× D × R, Φ(t, x, u) is a linear form on H. When Φ does not depend on u, the noise is said to be additive, otherwise it is multiplicative.

With such a source term, (1.2) falls into the realm of infinite dimensional stochastic differential equations.

Viscous scalar conservation laws with stochastic forcing

Well-posedness

Brief survey

Stochastic calculus is certainly the most appropriate framework to study Equation (1.2) forced with a white noise. Starting from the 1960's, efforts have been made to interpret evolutionary PDEs with random noise as generalised stochastic differential equations (SDEs) where the state space is Hilbert or Banach. Extensive results have been obtained since then and the essence of the theory is contained in Da Prato and Zabczyk's reference book [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF] in the domain of evolutionary stochastic partial differential equations (SPDEs). In [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF], the notion of infinite dimensional Wiener process is introduced together with the associated Itô integral. The Itô formula (the most important tool) is established for Hilbertvalued stochastic processes. Furthermore, well-posedness results are given for linear and semi-linear SPDEs.

In this framework, Equation (1.2) shall be expressed under the formulation du = (-div(A(u)) + ν∆u) dt + Φ(t, •, u)dW (t), t ≥ 0.

(1.4)

The sought solution is a stochastic process (u(t)) t≥0 taking values in a functional state space E, where E is a Hilbert or a Banach space. Observe that the mapping u → -div(A(u)) + ν∆u constitutes the drift of the (generalised) SDE. There are several possible definitions for the solution of Equation (1.4). Let us mention the three main ones.

• The most basic one is the strong solution, that is, a solution (u(t)) t≥0 with an initial condition u(0) = u 0 and which satisfies the strong formulation u(t) = u 0 + t 0 (-div(A(u)) + ν∆u)ds + t 0 Φ(s, x, u)dW (s), t ≥ 0, almost surely, where the second term of the right-hand side is a Bochner integral and the third term is an infinite dimensional Itô integral. Notice in particular that the solution u must have enough spatial regularity for the Laplacian ∆u to have a classical meaning. Suitable state spaces for the stochastic process (u(t)) t≥0 are for instance the Sobolev space W m,p (D), with m ≥ 2, p ≥ 1, or the space C m (D), m ≥ 2.

• A weaker definition of solution is given by the mild formulation u(t) = S t u 0 -t 0 S t-s div(A(u))ds + t 0 S t-s Φ(s, x, u)dW (s), t ≥ 0, almost surely, where (S t ) t≥0 is the semigroup generated by the operator ν∆ (i.e. the heat semigroup). It can be observed that in this formulation, the Laplacian need not exist as the spatial diffusion acts on each term of the dynamics through the semigroup (S t ) t≥0 . Therefore, a mild solution might be defined on a larger class of state spaces than a strong solution, e.g. in W m,p (D), m ≥ 1.

• Finally, the weak formulation generally writes: for all ϕ ∈ C 2 c (D), for all t ≥ 0, almost surely, ϕ(x)Φ(s, x, u(s, x))dxdW (s).

In the case where D has boundaries, the test function ϕ is usually needed to satisfy some Dirichlet boundary conditions. Notice that the weak formulation asks no spatial regularity on u. The state spaces for such solutions are often the spaces L p (D), p ≥ 1.

A particular observation arising from the mild formulation is that it looks like a mollified version of the hyperbolic counterpart. As in the deterministic case, weak, mild or strong solutions to Equation (1.4) are used to approximate solutions of du = -div(A(u))dt + Φ(t, x, u)dW (t), t ≥ 0, x ∈ D, (1.5) and the vanishing viscosity method is still a standard tool. Two main notions of solution are used in practice to establish the well-posedness of (1.5). Those are the generalisations to the stochastic case of the solutions in the entropic and kinetic sense. Let us give the formal definitions.

• To establish an entropic formulation of (1.5), one has first to define a set of entropy-entropy flux functions. That is a family of couples (η, q) where η is a real convex function at least twice differentiable, called entropy, and q is a continuously differentiable function from R to R d satisfying q (v) = η (v)A (v), for all v ∈ R, and which represents the flux of the entropy η. An entropic solution of (1.5) satisfies: for all entropy-entropy flux (η, q), and all non-negative test functions ϕ ∈ C 2 c (D), for all t ≥ 0, almost surely,

0 ≤ D η(u 0 (x))ϕ(x)dx - D η(u(t, x))ϕ(x)dx + t 0 D
q(u(s, x)) • ∇ϕ(x)dxds

+ t 0 D
η (u(s, x))Φ(s, x, u(s, x))ϕdx dW (s)+ 1 2 t 0 D |Φ(s, x, u(s, x))| 2 η (u(s, x))ϕdxds.

• The kinetic formulation is expressed through the characteristic function f (x, t, ξ) := 1 u(x,t)>ξ . More precisely, u is said to be a kinetic solution of (1.5) if for all T > 0, there exists a random where •, • denotes the L2 (D × R) scalar product.

In Figure 1.1, we recorded, in the chronological order, a list of well-posedness results for scalar conservation laws with stochastic forcing where we give some details on the setting. Some precautions must however be taken while examining this table:

• In some works, the equation considered is more general than a scalar conservation law. In these cases, the assumptions we mention in Figure 1.1 are those that remain after restricting the general equation to the particular case of a scalar conservation law. For instance, in [START_REF] Gyöngy | Existence and uniqueness results for semilinear stochastic partial differential equations[END_REF][START_REF] Gyöngy | On the Stochastic Burgers' Equation in the Real Line[END_REF][START_REF] Gyöngy | On L p -solutions of semilinear stochastic partial differential equations[END_REF], a nonlinear deterministic source term is added in the equation; in [START_REF] Gyöngy | On L p -solutions of semilinear stochastic partial differential equations[END_REF][START_REF] Hofmanová | Strong solutions of semilinear stochastic partial differential equations[END_REF], the viscous term consists of a uniformly elliptic diffusion operator instead of a Laplacian; in [START_REF] Hofmanová | Strong solutions of semilinear stochastic partial differential equations[END_REF], the non-linear term is defined in a more general way than the ordinary flux term; in [START_REF] Hofmanová | Degenerate parabolic stochastic partial differential equations[END_REF][START_REF] Debussche | Degenerate parabolic stochastic partial differential equations: Quasilinear case[END_REF][START_REF] Gess | Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE[END_REF], a positive semi-definite diffusion operator holds in place of the viscous term and thus, depending on the nature of this operator, the equation may be of the first or of the second order (such equations are usually called quasilinear degenerate parabolic-hyperbolic SPDEs).

• As much as it aims to be extensive, the list is still non-exhaustive.

• Only some particular assumptions are specified. For instance, we do not give details on the noise coefficients. Furthermore, assumptions of polynomial growth on the flux usually concern the derivatives of the flux function, we do not specify up to which order these derivatives are subject to growth constraints.

• In the cited articles, the aspects of the results that may go beyond well-posedness are not mentioned (such aspects concern for instance the regularity of the solutions, the dependence on initial conditions, the existence and uniqueness of an invariant measure, the Markov and/or Feller properties satisfied by the solution, etc...)

Another result of well-posedness

The work contained in [START_REF] Martel | Viscous scalar conservation law: strong solution and invariant measure[END_REF] (see the last line of Table 1.1) consitutes the second chapter of this manuscript. In the first part of this chapter, we are concerned with the well-posedness of Equation (1.4) in the following setting: the domain D is the one-dimensional torus T and the noise is additive and white in time. More precisely, given a cylindrical Brownian motion (W (t)) t≥0 in a Hilbert space H with the expression W (t) = k≥1 W k (t)e k , our noise operator Φ is defined by the coefficients

g k (x) := Φ(x)e k , x ∈ T, k ≥ 1.
In short, the equation becomes For each k ≥ 1, g k is a continuous function from T to R such that T g k (x)dx = 0. This condition implies in particular that any solution of (1.6) satisfies d dt T u(t, x)dx = 0. Thus, we may restrict our attention to solutions whose spatial average is zero. To this end, we will denote by W m,p 0 (T) the space of functions v ∈ W m,p (T) satisfying T v(x)dx = 0, and we endow it with the norm

du = (-∂ x A(u) + ν∂ xx u) dt + k≥1 g k dW k (t), t ≥ 0, x ∈ T. (1.6)
• W m,p 0 (T) = ∂ m x • L p (T)
. Incidentally, we will write H m 0 (T) := W m,2 0 (T). Note that the Poincaré inequality implies that

• L 2 (T) ≤ • H 1 0 (T) ≤ • H 2 0 (T) ≤ • • • . Given this setting, we assume that k≥1 g k 2 H 2 0 (T) =: D 0 < +∞.
Moreover, the function A is assumed to be of class C 2 , its first derivative A to have at most polynomial growth, and its second derivative A to be locally Lipschitz continuous. Under these assumptions, we show the following Theorem (Theorem 2.4). There exists a unique H 2 0 (T)-valued stochastic process (u(t)) t≥0 satisfying Equation (1.4) in the strong sense.

The proof follows a standard approach (used for instance in [START_REF] Da Prato | Stochastic Burgers' equation[END_REF] or [START_REF] Da | Ergodicity for Infinite Dimensional Systems[END_REF]Section 14.2]) but is driven by regularity issues required for establishing a strong solution (such issues are also addressed for instance in [15, Appendice A] or [START_REF] Hofmanová | Strong solutions of semilinear stochastic partial differential equations[END_REF]). Let us formally explain the overall approach of the proof. As the non-linear term is locally Lipschitz and thanks to the regularising properties of the heat semigroup, it is possible to establish, via a fixed point argument applied for each trajectory, the existence and uniqueness of a local-in-time mild solution

(u(t)) t∈[0,τ ] ∈ C([0, τ ], H 1 0 (T)) to (1.6)
, where τ is an almost surely positive stopping time. The mild formulation is best suited to apply the fixed point theorem, but is actually sufficient even if we ultimately seek well-posedness in the strong sense, as the following property holds: Proposition (Proposition 2.12). Any mild solution to (1.6) (local or global in time) with a regular enough initial condition is a strong solution.

To extend the local solution to a global solution, one usually needs a priori estimates that prevent any blow-up in finite time. Since the non-linearity is not globally Lipschitz, such estimates are not forthright. However, the viscosity endows Equation (1.6) with a decisive property for the stability of the solution.

Proposition (Dissipativity). The drift function in the SPDE (1.6) satisfies:

-∂ x A(v) + ν∂ xx v, v L 2 (T) = -ν v 2 H 1 0 (T) ≤ 0, v ∈ H 2 0 (T).
This property is actually quite standard for semi-linear equations and the proof follows from simple computations. To see its significance, let us apply the Itô formula to (1.6) with the squared L 2 -norm. We get

d u 2 L 2 (T) = 2 -∂ x A(u) + ν∂ xx u, u L 2 (T) dt + 2 k≥1 g k , u L 2 (T) dW k (t) + k≥1 g k 2 L 2 (T) dt.
Taking the expectation and applying the dissipativity property, it appears that

d dt E u(t) 2 L 2 (T) ≤ -2νE u(t) 2 H 1 0 (T) + D 0 , (1.7)
meaning that the growth of the L 2 -norm of the solution is controlled. We should notice that, at this point, we formally have the existence and uniqueness of an L 2 0 (T)-valued weak solution to (1.6). Indeed, Chapter 1. Introduction our local mild solution (u(t)) t∈[0,τ ] with values in H 1 0 (T) is even more so a local weak solution with values in L 2 0 (T). The L 2 estimate arising from the growth control (1.7) ensures that this weak local solution does not blow-up and is actually global. However, such estimates do not give information on the behaviour of the H 1 0 -norm, whose growth needs to be controlled in order to consider a global mild solution. In this regard, the dissipativity property is not sufficient but it can be strengthened to the following Proposition (Generalised dissipativity). The drift function in the SPDE (1.6) satisfies for all positive even number p:

-∂ x A(v) + ν∂ xx v, v p-1 L 2 (T) = - 4ν(p -1) p 2 v p/2 H 1 0 (T) ≤ 0, v ∈ H 2 0 (T).
When p = 2, this corresponds to the usual dissipativity property. This proposition is not plainly stated in the content of Chapter 2 but appears covertly in the step 3 of the proof of Lemma 2.15. It yields the a priori L p bounds given in the statement of Lemma 2.15, which eventually leads, using the polynomial growth of A , to have a control over the H 1 0 -norm, extending thereby the local solution to a global one.

Invariant measure

After well-posedness, the most addressed topic regarding stochastic conservation laws is the existence and uniqueness of an invariant measure.

Definition (Invariant measure). Let (X t ) t≥0 be a Markov process on a state space E. A probability measure µ on E is said to be an invariant measure of (X t ) t≥0 if

X 0 ∼ µ =⇒ ∀t ≥ 0, X t ∼ µ.
The analysis of invariant measures sheds light on the long time behaviour of solutions, the ergodic properties, and the statistical properties that are inherent to the system and do not depend on the initial conditions. More details on this last point will be discussed in Subsection 1.4. We mention here some of the previous works done in this direction but we refer the reader to [START_REF] Maslowski | Invariant measures for nonlinear SPDE's: uniqueness and stability[END_REF][START_REF] Gui-Qiang | Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing[END_REF] for more detailed reviews on the subject. Some of the works mentioned in Table 1.1 contain results regarding the invariant measure. For the stochastic Burgers equation, existence is proved by Da Prato, Debussche and Temam in [START_REF] Da Prato | Stochastic Burgers' equation[END_REF], by Dong and Xu in [START_REF] Dong | One-dimensional stochastic Burgers equation driven by Lévy processes[END_REF], by Hausenblas and Giri in [START_REF] Hausenblas | Stochastic Burger's equation with polynomial nonlinearity driven by Lévy process[END_REF]. Existence and uniqueness are proved by Da Prato and Gatarek in [START_REF] Da | Stochastic Burgers equation with correlated noise[END_REF], by Da Prato and Zabczyk in [START_REF] Da | Ergodicity for Infinite Dimensional Systems[END_REF]Chapter 14], by E, Khanin, Mazel and Sinai in [START_REF] Weinan | Invariant Measures for Burgers Equation with Stochastic Forcing[END_REF]. For more general conservation laws, existence and uniqueness of an invariant measure have been proved by Boritchev in [START_REF] Boritchev | Generalised Burgers equation with random force and small viscosity[END_REF]Chapter 4], by Debussche and Vovelle in [START_REF] Debussche | Invariant measure of scalar first-order conservation laws with stochastic forcing[END_REF] in a setting very close to [START_REF] Debussche | Scalar conservation laws with stochastic forcing[END_REF] but where the flux function satisfies a non-degeneracy condition. For stochastic anisotropic parabolic-hyperbolic equations, existence and uniqueness of an invariant measure have been proved by Chen and Pang in [START_REF] Gui-Qiang | Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing[END_REF].

An invariant measure for (1.6)

The second main result of the second chapter is Theorem (Theorem 2.7). The strong solution (u(t)) t≥0 of Equation (1.6) admits a unique invariant measure.

As for the well-posedness result presented in Subsection 1.2.1, the proof follows a standard approach but has to deal with regularity concerns. The usual method to address existence of an invariant measure is to use the Krylov-Bogoliubov theorem (see e.g. [START_REF] Da | Ergodicity for Infinite Dimensional Systems[END_REF]Theorem 3.1.1]):

Theorem (Krylov-Bogoliubov). A Feller process (X t ) t≥0 on a Polish space E admits at least one invariant measure if for some initial condition X 0 , there exists an increasing sequence of positive numbers (T n ) n≥1 tending to +∞ such that the family of probability measures {µ Tn : n ≥ 1} defined by

µ Tn : Γ ∈ B(E) -→ 1 T n Tn 0 P (X t ∈ Γ) dt ∈ [0, 1]
(where B(E) denotes the Borel σ-algebra on E) is tight.

The proof of the Krylov-Bogoliubov theorem is done in two steps. The first step is merely an application of the Prokhorov theorem stating that the sequence (µ Tn ) n≥1 admits a limit µ in the weak sense. The second step consists in showing that the limit µ is invariant for the process (X t ) t≥0 .

In our case, if we consider the strong solution (u(t)) t≥0 as an L 2 0 (T)-valued process, the conditions of the Krylov-Bogoliubov theorem are easily satisfied as the viscosity induces regularity in space and thus compactness. Indeed, from Equation (1.7), taking the average in time, we get

1 T T 0 E u(t) 2 H 1 0 (T) dt ≤ 1 2νT E u 0 2 L 2 (T) + D 0 .
It follows from the Markov inequality that for all ε > 0,

1 T T 0 P u(t) 2 H 1 0 (T) > ε dt ≤ ε 1 2νT E u 0 2 L 2 (T) + D 0 .
Hence, since the space H 1 0 (T) is compactly embedded in L 2 0 (T), the requirements of the Krylov-Bogoliubov theorem are met. However, (u(t)) t≥0 is not stricto sensu a Markov process on L 2 0 (T) but on H 2 0 (T). The weak limit of the sequence (µ Tn ) n∈N we obtain by this method do not qualify to be an invariant measure as we do not know if it gives full weight to H 2 0 (T). That is where higher regularity estimates need to intervene and where the original proof of the Krylov-Bogoliubov theorem needs to be adapted in order to conclude for the existence part of our proof (Lemma 2.23).

The proof of uniqueness relies on the following standard property for scalar conservation laws:

Proposition (L 1 -contraction, Proposition 2.21). Let (u(t)) t≥0 and (v(t)) t≥0 be two strong solutions of Equation (1.6). Then, for all 0 ≤ s ≤ t, almost surely,

u(t) -v(t) L 1 (T) ≤ u(s) -v(s) L 1 (T) .
That is to say, two solutions of (1.6) with possibly different initial conditions, but driven by the same noise, get closer (or at least do not move away from each other) with respect to the L 1 distance. Recall however that the dissipativity property induces that each of these solutions drifts towards the center of the space L 2 0 (T). This type of behaviour is the starting point for the following coupling argument: the two solutions are attracted to the center of L 2 0 (T) and thus, they get closer to one another for the L 2 distance and even more so for the L 1 distance, and each time they get closer, they stay closer as the L 1 -contraction prevents them to move apart. Applying this argument leads to the reinforced Proposition (L 1 -confluence, Equation (2.56)). Let (u(t)) t≥0 and (v(t)) t≥0 be two strong solutions of Equation (1.6). Then, almost surely,

lim t→∞ u(t) -v(t) L 1 (T) = 0.
Assuming that there exist two different invariant measures µ 1 and µ 2 for (1.6), and choosing two solutions (u(t)) t≥0 and (v(t)) t≥0 such that u 0 ∼ µ 1 and v 0 ∼ µ 2 , the above proposition yields promptly that µ 1 = µ 2 .

Chapter 1. Introduction

Authors and reference Equation Space discretisation

Time discretisation

Bréhier [START_REF] Bréhier | Approximation of the invariant measure with an Euler scheme for Stochastic PDE's driven by Space-Time White Noise[END_REF] Semi-linear parabolic SPDE

Galerkin method

Semi-implicit3 Euler scheme Bréhier, Kopec [START_REF] Bréhier | Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme[END_REF] Semi-linear parabolic SPDE

Spectral Galerkin method

Semi-implicit Euler scheme

Chen, Hong, Wang [START_REF] Chen | Approximation of invariant measure for damped stochastic nonlinear Schrödinger equation via an ergodic numerical scheme[END_REF] Damped stochastic non-linear Schrödinger equation

Spectral Galerkin method

Modified implicit Euler scheme Chen, Gan, Wang [START_REF] Chen | A full-discrete exponential Euler approximation of the invariant measure for parabolic stochastic partial differential equations[END_REF] Semi-linear parabolic SPDE Spectral Galerkin method Exponential Euler scheme Cui, Hong, Sun [START_REF] Cui | Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE[END_REF] Semi-linear parabolic SPDE with cubic polynomial non-linearity Spectral Galerkin method Implicit Euler scheme Boyaval, Martel, Reygner [START_REF] Boyaval | Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law[END_REF] Stochastic viscous scalar conservation law

Finite Volume method

Split-step Euler scheme

Numerical approximation

Most of the numerical schemes developed to approximate solutions of SPDEs in finite time are adapted from the usual deterministic methods. For semi-linear SPDEs such as Equation (1.4), a common approach is the finite element approximation and in particular the Spectral Galerkin method (see for instance [START_REF] Gabriel | An Introduction to Computational Stochastic PDEs[END_REF][START_REF] Wang | Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus[END_REF][START_REF] Andersson | Weak convergence for a spatial approximation of the nonlinear stochastic heat equation[END_REF][START_REF] Conus | Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients[END_REF][START_REF] Andersson | Duality in refined Sobolev-Malliavin spaces and weak approximation of SPDE[END_REF][START_REF] Kruse | Strong and Weak Approximation of Semilinear Stochastic Evolution Equations[END_REF]). This method is not suited for the inviscid equation (1.5). In this hyperbolic case, the discontinuous Galerkin method may be considered [START_REF] Li | A Discontinuous Galerkin Method for Stochastic Conservation Laws[END_REF], but the most common approach is the finite volume method [START_REF] Bauzet | Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation[END_REF][START_REF] Bauzet | Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise[END_REF][START_REF] Dotti | Convergence of the Finite Volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation[END_REF][START_REF] Sylvain | Numerical approximation of hyperbolic stochastic scalar conservation laws[END_REF]. A notable advantage of finite volumes is that they are well suited for conservation laws in general, should they be parabolic or hyperbolic [START_REF] Eymard | Finite volume methods[END_REF].

The numerical approximation results for the invariant measure of an SPDE are quite recent. In Figure 1.2, we recorded, in the chronological order, a list of these results. As far as we are aware, the list is exhaustive.

The last line of Figure 1.2 corresponds to the content of Chapter 3, which is devoted to the approximation of the invariant measure of the solution of Equation (1.6) with a finite volume method for the space discretisation and a split-step backward Euler method for the time discretisation. We should point out that in all the other mentioned references of Figure 1.2, convergence rates are established for the weak error of discretisation. In our case, the lack of such results is essentially due the non-globally Lipschitz nature of the flux term, but we will get back to this point after introducing our numerical scheme.

In order to discretise (1.6) with respect to the space variable, we first define the following regular mesh on the torus:

i -1 N , i N , i ∈ Z/N Z .
Averaging in (1.6) over each cell of the mesh, we get

d N i N i-1 N u(t, x)dx = -N A u t, i N -A u t, i -1 N dt + νN ∂ x u t, i N -∂ x u t, i -1 N dt + k≥1 N i N i-1 N g k (x)dxdW k (t), i ∈ Z/N Z. (1.8)
Finite-volume schemes aim to approximate the dynamics of the average value of the solution over each cell of the mesh. This leads to the introduction of a numerical flux function A(u, v) approximating the flux of the conserved quantity at the interface between two adjacent cells. As regards the viscous term in (1.8), we replace the space derivatives by their finite difference approximations. As for the noise coefficients, we introduce the shorthand notation

σ k i := N i N i-1 N g k (x)dx, k ≥ 1, i ∈ Z/N Z,
and we write σ k := (σ k 1 , . . . , σ k N ). In particular, σ k belongs to the space

R N 0 := {u = (u 1 , . . . , u N ) ∈ R N : u 1 + • • • + u N = 0}.
These operations result in the following stochastic differential equation

dU i (t) = -N A (U i (t), U i+1 (t)) -A (U i-1 (t), U i (t)) dt + νN 2 (U i+1 (t) -2U i (t) + U i-1 (t)) dt + k≥1 σ k i dW k (t), i ∈ Z/N Z, t ≥ 0, (1.9)
as a semi-discrete finite-volume approximation of (1.6) in the sense that U i (t) is meant to be an approximation of the spatial average

N i N i-1 N u(t, x)dx.
In order to be consistent with Equation (1.6), the numerical flux function has to satisfy A(v, v) = A(v) for all v ∈ R. Moreover, we make the assumption of a monotone numerical flux, i.e. we assume that A is non-decreasing with respect to the first variable and non-increasing with respect to the second.

Denoting by b = (b 1 , . . . , b N ) the function defined from

R N 0 to R N 0 by b i (v) := -N A(v i , v i+1 ) -A(v i-1 , v i ) + νN 2 (v i+1 -2v i + v i-1 ), i ∈ Z/N Z, v ∈ R N 0 ,
we can express (1.9) in the vectorised form

dU(t) = b(U(t))dt + k≥1 σ k dW k (t), t ≥ 0. (1.10)
Let us now discretise (1.10) with respect to the time variable. For this, we introduce a time step ∆t > 0, and we write

∆W k n := W k (n∆t) -W k ((n -1)∆t) for all k, n ≥ 1.
As it was already noticed in [START_REF] Mattingly | Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise[END_REF], explicit numerical schemes for SDEs with non-globally Lipschitz continuous coefficients do not preserve in general the large time stability, whereas implicit schemes are more robust. Therefore, since our main focus in this thesis is to approximate invariant measures, we propose the following split-step backward Euler method :

       U n+ 1 2 = U n + ∆tb U n+ 1 2 , U n+1 = U n+ 1 2 + k≥1 σ k ∆W k n+1 .
(1.11)

The first result of Chapter 3 is the following Theorem (Proposition 3.15, Proposition 3.23 and Theorem 3.5). Let U 0 be an R N 0 -valued random variable. Then, (i) there exists a unique R N 0 -valued stochastic process (U(t)) t≥0 solution to (1.10) with initial condition U 0 ; Chapter 1. Introduction (ii) there exists a unique R N 0 -valued Markov chain (U n ) n∈N solution to (1.11) with initial condition U 0 . Furthermore, (iii) (U(t)) t≥0 admits a unique invariant measure ν N ;

(iv) (U n ) n∈N admits a unique invariant measure ν N,∆t .

The proof of this theorem relies on properties analogous to the SPDE. In particular, the monotonicity of the numerical flux ensures the following Proposition (Lemma 3.13). For any u, v ∈ R N 0 , the function b satisfies

i∈Z/N Z u i b i (u) ≤ -νN 2 i∈Z/N Z (u i+1 -u i ) 2 (dissipativity) (1.12) and i∈Z/N Z sign(u i -v i ) (b i (u) -b i (v)) ≤ 0. (1.13)
The well-posedness of (1.10) is proved by use of Inequality (1.12) in a similar way as for the SPDE, while (1.12) and (1.13) allow to prove respectively the existence and uniqueness of a solution (U n ) n∈N to (1.11) and more than that, to prove respectively the existence and uniqueness of an invariant measure for both the processes (U(t)) t≥0 and (U n ) n∈N . In particular, thanks to (1.13), the L 1 -contraction has been preserved by the dicretisation: Proposition (Discrete L 1 -contraction, Proposition 3.17 and Lemma 3.24). Let (U(t)) t≥0 and (V(t)) t≥0 be two solutions of (1.10), with possibly different initial conditions, but perturbed by the same Wiener process. Then, for all 0 ≤ s ≤ t, almost surely

U(t) -V(t) 1 ≤ U(s) -V(s) 1 .
Similarly, two solutions (U n ) n∈N and (V n ) n∈N to (1.11) satisfy for all n ∈ N, almost surely,

U n+1 -V n+1 1 ≤ U n -V n 1 .
From the numerical scheme (1.11), we now reconstruct a piecewise constant approximation (u N (t)) t≥0 to the solution (u(t)) t≥0 of (1.6) by setting

u N (t, x) := i∈Z/N Z U i (t)1 ( i-1 N , i N ] (x).
(1. [START_REF] Boritchev | Turbulence de Burgers en 1D : un cas modèle pour la théorie de Kolmogorov[END_REF] This way (u N (t)) t≥0 is an L 2 0 (T)-valued process with an invariant probability measure µ N on L 2 0 (T). This measure µ N is actually the embedding of ν N in L 2 0 (T) via the reconstruction given by (1.14), and in the same way, we can define the measure µ N,∆t on L 2 0 (T) as the pushforward of the measure ν N,∆t via the same reconstruction.

It is now possible to compare µ N,∆t , µ N and µ as all of these measures belong to the space P(L 2 0 (T)) of probability measures over L 2 0 (T). Note that P(L 2 0 (T)) (resp. P(R N 0 )) is naturally endowed with the topology associated to the weak convergence. The subset of P(L 2 0 (T)) (resp. P(R N 0 )) containing all the measures with finite second order moment is denoted P 2 (L 2 0 (T)) (resp. P 2 (R N 0 )). We may endow this subspace with the topology associated to the following metric: Let (E, • E ) be a normed vector space. Let α and β be two probability measures on E with finite second order moment. The second order Wasserstein distance between α and β is defined by

U n ν N,∆t Invariant measure U(t) ν N Invariant measure ∆t → 0 u(t) µ Invariant measure N → +∞
W 2 (α, β) := inf u∼α v∼β E u -v 2 E 1/2 .
Let us now state the second result of Chapter 3, and main result of this thesis:

Theorem (Convergence of invariant measures, Theorem 3.7). The following equality holds:

lim N →∞ lim ∆t→0 W 2 (µ N,∆t , µ) = 0.
As illustrated by Figure 1.3, a specific order for the discretisation leads to the proof of the above theorem. In particular, we show that: A) µ N converges towards µ as N → +∞; B) for any integer N ≥ 1, ν N,∆t (or µ N,∆t ) converges towards ν N (or µ N ) as ∆t → 0.

The step A is the subject of Section 3.3 while the step B is addressed in Section 3.4. Each of these steps is itself divided in two substeps as we will show that A.1) the family (µ N ) N ≥1 is relatively compact in P 2 (L 2 0 (T));

A.2) any subsequential µ * is invariant for the solution (u(t)) t≥0 of (1.6), and thus µ * = µ;

B.1) the family (ν N,∆t ) ∆t>0 is relatively compact in P 2 (R N 0 ); B.2) any subsequential limit ν * N (as ∆t → 0) is invariant for the solution (U(t)) t≥0 of (1.10), and thus

ν * N = ν N .
As the proofs of A.1 and A.2 are very much alike those of B.1 and B.2 respectively, let us give only the intuition regarding the step A. The proof of relative compactness of the family (µ N ) N ≥1 in P 2 (L 2 0 (T)) involves the Prokhorov theorem and the following uniform discrete H 1 0 (T) estimate (Lemma 3.31):

E   N i∈Z/N Z (U i+1 -U i ) 2   ≤ D 0 2ν , U ∼ ν N .
Chapter 1. Introduction Now, let µ * be the limit in P 2 (L 2 0 (T)) of some subsequence (µ N j ) j∈N . By virtue of the Skorokhod representation theorem, we choose random variables u(0) ∼ µ * and u N j (0) ∼ µ N j such that u N j (0) converges almost surely in L 2 0 (T) towards u(0) as j → +∞. Consequently, we can define on the same probability space the processes (u(t)) t≥0 and (u N j (t)) t≥0 respectively as the solution of (1.6) with initial condition u(0) and the reconstructed solution of (1.10) via (1.14) with initial condition u N j (0). In this setting, we show the following finite time convergence result (Proposition 3.36):

∀t ≥ 0, lim j→∞ E u N j (t) -u(t) 2 L 2 0 (T) = 0. (1.15)
Since for all t ≥ 0, the law of u N j (t) is µ N j , it follows immediately that W 2 (µ N j , L(u(t))) converges to 0 as j → +∞. Thus, W 2 (µ * , L(u(t))) = 0 for all t ≥ 0, which means that µ * is actually invariant for the process (u(t)) t≥0 .

In establishing the finite time approximation (1.15), we had to deal with a non-Lipschitz flux term. Our control of the strong error relied on moment estimates, which themselves relied on the assumption of polynomial growth of the flux function. The approximation in time is treated in the same way. A drawback of this approach is that no convergence rate of the discretisation error can be derived. A possible way to establish orders of convergence of the invariant measure for the W 2 -distance would be to make the stronger assumption that A is globally Lipschitz and start from the convergence rates for the strong error in finite time that would come up. One could also follow a more standard procedure, and analyse the weak error, in particular when ∆t → 0, with the techniques that were pioneered by Talay [START_REF] Talay | Second order discretization schemes of stochastic differential systems for the computation of the invariant law[END_REF][START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF].

Application to turbulence

The rigorous understanding of the turbulence phenomenon is one of the most important open problems in mathematical physics. Among the extensive research efforts that have been made in this field, stochastic scalar conservation laws, and more specifically the stochastic Burgers equation, were found to play some role. To explain how these SPDEs were thrown under the light of such physical motivations, some context ought to be introduced.

The most common model for describing the motion of an incompressible viscous fluid flow is given by the Navier-Stokes equations. Derived from the fundamental principles of physics that are the continuum hypothesis (see the beginning of Section 1.1) and Newton's second law, and from taking into account the internal friction (i.e. the viscosity), this system of equations is given by:

∂ t u + (u • ∇)u + ∇p = ν∆u + f div(u) = 0, (1.16)
where the unkowns are the velocity field u(t, x) and the pressure p(t, x), x ∈ R 3 (see for instance [START_REF] Landau | Fluid Mechanics[END_REF] for the complete derivation of the system). The parameters include the kinematic viscosity coefficient ν > 0 and the external forces f (t, x) acting on the fluid. To study the general features of the flow that do not depend on the time and space scales, one may non-dimensionalise (1.16). In this regard, a most notable indicator of the behaviour of the flow is the dimensionless Reynolds number Re = U L ν , where U and L are respectively the typical velocity and length scales. When the Reynolds number is low enough, the flow is characterised by an orderly and predictable motion. The fluid particles follow smooth paths that do not cross each other. In this case, the flow is said to be laminar. Such a stable behaviour contrasts drastically with high Reynolds number flows. Indeed, when Re is high enough (usually around the order Re ≈ 10 3 and beyond), the flow follows a quite unsteady motion. Vortices of many sizes appear in a disorganised manner. In particular, fluctuations in the fluid occur at a wide range of space and time scales. A flow displaying such a chaotic behaviour is said to be turbulent.

Application to turbulence

Motivated by industrial and theoretical issues, hydrodynamic turbulence constitutes an active field of research. Numerous works testing numerical simulations against experimental results have attested the relevance of the Navier-Stokes model (for turbulent flows as much as for laminar ones).

Groundbreaking advances in the understanding of turbulence came forth in 1941 with the works of Kolmogorov [START_REF] Andreï | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF][START_REF] Andreï | On degeneration of isotropic turbulence in an incompressible viscous liquid[END_REF][START_REF] Andreï | Dissipation of Energy in Locally Isotropic Turbulence[END_REF] (see also [START_REF] Frisch | Turbulence: The Legacy of A. N. Kolmogorov[END_REF]). Using scaling arguments, Kolmogorov postulated that the (seemingly chaotic) turbulent flows actually display some universal behaviour, i.e. they show particular features that depend only on Re4 . His framework relies on several circumstantial assumptions. In particular, the velocity u(t, x) is assumed to be a random field whose space and time increments are pairwise independent and invariant with respect to rotations and reflections in some domain (the turbulence is said locally homogeneous and isotropic). Let us give some physical insight as regards these universality predictions. The external forces applied on the flow supply an amount of kinetic energy which is described by large space scales (of an order L). Due to the convection happening within the fluid, the energy is transported to smaller and smaller scales, down to a scale l, under which the energy is dissipated by the effect of viscosity. Between the energy scale L and the dissipation scale l, the transfer of kinetic energy does not depend on the forcing nor on the viscosity. Thus, in this interpretation of turbulence as an energy cascade, the emergence of universal properties in a particular scale range seems conceivable. The universality prediction is asserted via quantitative estimates such as the following:

(u(t, x + r) -u(t, x)) • r r 2 n ∼ C n r n 3 2 , l r 2 L, n ≥ 0, (1.17)
where • denote the expectation of the underlying random variable, and where C n is a universal constant. When n = 2, (1.17) may be expressed in the Fourier space, which gives the following relation for the space Fourier tranform û of u:

k< k 2 ≤k+1 û(k) 2 2 ∼ k -5 3 , L -1 k l -1 , (1.18)
which is the well-known Kolmogorov spectrum.

It should be noticed that (1.17), and thus (1.18), have been derived solely from scaling analysis and physical arguments. In particular, they do not rely on the Navier-Stokes equations. By the way, their mathematical status remains at the conjecture level. A rigorous analysis of the universal properties of turbulence ought to be achieved through (1.16). For instance, one could consider the stochastic process (u(t, x)) t≥0 solution to (1.16) with a random initial condition. In this perspective, one could furthermore study the invariant measures for (1.16) and compute the energy spectrum of stationary solutions. However, the well-posedness of (1.16) remains a notorious open problem and despite many efforts to overcome it, there are only very limited partial results on this topic [START_REF] Fefferman | Existence and smoothness of the Navier-Stokes equation[END_REF].

Nonetheless, the concept of energy cascade may be addressed via simpler models. In this perspective, if we bring Equation (1.16) to one space dimension, thereby dropping the pressure term and incompressibility condition which do not have a meaning in 1D, it boils down to

∂ t u + u∂ x u = ν∂ xx u + f,
that is, the viscous Burgers equation. This equation was actually studied by J. M. Burgers as a toy model for hydrodynamic turbulence [START_REF] Burgers | Hydrodynamics. -Application of a model system to illustrate some points of the statistical theory of free turbulence[END_REF]. Many works followed in this direction and some adjustments were made to this initial equation, by considering for instance a random initial condition or a random forcing. Indeed, the restriction from three to one space dimension induces the loss of the chaotic behaviour in the model. The stochastic forcing allows to maintain a steady yet unpredictable supply of energy in the system and thus to keep a cascade-like energy transfer.

Chapter 1. Introduction

More than that, what was originally meant to be a testing ground for the Navier-Stokes equations (for theoretical as much as for numerical concerns) turned out to be a prototype for a wide range of physical systems where the non-linear effects induce a non-trivial flux of energy across scales. Applications were found for instance in cosmology [START_REF] Ya | Gravitational instability: An Approximate theory for large density perturbations[END_REF][START_REF] Gurbatov | Probability distributions and spectra of potential hydrodynamic turbulence[END_REF][START_REF] Vergassola | Burgers' equation, devil's staircases and the mass distribution for large-scale structures[END_REF][START_REF] Albeverio | Stratified structure of the Universe and Burgers' equation-a probabilistic approach[END_REF], vehicle traffic models [START_REF] Chowdhury | Statistical physics of vehicular traffic and some related systems[END_REF], vortices in superconductors [START_REF] Blatter | Vortices in high-temperature superconductors[END_REF]... The stochastic Burgers equation has also close ties with the KPZ equation [START_REF] Bertini | Stochastic Burgers and KPZ equations from particle systems[END_REF][START_REF] Gubinelli | KPZ Reloaded[END_REF] and in this respect, happened to be an appropriate model for growing interfaces and directed polymers [START_REF] Kardar | Scaling of directed polymers in random media[END_REF][START_REF] Bouchaud | Scaling and intermittency in Burgers turbulence[END_REF][START_REF] Kifer | The Burgers equation with a random force and a general model for directed polymers in random environments[END_REF].

The mathematical questions raised by the randomly forced Burgers equation are the well-posedness, the invariant measures and the regularity of solutions. These were discussed in Section 1.2. Questions of a more physical nature concern the energy spectrum and the small-scale characteristics [START_REF] David | Karhunen-Loève expansion of Burgers' model of turbulence[END_REF][START_REF] Choi | Feedback control for unsteady flow and its application to the stochastic Burgers equation[END_REF][START_REF] Jeng | Forced Model Equation for Turbulence[END_REF][START_REF] Gotoh | Statistics of decaying Burgers turbulence[END_REF] as well as the asymptotic behaviour of probability distribution functions [START_REF] Weinan | Probability Distribution Functions for the Random Forced Burgers Equation[END_REF][START_REF] Weinan | On the statistical solution of the Riemann equation and its implications for Burgers turbulence[END_REF][START_REF] Weinan | Statistical theory for the stochastic Burgers equation in the inviscid limit[END_REF][START_REF] Weinan | Asymptotic theory for the probability density functions in Burgers turbulence[END_REF]. For general surveys, see [START_REF] Frisch | Burgulence[END_REF][START_REF] Bec | Burgers turbulence[END_REF][START_REF] Weinan | Stochastic hydrodynamics[END_REF][START_REF] Weinan | Stochastic PDEs in turbulence theory[END_REF][START_REF] Weinan | Recent results on mathematical and statistical hydrodynamics[END_REF].

Let us mention here an analogous result to the Kolmogorov spectrum (1.18) in the framework of Equation (1.6):

Theorem (Boritchev,[START_REF] Boritchev | Generalised Burgers equation with random force and small viscosity[END_REF]Theorem 4.7.3]). Let µ be the invariant measure for (1.6) where the flux function A is C ∞ and strictly convex, and let u be a random variable with distribution µ. Then the Fourier transform û of u satisfies for some M > 0

E   kM -1 ≤n≤kM |û n |   ∼ k -2 .
In particular, the Burgers energy spectrum has a decay rate of order -2. In Chapter 4, numerical experiments are realised on Burgers' equation with the finite volume scheme introduced in Section 1.3. After testing the stationarity of the scheme and establishing empirical convergence rates in space and in time, we compute some small-scale characteristics regarding turbulence. Notably, the slope of the energy spectrum is plotted and the result of the above theorem is recovered numerically.

Chapter 2

Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure Résumé. Ce chapitre correspond à la pré-publication [START_REF] Martel | Viscous scalar conservation law: strong solution and invariant measure[END_REF], écrite en collaboration avec J. Reygner. On s'intéresse aux lois de conservation scalaires avec de la viscosité et un bruit blanc en temps mais spatialement correlé. Le domaine spatial considéré est de dimension 1 et périodique. La fonction de flux est supposée localement lipschitzienne et à croissance polynomiale. Aucune hypothèse de nondégénerescence n'est imposée au flux ou au bruit. Dans un premier temps, on prouve l'existence et l'unicité d'une solution globale au sens fort. Dans un second temps, on établit l'existence et l'unicité d'une mesure invariante pour cette solution forte.

Abstract. This chapter corresponds to the preprint [START_REF] Martel | Viscous scalar conservation law: strong solution and invariant measure[END_REF], written in collaboration with J. Reygner. We are interested in viscous scalar conservation laws with a white-in-time but spatially correlated stochastic forcing. The equation is assumed to be one-dimensional and periodic in the space variable, and its flux function to be locally Lipschitz continuous and have at most polynomial growth. Neither the flux nor the noise need to be non-degenerate. In a first part, we show the existence and uniqueness of a global solution in a strong sense. In a second part, we establish the existence and uniqueness of an invariant measure for this strong solution.

Introduction

Stochastic viscous scalar conservation law

We are interested in the existence, uniqueness, regularity and large time behaviour of solutions of the following viscous scalar conservation law with additive and time-independent stochastic forcing

du = -∂ x A(u)dt + ν∂ xx udt + k≥1 g k dW k (t), x ∈ T, t ≥ 0, (2.1)
where (W k (t)) t≥0 , k ≥ 1, is a family of independent Brownian motions. Here, T denotes the onedimensional torus R/Z, meaning that the sought solution is periodic in space. The flux function A is assumed to satisfy the following set of conditions.

Assumption 2.1 (on the flux function). The function A : R → R is C 2 on R, its first derivative has at most polynomial growth:

∃C 1 > 0, ∃p A ∈ N * , ∀v ∈ R, |A (v)| ≤ C 1 (1 + |v| p A ) , (2.2)
Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure and its second derivative A is locally Lipschitz continuous on R.

The parameter ν > 0 is the viscosity coefficient. In order to present our assumptions on the family of functions g k : T → R, k ≥ 1, which describe the spatial correlation of the stochastic forcing of (2.1), we first introduce some notation. For any p ∈ [1, +∞], we denote by L p 0 (T) the subset of functions v ∈ L p (T) such that T vdx = 0.

The L p norm induced on L p 0 (T) is denoted by • L p 0 (T) . For any integer m ≥ 0, we denote by H m 0 (T) the intersection of the Sobolev space H m (T) with L 2 0 (T). Equipped with the norm

v H m 0 (T) := T |∂ m x v| 2 dx 1/2
, and the associated scalar product •, • H m 0 (T) , it is a separable Hilbert space. On the one-dimensional torus, the Poincaré inequality implies that

H m+1 0 (T) ⊂ H m 0 (T) and • H m 0 (T) ≤ • H m+1 0 (T)
. Actually, the following stronger inequality holds: if v ∈ H 1 0 (T), then v ∈ L ∞ 0 (T) and for all p ∈ [1, +∞),

v L p 0 (T) ≤ v L ∞ 0 (T) ≤ v H 1 0 (T) . (2.3)
The spaces H m 0 (T), m ≥ 0, generalise to the class of fractional Sobolev spaces H s 0 (T), where s ∈ [0, +∞), which will be defined in Section 2.2.1. We may now state: Assumption 2.2 (on the noise functions). For all k ≥ 1, g k ∈ H 2 0 (T) and

D 0 := k≥1 g k 2 H 2 0 (T) < +∞.
(2.4)

Let (Ω, F, P) be a probability space, equipped with a normal filtration (F t ) t≥0 in the sense of [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF]Section 3.3], on which (W k ) k≥1 is a family of independent Brownian motions. Under Assumption 2.2, the series k g k W k converges in L 2 (Ω, C([0, T ], H 2 0 (T))), for any T > 0, towards an H 2 0 (T)-valued Wiener process (W Q (t)) t∈[0,T ] with respect to the filtration (F t ) t≥0 , defined in the sense of [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF]Section 4.2], with the trace class covariance operator Q :

H 2 0 (T) → H 2 0 (T) given by ∀u, v ∈ H 2 0 (T), u, Qv H 2 0 (T) = k≥1 u, g k H 2 0 (T) v, g k H 2 0 (T) . (2.5)
Thus, almost surely, t → W Q (t) is continuous in H 2 0 (T) and for all u ∈ H 2 0 (T), the process

( W Q (t), u H 2 0 (T) ) t≥0 is a real-valued Wiener process with variance E W Q (t), u 2 H 2 0 (T) = t k≥1 g k , u 2 H 2 0 (T) .
(2.6)

Main results and previous works

First, we are interested in the well-posedness in the strong sense of Equation (2.1). In particular, we look for solutions that admit at least a second spatial derivative in order to give a classical meaning to the viscous term, in the sense of the following definition:

Definition 2.3 (Strong solution to (2.1)). Let u 0 ∈ H 2 0 (T)
. Under Assumptions 2.1 and 2.2, a strong solution to Equation (2.1) with initial condition u 0 is an (F t ) t≥0 -adapted process (u(t)) t≥0 with values in H 2 0 (T) such that, almost surely:

1. the mapping t → u(t) is continuous from [0, +∞) to H 2 0 (T);

2. for all t ≥ 0, the following equality holds:

u(t) = u 0 + t 0 (-∂ x A (u(s)) + ν∂ xx u(s)) ds + W Q (t). (2.7)
In the above definition, the first condition ensures that the time integral in Equation (2.7) is a well-defined Bochner integral in L 2 0 (T). For a careful introduction of the general concepts of random variables and stochastic processes in Hilbert spaces, the reader is referred to the third and fourth chapters of the reference book [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF].

Our first result is the following:

Theorem 2.4 (Well-posedness). Let u 0 ∈ H 2 0 (T)
. Under Assumptions 2.1 and 2.2, there exists a unique strong solution (u(t)) t≥0 to Equation (2.1) with initial condition u 0 . Moreover, the solution depends continuously on initial data in the following sense: if (u

(j) 0 ) j≥1 is a sequence of H 2 0 (T) satisfying lim j→∞ u 0 -u (j) 0 H 2 0 (T) = 0,
then, denoting by (u (j) (t)) t≥0,j≥1 the family of associated solutions, for any T ≥ 0, we have almost surely lim

j→∞ sup t∈[0,T ] u(t) -u (j) (t) H 2 0 (T) = 0.
Similar results have already been established: the case where the flux A is strictly convex is treated in [15, Appendix A], and the case where A is globally Lipschitz continuous is treated in [START_REF] Hofmanová | Strong solutions of semilinear stochastic partial differential equations[END_REF]. Furthermore, the case of mild solutions (in L p spaces) has been looked at in [START_REF] Gyöngy | On L p -solutions of semilinear stochastic partial differential equations[END_REF]. Here, no global Lipschitz continuity assumption nor restrictions on the convexity of the flux function are made. We can also point out that the well-posedness of stochastically forced conservations laws in the inviscid case (i.e. when ν = 0) has been under a great deal of investigation in the recent years. In this "hyperbolic" framework, the appearance of shocks prevents the solutions to be smooth enough to be considered in a strong sense as in our present work. Therefore, the study of entropic solutions [START_REF] Feng | Stochastic scalar conservation laws[END_REF] or kinetic solutions [START_REF] Debussche | Scalar conservation laws with stochastic forcing[END_REF] to the SPDE have been the two main approaches, both of which rely on a vanishing viscosity argument: the entropic or kinetic solution is sought as the limit of its viscous approximation as the viscosity coefficient tends to 0.

Let C b (H 2 0 (T)) denote the set of continuous and bounded functions from H 2 0 (T) to R. As a consequence of Theorem 2.4, we can define a family of functionals (P t ) t≥0 on C b (H 2 0 (T)) by writing

P t ϕ(u 0 ) := E u 0 [ϕ(u(t))] , t ≥ 0, u 0 ∈ H 2 0 (T),
where the notation E u 0 indicates that the random variable u(t) is the solution to (2.1) at time t starting from the initial condition u 0 .

Corollary 2.5. Under Assumptions 2.1 and 2.2, the family (P t ) t≥0 is a Feller semigroup and the process (u(t)) t≥0 is a strong Markov process in H 2 0 (T) with semigroup (P t ) t≥0 .

Proof. The uniqueness of a strong solution and the fact that, for all t ≥ 0, the processes (W Q (t + s) -W Q (t)) s≥0 and (W Q (s)) s≥0 have the same distribution, ensure that (P t ) t≥0 is a semigroup, and therefore that (u(t)) t≥0 is a Markov process. The Feller property is a straightforward consequence of the result of continuous dependence on initial conditions given in Theorem 2.4, whereas it is a classical result that the strong Markov property of (u(t)) t≥0 follows from the Feller property of (P t ) t≥0 (see for instance the proof of [START_REF] Breiman | Probability. Addison-Wesley series in statistics[END_REF]Theorem 16.21]).

Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure Let B(H 2 0 (T)) denote the Borel σ-algebra of the metric space H 2 0 (T), and P(H 2 0 (T)) refer to the set of Borel probability measures on H 2 0 (T). The Markov property allows us to extend the notion of strong solution to (2.1) by considering not only a deterministic initial condition but any F 0 -measurable random variable u 0 on H 2 0 (T). In this perspective, we define the dual semigroup (P * t ) t≥0 of (P t ) t≥0 by

P * t α(Γ) := H 2 0 (T) P u 0 (u(t) ∈ Γ) dα(u 0 ), t ≥ 0, α ∈ P H 2 0 (T) , Γ ∈ B H 2 0 (T) .
In particular, P * t α is the law of u(t) when u 0 is distributed according to α.

Definition 2.6 (Invariant measure). We say that a probability measure µ ∈ P(H 2 0 (T)) is an invariant measure for the semigroup (P t ) t≥0 (or equivalently for the process (u(t)) t≥0 ) if and only if

∀t ≥ 0, P * t µ = µ.
Theorem 2.7 (Existence, uniqueness and estimates on the invariant measure). Under Assumptions 2.1 and 2.2, the process (u(t)) t≥0 solution to the SPDE (2.1) admits a unique invariant measure µ.

Besides, if u ∈ H 2 0 (T) is distributed according to µ, then E[ u 2 H 2 0 (T) ] < +∞ and, for all p ∈ [1, +∞), E[ u p L p 0 (T) ] < +∞.
A few similar results exist in the literature. Da Prato, Debussche and Temam [START_REF] Da Prato | Stochastic Burgers' equation[END_REF] have studied the viscous Burgers equation (which corresponds to the flux function A(u) = u 2 /2) perturbed by an additive space-time white noise whereas Da Prato and Gatarek [START_REF] Da | Stochastic Burgers equation with correlated noise[END_REF] studied the same equation but with a multiplicative white noise. Both showed the well-posedness of the equation as well as the existence of an invariant measure. These results are moreover put in a much detailed context in the two reference books [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF][START_REF] Da | Ergodicity for Infinite Dimensional Systems[END_REF]. Boritchev [START_REF] Boritchev | Turbulence de Burgers en 1D : un cas modèle pour la théorie de Kolmogorov[END_REF][START_REF] Boritchev | Generalised Burgers equation with random force and small viscosity[END_REF][START_REF] Boritchev | Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation[END_REF] showed the existence and uniqueness of an invariant measure for the viscous generalised Burgers equation (which corresponds to the case of strictly convex flux function) perturbed by a white-in-time and spatially correlated noise. E, Khanin, Mazel and Sinai [START_REF] Weinan | Invariant Measures for Burgers Equation with Stochastic Forcing[END_REF] showed the existence and uniqueness of an invariant measure for the inviscid Burgers equation with a white-in-time and spatially correlated noise. Debussche and Vovelle [START_REF] Debussche | Invariant measure of scalar first-order conservation laws with stochastic forcing[END_REF] generalised this last result by extending it to non-degenerate flux functions (roughly speaking, there is no non-negligible subset of R on which A is linear). Besides, the fact that these results from [START_REF] Weinan | Invariant Measures for Burgers Equation with Stochastic Forcing[END_REF][START_REF] Debussche | Invariant measure of scalar first-order conservation laws with stochastic forcing[END_REF] also hold when ν = 0 makes them quite powerful: it shows indeed that the presence of a viscous term is not a necessary condition for the solution to be stationary.

The stochastic Burgers equation is mainly studied as a one-dimensional model for turbulence. By showing a stable behaviour at large times, this model manages, to some extent, to fit the predicitions of Kolmogorov's "K41" theory about the universal properties of a turbulent flow [START_REF] Andreï | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF][START_REF] Andreï | On degeneration of isotropic turbulence in an incompressible viscous liquid[END_REF]. Whether it is modelled by the Burgers equation or a by more general process such as Equation (2.1), turbulence is then described through the statistics of some particular small-scale quantities in the stationary state [START_REF] Weinan | Stochastic hydrodynamics[END_REF][START_REF] Weinan | Stochastic PDEs in turbulence theory[END_REF]. Sharp estimates were given by Boritchev for these small-scale quantities [START_REF] Boritchev | Generalised Burgers equation with random force and small viscosity[END_REF], which were furthermore shown to be independent of the viscosity coefficient. One of the purposes of this chapter is to lay the groundwork for the numerical analysis of Equation (2.1). In Chapter 3, we introduce a finite-volume approximation of (2.1) which allows to approximate the invariant measure µ. Generating random variables with distribution µ shall eventually lead us to compute said small-scale quantities and analyse the development of turbulence in the model established by Equation (2.1).

Outline of the chapter

The proofs of Theorems 2.4 and 2.7 are respectively detailed in Sections 2.2 and 2.3.

Well-posedness and regularity

29

Well-posedness and regularity

This section is dedicated to the proof of Theorem 2.4. This proof is decomposed as follows. In Subsection 2.2.1, we introduce a weaker formulation of Equation (2.1), the so-called mild formulation. In Subsection 2.2.2, we show that Equation (2.1) is well-posed locally in time both in the mild and in the strong sense. In Subsection 2.2.3, we give higher bounds for the Lebesgue and Sobolev norms of this local solution. Eventually, these estimates allow us to extend the local solution to a global-in-time solution, and thus to prove Theorem 2.4 in Subsection 2.2.4.

Mild formulation of (2.1)

In this subsection, we collect preliminary results which shall enable us to provide a mild formulation of Equation (2.1), for which we prove the existence and uniqueness of a solution on a small interval. The proofs of several results are postponed to Subsection 2.2.5.

Fractional Sobolev spaces

For all m ≥ 1, let us define λ 2m -1 = λ 2m = -(2πm ) 2 , and e 2m -1 (x) = √ 2 sin(2πm x), e 2m (x) = √ 2 cos(2πm x). The family (e m ) m≥1 is a complete orthogonal basis of L 2 0 (T) such that, for all m ≥ 1, e m is C ∞ on T and ∂ xx e m = λ m e m .
With respect to this basis, we define the fractional Sobolev space

H s 0 (T), for any s ∈ [0, +∞), as the space of functions v ∈ L 2 0 (T) such that v H s 0 (T) :=   m≥1 (-λ m ) s v, e m 2 L 2 0 (T)   1/2 < +∞. (2.8)
We take from [15, Appendice A] the following proposition and adapt it to our case of a flux function satisfying Assumption 2.1:

Proposition 2.8. Under Assumption 2.1, for any s ∈ [1, 2], the mapping v ∈ H s 0 (T) -→ ∂ x A(v) ∈ H s-1 0 (T)
is bounded on bounded subsets of H s 0 (T). Moreover, when s = 1 or s = 2, it is Lipschitz continuous on bounded subsets of H s 0 (T).

The proof of Proposition 2.8 is postponed to Subsection 2.2.5. By virtue of Proposition 2.8, for all m ≥ 1, we denote by

C (m) 2
and C

(m) 3

two finite constants such that:

• for all v ∈ H 1 0 (T) such that v H 1 0 (T) ≤ m, ∂ x A(v) L 2 0 (T) ≤ C (m) 2 ; • for all v 1 , v 2 ∈ H 1 0 (T) such that v 1 H 1 0 (T) ∨ v 2 H 1 0 (T) ≤ m, ∂ x A(v 1 )-∂ x A(v 2 ) L 2 0 (T) ≤ C (m) 3 v 1 - v 2 H 1 0 (T) .

Heat kernel

Let us denote by (S t ) t≥0 the semigroup generated by the operator ν∂ xx :

S t v := m≥1 e νλmt v, e m L 2 0 (T) e m , v ∈ L 2 0 (T), t ≥ 0.
(2.9) Some of its properties are gathered in the following proposition.

Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure Proposition 2.9 (Properties of the heat kernel). The semigroup (S t ) t≥0 satisfies the following properties.

1. For any s ≥ 0, for any v ∈ H s 0 (T), for any t ≥ 0, S t v ∈ H s 0 (T) and

S t v H s 0 (T) ≤ v H s 0 (T) ; besides, the mapping t → S t v ∈ H s 0 (T) is continuous on [0, +∞).
2. For all 0 ≤ s 1 ≤ s 2 , there exists a constant

C 4 = C 4 (s 1 , s 2 ) > 0 such that ∀v ∈ H s 1 0 (T), ∀t ≥ 0, S t v H s 2 0 (T) ≤ C 4 t s 1 -s 2 2 v H s 1 0 (T) . 3. For any s ∈ [0, +∞), T > 0 and (v(t)) t∈[0,T ] ∈ C([0, T ], H s 0 (T)), the process ( t 0 S t-r v(r)dr) t∈[0,T ] belongs to C([0, T ], H s+3/2 0 (T)).
The proof of Proposition 2.9 is postponed to Subsection 2.2.5.

Stochastic convolution and mild formulation of (2.1)

Let (F t ) t≥0 be a normal filtration on the probability space (Ω, F, P) and (W

Q (t)) t≥0 be a Q-Wiener
process in H 2 0 (T) with respect to this filtration. Given that the orthonormal basis (e m ) m≥1 of the space L 2 0 (T) satisfies ∂ xx e m = λ m e m , the family (e m /λ m ) m≥1 is an orthonormal basis of H 2 0 (T). We set

W m (t) := W Q (t), e m λ m H 2 0 (T) , m ≥ 1, t ≥ 0, so that by (2.6), (W m (t)) t≥0 is a real-valued Brownian motion with variance k≥1 g k , e m /λ m 2 H 2 0 (T) . Next, we write w m (t) := t 0 e νλm(t-s) dW m (s), m ≥ 1, t ≥ 0.
Proposition 2.10. Under Assumption 2.2, for all T > 0, the series

m≥1 e m λ m (w m (t)) t∈[0,T ] converges in L 2 (Ω, C([0, T ], H 2 0 (T)))
, and its sum defines an (F t ) t≥0 -adapted, H 2 0 (T)-valued process (w(t)) t≥0 almost surely continuous.

The proof of Proposition 2.10 is postponed to Subsection 2.2.5. The process (w(t)) t≥0 is called the stochastic convolution associated to the Q-Wiener process (W

Q (t)) t≥0 .
In the sequel, we let τ be a (F t ) t≥0 -stopping time, almost surely finite. We shall say that a process

(u(t)) t∈[0,τ ] is (F t ) t≥0 -adapted if for all t ≥ 0, the random variable u(t)1 t≤τ is F t -measurable.
Definition 2.11 (Local mild solution). Let u 0 be an F 0 -measurable, H 1 0 (T)-valued random variable. Under Assumptions 2.1 and 2.2, a (local) mild solution to the SPDE

du(t) = -∂ x A(u(t))dt + ν∂ xx u(t)dt + dW Q (t) (2.10) on [0, τ ] is an H 1 0 (T)-valued, (F t ) t≥0 -adapted process (u(t)) t∈[0,τ ] such that, almost surely: 1. the mapping t → u(t) ∈ H 1 0 (T) is continuous on [0, τ ];
2. for all t ∈ [0, τ ],

u(t) = S t u 0 - t 0 S t-s ∂ x A(u(s))ds + w(t). (2.11)
The combination of Propositions 2.8 and 2.9 ensures that all terms of the identity (2.11) are welldefined.

We now clarify the relationship between the notions of mild and strong solutions.

Proposition 2.12 (Mild and strong solutions). Under the assumptions of Definition 2.11, let (u(t)) t∈[0,τ ] be a mild solution to (2.10) on [0, τ ]. If u 0 ∈ H 2 0 (T), then:

1. for all t ∈ [0, τ ], u(t) ∈ H 2 0 (T) and the mapping t → u(t) ∈ H 2 0 (T) is continuous on [0, τ ];
2. for all t ∈ [0, τ ],

u(t) = u 0 + t 0 (-∂ x A (u(s)) + ν∂ xx u(s)) ds + W Q (t).
Conversely, any H 2 0 (T)-valued, (F t ) t≥0 -adapted process (u(t)) t∈[0,τ ] satisfying these two conditions almost surely is a mild solution to (2.10) on [0, τ ].

The proof of Proposition 2.12 is postponed to Subsection 2.2.5.

Existence and uniqueness of a mild solution on a small interval

For any integer m 0 ≥ 0, let us define

τ m 0 W Q = 1 8 C 1 C (m 0 +1) 3 2 ∧ inf t ≥ 0 : 2C 4 C (m 0 +1) 2 √ t + w(t) H 1 0 (T) ≥ 1 ,
where we recall that the constant C 4 is defined in Proposition 2.9, the constants

C (m) 2
and C

(m) 3 are defined after Proposition 2.8, and the constant C 1 is defined in (2.2).

Notice that τ m 0 (W Q ) ∈ (0, +∞), almost surely.

In the spirit of [START_REF] Da Prato | Stochastic Burgers' equation[END_REF][START_REF] Boritchev | Generalised Burgers equation with random force and small viscosity[END_REF], we obtain the existence and uniqueness of a mild solution to (2.10) on the "small" interval [0, τ m 0 (W Q )] by a fixed-point argument.

Lemma 2.13 (Local existence and uniqueness). Let u 0 and m 0 be two F 0 -measurable random variables taking values respectively in H 1 0 (T) and N such that u 0 H 1 0 (T) ≤ m 0 . Furthermore, let us set τ := τ m 0 (W Q ). Then, under Assumptions 2.1 and 2.2, there is a unique mild solution (u(t)) t∈[0,τ ] to (2.10)

on [0, τ ].
Proof. Let us introduce the random set

Σ := (v(t)) t∈[0,τ ] ∈ C [0, τ ], H 1 0 (T) : ∀t ∈ [0, τ ], v(t) H 1 0 (T) ≤ m 0 + 1 .
Thanks to Propositions 2.9 and 2.10, we may define the random operator

G : C([0, τ ], H 1 0 (T)) → C([0, τ ], H 1 0 (T)) by (Gv)(t) = S t u 0 - t 0 S t-s ∂ x A(v(s))ds + w(t), t ∈ [0, τ ],
and notice that any v ∈ C([0, τ ], H 1 0 (T)) satisfies Equation (2.11) if and only if Gv = v. We first write, for some v ∈ C([0, τ ], H 1 0 (T)) and for any t

∈ [0, τ ], (Gv)(t) H 1 0 (T) ≤ S t u 0 H 1 0 (T) + t 0 S t-s ∂ x A(v(s)) H 1 0 (T) ds + w(t) H 1 0 (T) .
(2.12)

Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure

On the one hand, by the first assertion of Proposition 2.9, S t u 0 H 1 0 (T) ≤ u 0 H 1 0 (T) ≤ m 0 ; on the other hand, we know thanks to the second assertion of Proposition 2.9 that

S t-s ∂ x A(v(s)) H 1 0 (T) ≤ C 4 √ t -s ∂ x A(v(s)) L 2 0 (T) , (2.13) furthermore, thanks to Proposition 2.8, if v ∈ Σ, then ∂ x A(v) is bounded in L 2 0 (T) uniformly in time, i.e. for all s ∈ [0, τ ], ∂ x A(v(s)) L 2 0 (T) ≤ C (m 0 +1) 2
. Thus,

(Gv)(t) H 1 0 (T) ≤ m 0 + 2C 4 C (m 0 +1) 2 √ t + w(t) H 1 0 (T) , t ∈ [0, τ ]. (2.14)
By definition of τ , it follows that Gv ∈ Σ whenever v ∈ Σ.

We now take (v

1 (t)) t∈[0,τ ] , (v 2 (t)) t∈[0,τ ] ∈ Σ. Then, for any t ∈ [0, τ ], (Gv 1 )(t) -(Gv 2 )(t) H 1 0 (T) = t 0 S t-s (∂ x A(v 1 (s)) -∂ x A(v 2 (s))) ds H 1 0 (T) ≤ t 0 C 4 √ t -s ∂ x A(v 1 (s)) -∂ x A(v 2 (s)) L 2 0 (T) ds, (2.15)
where we have used the same arguments as above. Using now the Lipschitz continuity result in Proposition 2.8 and the definition of τ , we get for all t ∈ [0, τ ],

(Gv 1 )(t) -(Gv 2 )(t) H 1 0 (T) ≤ 2C 1 C (m 0 +1) 3 √ t sup s∈[0,t] v 1 (s) -v 2 (s) H 1 0 (T) ≤ 1 2 sup s∈[0,τ ] v 1 (s) -v 2 (s) H 1 0 (T) ,
meaning that G is a contraction mapping on Σ, which is complete. Then, by the Banach fixed-point theorem, G admits a unique fixed point (u(t)) t∈[0,τ ] in Σ. To show that this solution to Equation (2.11) is unique among all the H 1 0 (T)-valued continuous processes, let us first notice that our choice of τ implies ∀t < τ , u(t) H 1 0 (T) < m 0 + 1. Assume that there is another solution ( u(t)) t∈[0,τ ] of (2.11) not belonging almost surely to Σ. Then we have with positive probability

∃ τ < τ , u( τ ) H 1 0 (T) ≥ m 0 + 1. This means that the double inequality u ( τ ) H 1 0 (T) < m 0 + 1 ≤ u ( τ ) H 1 0 (T)
holds on some nonnegligible event. On this event, the fixed-point argument also holds in the set

Σ := (v(t)) t∈[0, τ ] : ∀t ∈ [0, τ ], v(t) H 1 0 (T) ≤ m 0 + 1 which is formally a subset of Σ. Thus, by uniqueness of the fixed point, we have u |[0, τ ] = u |[0, τ ] and in particular u ( τ ) = u ( τ ), which is absurd. As a consequence, (u(t)) t∈[0,τ ] is the only H 1 0 (T)-valued process with continuous trajectories satisfying Equation (2.11) on [0, τ ].
Finally, let v (0) = 0 and define the sequence of processes v 1) . It is clear from the definition of the operator G and from Proposition 2.10 that each process (v (j) (t)1 t≤τ ) t≥0 is (F t ) t≥0 -adapted. On the other hand, the Banach fixed-point theorem asserts that almost surely, the sequence

(j) ∈ C([0, τ ], H 1 0 (T)), j ≥ 1 by v (j) = Gv (j-
(v (j) (t)) t∈[0,τ ] converges to (u(t)) t∈[0,τ ] in C([0, τ ], H 1 0 (T)
). As a consequence, for any t ≥ 0, the sequence of F t -measurable random variables 1 t≤τ v (j) (t) converges almost surely to 1 t≤τ u(t), which makes this limit also F t -measurable. Thus, the process (1 t≤τ u(t)) t≥0 is (F t ) t≥0 -adapted.

Construction of a maximal solution to (2.1)

In this subsection, we use the notions introduced in Subsection 2.2.1 to prove the following existence and uniqueness result for (2.1).

Lemma 2.14 (Existence and uniqueness result of a maximal solution to (2.1)). Under Assumptions 2.1 and 2.2, for any u 0 ∈ H 1 0 (T), there exists a pair (T * , (u(t)) t∈[0,T * ) ) such that: 1. for any (F t ) t≥0 -stopping time T such that almost surely, T < +∞ and T ≤ T * , (u(t)) t∈[0,T ] is the unique mild solution to (2.1) on [0, T ];

2. almost surely, T * = +∞ or lim sup t→T * u(t) H 1 0 (T) = +∞. The random time T * is called the explosion time and the process (u(t)) t∈[0,T * ) is called the maximal solution to (2.1).

Proof. Let u 0 ∈ H 1 0 (T). Let m (0) 0 = u 0 H 1 0 (T)
. By Lemma 2.13, Equation (2.1) possesses a unique mild solution (u(t)) t∈[0,τ (0) ] on [0, τ (0) ], where τ (0) = τ m (0) 0 . We now define the filtration (F

(1) t ) t≥0 by F (1) t = F τ (0) +t = B ∈ F : ∀s ≥ 0, B ∩ {τ (0) + t ≤ s} ∈ F s ,
and recall that the process W Q, (1)

defined by W Q,(1) (t) = W Q (τ (0) + t) -W Q (t) is a Q-Wiener process with respect to (F (1)
t ) t≥0 . Therefore, applying Lemma 2.13 again with this Q-Wiener process, and initial condition u (1) on [0, τ (1) ], where τ (1) = τ m (1) 0 (W Q, (1) ). It is then easily checked that defining T (1) = τ (0) + τ (1) and u(t + τ (0) ) = u (1) (t) for any t ∈ (0, τ (1) ], we obtain a unique mild solution (u(t)) t∈[0,T (1) ] to Equation (2.1) on [0, T (1) ].

(1) 0 = u(τ (0) ) and m (1) 0 = u(τ (0) ) H 1 0 (T) ∨ m (0) 0 , we obtain a mild solution (u (1) (t)) t∈[0,τ (1) ] of du = -∂ x A(u)dt + ν∂ xx udt + dW Q,
We now proceed by induction and set for all n ≥ 1,

T (n) := n i=0 τ (i) , m (n+1) 0 := u T (n) H 1 0 (T) ∨ m (n) 0 , τ (n+1) := τ m (n+1) 0 W Q T (n) + • -W Q T (n) , T * := sup n≥1 T (n) ,
where at each iteration we use Lemma 2.13 to extend the process (u(t)) t∈[0,T (n) ] to the unique mild

solution of Equation (2.1) on [0, T (n) ]. It is then clear that (u(t)) t∈[0,T * )
satisfies the first assertion of Lemma 2.14.

Since the sequence of integers (m

(n) 0 ) n≥0 is nondecreasing, sup n≥0 m (n)
0 < +∞ if and only if there exists n 0 ≥ 0 and m ≥ 0 such that, for all n ≥ n 0 , m (n) 0 = m. Hence, we can write

T * < +∞, sup n≥0 m (n) 0 < +∞ = n 0 ≥0,m≥0 ∞ n=0 τ (n) < +∞, ∀n ≥ n 0 , m (n) 0 = m = n 0 ≥0,m≥0 ∞ n=n 0 +1 τ (n) < +∞, ∀n ≥ n 0 , m (n) 0 = m ⊂ n 0 ≥0,m≥0 ∞ n=n 0 +1 τ m W Q T (n) + • -W Q T (n) < +∞ .
Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure However, by the strong Markov property, for any m ≥ 0, the random variables

τ m (W Q (T (n) + •) - W Q (T (n)
)), n ≥ 1, are independent and identically distributed, and by the definition of τ m (•), they are almost surely positive. As a consequence, by Borel's 0-1 law,

∀n 0 , m ≥ 0, P ∞ n=n 0 +1 τ m W Q T (n) + • -W Q T (n) < +∞ = 0.
As the countable union of negligible events is still negligible, we get

P T * < +∞, sup n≥0 m (n) 0 < +∞ = 0.
This implies that almost surely, if

T * < +∞ then sup n≥0 m (n) 0 = +∞, so that lim sup n→∞ u(T (n) ) H 1 0 (T)
= +∞, which is the wanted result.

Estimates on the maximal solution

Let u 0 ∈ H 2 0 (T). Let (T * , (u(t)) t∈[0,T * )
) be the maximal solution to Equation (2.1) given by Lemma 2.14. By Proposition 2.12, (u(t)) t∈[0,T * ) is a continuous H 2 0 (T)-valued process. Besides, Lemma 2.14 allows us to define, for any r ≥ 0, the stopping time

T r := inf t ∈ [0, T * ) : u(t) 2 H 1 0 (T) ≥ r , (2.16)
which always satisfies T r ≤ T * . In the sequel, we shall prove that lim r→∞ T r = +∞, which shall imply that T * = +∞, almost surely.

Lemma 2.15. Under Assumptions 2.1 and 2.2, for any p ∈ 2N * and for all t ≥ 0, we have:

4ν p (p -1)E t∧Tr 0 T ∂ x u(s) p/2 2 dxds ≤ u 0 p L p 0 (T) + p(p -1) 2 D 0 E t∧Tr 0 u(s) p-2 L p-2 0 (T)
ds .

(2.17) Moreover, there exist two constants C (2.18)

Proof. Let p ∈ 2N * . We want to apply Itô's formula on [0, t ∧ T r ] to the H 2 0 (T)-valued process (u(t)) t∈[0,T * ) with the function F p : u → u p L p 0 (T) . Since this process writes

u(t) = u 0 + t 0 ϕ(s)ds + W Q (t) with ϕ(t) = -∂ x A(u(t)) + ν∂ xx u(t) ∈ L 2 0 (T)
, the standard formulation of Itô's formula in Hilbert spaces [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF]Theorem 4.32] requires at least F p to be continuous on L 2 0 (T), which is not the case for p > 2 here. Hence, we shall proceed to approximate F p with a sequence of smooth functions F M,p , M ≥ 1, apply Itô's formula to the functions F M,p and then take the limit M → +∞.

Step 1. Approximation of the L p 0 (T)-norm. Let ρ be a C ∞ function from R to R + such that R ρ(u)du = 1 and whose support is contained in the interval (-1 2 , 1 2 ). For any M ≥ 1, we set the regularised Heaviside function ψ M := 1 (-∞,M+ 1 2 ] * ρ and its antiderivative

φ M : u ∈ R + -→ u 0 ψ M (v)dv ∈ R + .
2.2. Well-posedness and regularity

35

We now define a truncated L p 0 (T)-norm by setting

F M,p : L 2 0 (T) -→ R + v -→ T φ M (v(x) p ) dx.
The first differential DF M,p and the second differential D 2 F M,p have the following expressions:

∀v, h ∈ L 2 0 (T), DF M,p (v), h L 2 0 (T) = p T h(x)v(x) p-1 φ M (v(x) p ) dx, D 2 F M,p (v) • h, h L 2 0 (T) = p(p -1) T h(x) 2 v(x) p-2 φ M (v(x) p ) dx + p 2 T h(x) 2 v(x) 2(p-1) φ M (v(x) p ) dx.
Step 2. Itô's formula. First, let us notice that the process (W Q (t)) t≥0 can be seen as an L 2 0 (T)-valued Q -Wiener process where the operator

Q : L 2 0 (T) → L 2 0 (T) has covariance u, Q v L 2 0 (T) = k≥1 g k , u L 2 0 (T) g k , v L 2 0 (T) . Indeed, Assumption 2.2 ensures that Q (L 2 0 (T)) ⊂ H 2 0 (T) and Q |H 2 0 (T) = Q. We now have Tr D 2 F M,p (v)Q = k≥1 D 2 F M,p (v)g k , g k L 2 0 (T) ,
so that we can apply Itô's formula [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF]Theorem 4.32] for the real-valued process (F M,p (u(t))) t∈[0,T * ) , which leads to

F M,p (u(t)) =F M,p (u 0 ) + p t 0 T (-∂ x A(u(s)) + ν∂ xx u(s)) u(s) p-1 φ M (u(s) p ) dxds + t 0 DF M,p (u(s)), dW Q (s) L 2 0 (T) + 1 2 p(p -1) k≥1 t 0 T g 2 k u(s) p-2 φ M (u(s) p )dxds + 1 2 p 2 k≥1 t 0 T g 2 k u(s) 2(p-1) φ M (u(s) p )dxds.
Since the L 2 0 (T)-norm of DF M,p (u(s)) is bounded uniformly in time, the third term of the right-hand side is a square integrable martingale [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF]Theorem 4.27]. Thus, for t ≥ 0, integrating in time up to t ∧ T r and taking the expectation, we get

E [F M,p (u(t ∧ T r ))] =F M,p (u 0 ) -pE t∧Tr 0 T ∂ x A(u(s))u(s) p-1 φ M (u(s) p ) dxds (2.19) + pE t∧Tr 0 T ν∂ xx u(s)u(s) p-1 φ M (u(s) p ) dxds (2.20) + 1 2 p(p -1)E   k≥1 t∧Tr 0 T g 2 k u(s) p-2 φ M (u(s) p )dxds   (2.21) + 1 2 p 2 E   k≥1 t∧Tr 0 T g 2 k u(s) 2(p-1) φ M (u(s) p )dxds   . (2.22)
Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure

Step 3. Passing M → +∞. We want now to pass to the limit M → +∞. Regarding the left-hand side in the above equation, the family of functions φ M is non-decreasing with respect to M , so that the monotone convergence theorem yields

lim M →∞ E [F M,p (u(t ∧ T r ))] = E T lim M →∞ φ M (u(t ∧ T r ) p ) dx = E u(t ∧ T r ) p L p 0 (T) .
For the flux term, we have almost surely, for all s ∈ [0, t∧T r ] and for all M ≥ 0,

∂ x A(u(s))u(s) p-1 φ M (u(s) p ) ≤ |∂ x A(u(s))||u(s)| p-1 . Furthermore, E t∧Tr 0 T |∂ x A(u(s))| |u(s)| p-1 dxds ≤ E sup s∈[0,t∧Tr] u(s) p-1 L ∞ 0 (T) t∧Tr 0 T |∂ x A(u(s))| dxds ≤ r p-1 2 E t∧Tr 0 ∂ x A(u(s)) L 2 0 (T) ds (from (2.3) and (2.16)) ≤ r p-1 2 L r E t∧Tr 0 u(s) L 2 0 (T) ds (from Proposition 2.8) ≤ L r r p 2 t < +∞.
Thus, the dominated convergence theorem applies and yields

lim M →∞ pE t∧Tr 0 T ∂ x A(u(s))u(s) p-1 φ M (u(s) p ) dxds = pE t∧Tr 0 T ∂ x A(u(s))u(s) p-1 dxds .
We now integrate by parts the viscous term:

pνE t∧Tr 0 T ∂ xx u(s)u(s) p-1 φ M (u(s) p ) dxds = -pνE t∧Tr 0 T ∂ x u(s) ∂ x u(s) p-1 φ M (u(s) p ) + u(s) p-1 ∂ x φ M (u(s) p ) dxds = -pνE t∧Tr 0 T (∂ x u(s)) 2 (p -1)u(s) p-2 φ M (u(s) p ) + pu(s) 2(p-1) φ M (u(s) p ) dxds ,
and this last integrand is dominated uniformly in M by (∂ x u(s)) 2 (p -1)u(s) p-2 + κpu(s) 2(p-1) , where κ = sup R |ρ|. Furthermore, thanks to (2.16), we have

E t∧Tr 0 T (∂ x u(s)) 2 (p -1)u(s) p-2 + κpu(s) 2(p-1) dxds ≤ E (p -1) sup s∈[0,t∧Tr] u(s) p-2 L ∞ 0 (T) + κp sup s∈[0,t∧Tr] u(s) 2(p-1) L ∞ 0 (T) t∧Tr 0 u(s) 2 H 1 0 (T) ds ≤ (p -1)r p-2 2 + κpr p-1 rt < +∞.
Thus, we get from the dominated convergence theorem,

lim M →∞ pE t∧Tr 0 T ν∂ xx u(s)u(s) p-1 φ M (u(s) p ) dxds = -νp(p-1)E t∧Tr 0 T (∂ x u(s)) 2 u(s) p-2 dxds .
With similar computations, for the noise term, we have lim

M →∞ p(p -1)E   k≥1 t∧Tr 0 T g 2 k u(s) p-2 φ M (u(s) p )dxds   = p(p -1)E   k≥1 t∧Tr 0 T g 2 k u(s) p-2 dxds   , 2
E u(t ∧ T r ) p L p 0 (T) = u 0 p L p 0 (T) -pE t∧Tr 0 T ∂ x A(u(s))u(s) p-1 dxds -νp(p -1)E t∧Tr 0 T (∂ x u(s)) 2 u(s) p-2 dxds + 1 2 p(p -1) k≥1 E t∧Tr 0 T u(s) p-2 g 2 k dxds . (2.23)
It turns out that the flux term disappears:

T u(s) p-1 ∂ x A(u(s))dx = T u(s) p-1 A (u(s))∂ x u(s)dx = T ∂ x (A p (u(s))) dx = 0, (2.24)
where A p is an antiderivative of v → v p-1 A (v). As regards the noise coefficients, we have

k≥1 g k (x) 2 ≤ k≥1 g k 2 L ∞ 0 (T) ≤ k≥1 g k 2 H 1 0 (T) ≤ D 0 ,
thanks to (2.3) and (2.4). As a consequence, we get from (2.23) the inequality

νp(p -1)E t∧Tr 0 T u(s) p 2 -1 ∂ x u(s) 2 dxds ≤ u 0 p L p 0 (T) + 1 2 p(p -1)D 0 E t∧Tr 0 u(s) p-2 L p-2 0 (T)
ds .

(2.25) Rewriting the integrand in the left-hand side, we get

4ν p (p -1)E t∧Tr 0 T ∂ x u(s) p/2 2 dxds ≤ u 0 p L p 0 (T) + p(p -1) 2 D 0 E t∧Tr 0 u(s) p-2 L p-2 0 (T)
ds .

(2.26) Since u(s) has a zero space average and is continuous in space (because it belongs to H 1 0 (T)), almost surely the function u(s) p/2 vanishes somewhere on the torus. Thus, we can apply the Poincaré inequality on the left-hand side which leads, after multiplying by p/(4ν(p -1)) on both sides, to the inequality

E t∧Tr 0 u(s) p L p 0 (T) ds ≤ p 4ν(p -1) u 0 p L p 0 (T) + p 2 D 0 8ν E t∧Tr 0 u(s) p-2 L p-2 0 (T)
ds .

(2.27)

For p = 2, we get

E t∧Tr 0 u(s) 2 L 2 0 (T) ds ≤ 1 2ν u 0 2 L 2 0 (T) + D 0 t 2ν ,
and the claimed result for arbitrary p ∈ 2N * follows by induction and from the inequalities u 0 [START_REF] Boritchev | Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation[END_REF]. By Jensen's inequality, the bound (2.18) also holds for any real number p ≥ 2.

p-2r L p-2r 0 (T) ≤ 1 + u 0 p L p 0 (T) and E[t ∧ T r ] ≤ t. Remark 2.
Lemma 2.17. Under Assumptions 2.1 and 2.2, there exist two constants C 7 , C 8 > 0 depending only on ν, p A , C 1 and D 0 , such that for all t ≥ 0 and all r ≥ 0,

E u(t ∧ T r ) 2 H 1 0 (T) + νE t∧Tr 0 u(s) 2 H 2 0 (T) ds ≤ u 0 2 H 1 0 (T) + C 7 1 + u 0 2p A +2 L 2p A +2 0 (T) + C 8 t.
Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure Proof. We want to apply Itô's formula to the squared H 1 0 (T)-norm of the process (u(t)) t∈[0,T * ) . As for the proof of Lemma 2.15, we proceed by truncation of this function.

Step 1. Approximation of the H 1 0 (T)-norm. We set

G M : L 2 0 (T) -→ R + v -→ M m=1 (-λ m ) v, e m 2 L 2 0 (T)
The first differential DG M and the second differential D 2 G M have the following expressions:

∀h ∈ L 2 0 (T), DG M (v), h L 2 0 (T) = -2 M m=1 λ m v, e m L 2 0 (T) h, e m L 2 0 (T) , D 2 G M (v) • h, h L 2 0 (T) = -2 M m=1 λ m h, e m 2 L 2 0 (T) .
Step 2. Itô's formula. Itô's formula applied to G M yields almost surely and for all r ≥ 0,

G M (u(t ∧ T r )) = G M (u 0 ) -2 t∧Tr 0 M m=1 λ m u(s), e m L 2 0 (T) -∂ x A(u(s)) + ν∂ xx u(s), e m L 2 0 (T) ds -2 t∧Tr 0 DG M (u(s)), dW Q (s) L 2 0 (T) -2 k≥1 t∧Tr 0 M m=1 λ m g k , e m 2 L 2 0 (T) ds. (2.28)
We first check that the third term of the right-hand side is a square-integrable martingale:

E t∧Tr 0 DG M (u(s)) 2 L 2 0 (T) ds = 4 M m=1 λ 2 m E t∧Tr 0 u(s), e m 2 L 2 0 (T) ds ≤ 4 M m=1 λ 2 m E t∧Tr 0 u(s) 2 L 2 0 (T) ds ≤ 4 M m=1 λ 2 m tr < +∞.
Thus, taking the expectation, the stochastic integral disappears and we get

E [G M (u(t ∧ T r ))] = G M (u 0 ) + 2E t∧Tr 0 M m=1 λ m u(s), e m L 2 0 (T) ∂ x A(u(s)), e m L 2 0 (T) ds -2E t∧Tr 0 M m=1 λ m u(s), e m L 2 0 (T) ν∂ xx u(s), e m L 2 0 (T) ds -E   k≥1 t∧Tr 0 M m=1 λ m g k , e m 2 L 2 0 (T) ds   .
(2.29)

On one hand, we can rewrite the viscous term as follows:

M m=1 λ m u(s), e m L 2 0 (T) ν∂ xx u(s), e m L 2 0 (T) = M m=1 λ m u(s), e m L 2 0 (T) νu(s), ∂ xx e m L 2 0 (T) = M m=1 λ m u(s), e m L 2 0 (T) νu(s), λ m e m L 2 0 (T) = ν M m=1 λ 2 m u(s), e m 2 L 2 0 (T) .
(2.30)

Well-posedness and regularity

39

On the other hand, applying Young's inequality on the flux term, we get

2E t∧Tr 0 M m=1 λ m u(s), e m L 2 0 (T) ∂ x A(u(s)), e m L 2 0 (T) ds ≤ 2νE t∧Tr 0 M m=1 λ 2 m u(s), e m 2 L 2 0 (T) ds + 1 2ν E t∧Tr 0 M m=1 ∂ x A(u(s)),
E [G M (u(t ∧ T r ))] ≤ G M (u 0 )+ 1 2ν E t∧Tr 0 M m=1 ∂ x A(u(s)), e m 2 L 2 0 (T) ds -E[t∧T r ] k≥1 M m=1 λ m g k , e m 2 L 2 0 (T) . (2.32)
Step 3. Passing M → +∞. From Proposition 2.8, for any r ≥ 0, there is a constant L r such that for all M ≥ 1, we have

M m=1 ∂ x A(u(s)), e m 2 L 2 0 (T) ≤ ∂ x A(u(s)) 2 L 2 0 (T) ≤ L r u(s) 2 H 1 0 (T) ≤ rL r .
Thus, we can use the dominated convergence theorem to let M go to infinity in (2.32) and we get

E u(t ∧ T r ) 2 H 1 0 (T) ≤ u 0 2 H 1 0 (T) + 1 2ν E t∧Tr 0 ∂ x A(u(s)) 2 L 2 0 (T) ds + E[t ∧ T r ] k≥1 g k 2 H 1 0 (T) . (2.33)
Since from Assumption 2.1, A has polynomial growth, we can bound the second term of the right-hand side: using (2.2) and (2.17) with p = 2 and p = 2p A + 2, we get

E t∧Tr 0 ∂ x A(u(s)) 2 L 2 0 (T) ds = E t∧Tr 0 T (∂ x u(s)) 2 A (u(s)) 2 dxds ≤ 2C 2 1 E t∧Tr 0 T (∂ x u(s)) 2 1 + |u(s)| 2p A dxds = 2C 2 1 E t∧Tr 0 u(s) 2 H 1 0 (T) ds + E t∧Tr 0 T (∂ x u(s)) 2 u(s) 2p A dxds ≤ C 2 1 ν u 0 2 L 2 0 (T) + D 0 E[t ∧ T r ] + 2 (2p A + 2)(2p A + 1) u 0 2p A +2 L 2p A +2 0 (T) + D 0 E t∧Tr 0 u(s) 2p A L 2p A 0 (T)
ds .

Applying now Lemma 2.15, we get

E t∧Tr 0 ∂ x A(u(s)) 2 L 2 0 (T) ds ≤ C 2 1 ν 2 1 + u 0 2p A +2 L 2p A +2 0 (T) + D 0 t + D 0 C (2p A ) 5 1 + u 0 2p A L 2p A 0 (T) + C (2p A ) 6 t .
Injecting this last bound in (2.33), we get the wanted result.

P (T r < t) ≤ 1 r E u(t ∧ T r ) 2 H 1 0 (T) .
We apply now Lemma 2.17 to get

P (T r < t) ≤ 1 r u 0 2 H 1 0 (T) + C 7 1 + u 0 2p A +2 L 2p A +2 0 (T) + C 8 t -→ r→∞ 0.
Since t has been chosen arbitrarily, it follows that almost surely, T r tends to +∞ as r → +∞. Then, since T r ≤ T * , we have T * = +∞ almost surely.

(j) 0 ) j≥1 is a sequence of H 2 0 (T) satisfying lim j→∞ u 0 -u (j) 0 H 2 0 (T) = 0,
then, denoting by (u (j) (t)) t≥0,j≥1 the family of associated solutions, for any T ≥ 0, we have almost surely lim

j→∞ sup t∈[0,T ] u(t) -u (j) (t) H 2 0 (T) = 0.
Proof. Let us fix a time horizon T > 0. Subtracting the mild formulations of (u(t)) t≥0 and (u (j) (t)) t≥0 given by Proposition 2.12 and taking the H 2 0 (T)-norm, we get by the triangle inequality and Proposition 2.9, for all t ∈ [0, T ],

u(t) -u (j) (t) H 2 0 (T) ≤ S t u 0 -u (j) 0 H 2 0 (T) + t 0 S t-s ∂ x A(u(s)) -A u (j) (s) H 2 0 (T) ≤ u 0 -u (j) 0 H 2 0 (T) + t 0 C 4 √ t -s ∂ x A(u(s)) -∂ x A u (j) (s) H 1 0 (T) ds. (2.34)
Now, for any M > 0, we define the stopping times

τ M := inf t ≥ 0 : u(t) H 2 0 (T) ≥ M , τ (j)
M := inf t ≥ 0 : u (j) (t) H 2 0 (T) ≥ M , j ∈ N, and we denote by L M , according to Proposition 2.8, the Lipschitz constant of the mapping v ∈ H 2 0 (T) → ∂ x A(v) ∈ H 1 0 (T) over the centered ball in H 2 0 (T) of radius M . For an arbitrarily fixed t ∈ [0, T ], the inequality (2.34) implies

u t ∧ τ M ∧ τ (j) M -u (j) t ∧ τ M ∧ τ (j) M H 2 0 (T) ≤ u 0 -u (j) 0 H 2 0 (T) + t∧τ M ∧τ (j) M 0 C 4 L M t ∧ τ M ∧ τ (j) M -s u(s) -u (j) (s) H 2 0 (T)
ds.

In the next step, we iterate this last inequality and apply the Fubini theorem on the double time integral:

u t ∧ τ M ∧ τ (j) M -u (j) t ∧ τ M ∧ τ (j) M H 2 0 (T) ≤ u 0 -u (j) 0 H 2 0 (T) 1 + 2 t ∧ τ M ∧ τ (j) M C 4 L M + C 2 4 L 2 M t∧τ M ∧τ (j) M 0 s 0 1 (t ∧ τ M ∧ τ (j) M -s)(s -r) u(r) -u (j) (r) H 2 0 (T) drds ≤ u 0 -u (j) 0 H 2 0 (T) 1 + 2 √ T C 4 L M + C 2 4 L 2 M t∧τ M ∧τ (j) M 0   t∧τ M ∧τ (j) M r 1 (t ∧ τ M ∧ τ (j) M -s)(s -r) ds   u(r) -u (j) (r) H 2 0 (T)
dr.

However, by a change of variable, we have

t∧τ M ∧τ (j) M s 1 (t ∧ τ M ∧ τ (j) M -r)(r -s) dr = 1 -1 1 1 -y 2 dy = π.
Hence, Grönwall's lemma yields the following control

u t ∧ τ M ∧ τ (j) M -u (j) t ∧ τ M ∧ τ (j) M H 2 0 (T) ≤ u 0 -u (j) 0 H 2 0 (T) 1 + 2 √ T C 4 L M e C 2 4 L 2 M πt∧τ M ∧τ (j) M .
It follows from this inequality that lim inf j→∞ τ (j)

M ≥ τ M ∧ T . Indeed, assuming the opposite, we would have (along a subsequence)

u τ (j) M -u (j) τ (j) M H 2 0 (T) ≤ u 0 -u (j) 0 H 2 0 (T) 1 + 2 √ T C 4 L M e C 2 4 L 2 M T -→ j→∞ 0,
which would imply

M ≤ lim j→∞ u (j) τ (j) M H 2 0 (T) = lim j→∞ u τ (j) M H 2 0 (T) < M.
Hence, necessarily, beyond a certain rank j, we have

u(t ∧ τ M ) -u (j) (t ∧ τ M ) H 2 0 (T) ≤ u 0 -u (j) 0 H 2 0 (T) 1 + 2 √ T C 4 L M e C 2 4 L 2 M t∧τ M .
Since the solutions of (2.7) do not explode, the stopping time τ M tends almost surely to +∞ as M tends to +∞. As a consequence, there exists M T > 0 such that T < τ M T almost surely, so that for all t ∈ [0, T ], u(t) -u (j) (t)

H 2 0 (T) ≤ u 0 -u (j) 0 H 2 0 (T) 1 + 2 √ T C 4 L M T e C 2 4 L 2 M T T .
Hence the result.

Proofs of preliminary results

In this subsection, we detail the proofs of the preliminary results from Subsection 2.2.1, namely Propositions 2.8, 2.9, 2.10 and 2.12.

Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure Proof of Proposition 2.8. We shall prove only the second claim when s = 2. For the first claim, we refer the reader to [15, Lemma A.0.5]. The second claim with s = 1 is proved in the same way as the case s = 2, but more easily. Let M > 0 and let u, v ∈ H 2 0 (T) such that u H 2 0 (T) ≤ M and v H 2 0 (T) ≤ M . Note in particular that using (2.3)

, all quantities u L ∞ 0 (T) , v L ∞ 0 (T) , ∂ x u L ∞ 0 (T) , ∂ x v L ∞ 0 (
T) are bounded from above by M . We write

∂ x A(u) -∂ x A(v) 2 H 1 0 (T) = T ∂ x ∂ x uA (u) -∂ x vA (v) 2 dx = T ∂ xx uA (u) + (∂ x u) 2 A (u) -∂ xx vA (v) -(∂ x v) 2 A (v) 2 dx ≤ 2 T ∂ xx uA (u) -∂ xx vA (v) 2 dx + 2 T (∂ x u) 2 A (u) -(∂ x v) 2 A (v) 2 dx ≤ 4 T (∂ xx u) 2 A (u) -A (v) 2 dx + 4 T (∂ xx u -∂ xx v) 2 A (v) 2 dx + 4 T (∂ x u) 4 A (u) -A (v) 2 dx + 4 T (∂ x u -∂ x v) 2 (∂ x u + ∂ x v) 2 A (v) 2 dx ≤ 4 u 2 H 2 0 (T) ess sup T A (u) -A (v) 2 + 4 u -v 2 H 2 0 (T) sup [-M,M ] A 2 + 4 T (∂ x u) 4 dx ess sup T A (u) -A (v) 2 + 4 ess sup T |∂ x u + ∂ x v| 2 sup [-M,M ] |A | 2 u -v 2 H 1 0 (T)
By Assumption 2.1, A and A are locally Lipschitz continuous on R (and thus locally bounded). Hence, there exist constants C M and L M such that

∂ x A(u) -∂ x A(v) 2 H 1 0 (T) ≤ 4M 2 L M u -v 2 L ∞ 0 (T) + 4C M u -v 2 H 2 0 (T) + 4M 4 L M u -v 2 L ∞ 0 (T) + 16C M M 4 u -v 2 H 1 0 (T) ≤ 4M 2 L M + 4C M + 4M 4 L M + 16C M M 4 u -v 2 H 2 0 (T) ,
where we used (2.3) thrice in the last line.

Proof of Proposition 2.9. The equations (2.8) and (2.9) yield the immediate estimate

S t v 2 H s 0 (T) = m≥1 (-λ m ) s e 2νλmt v, e m 2 L 2 0 (T) ≤ m≥1 (-λ m ) s v, e m 2 L 2 0 (T) = v 2 H s 0 (T) ,
which ensures that S t v ∈ H s 0 (T) and then implies the first assertion of Proposition 2.9 thanks to the dominated convergence theorem.

The second assertion is proved in [START_REF] Boritchev | Generalised Burgers equation with random force and small viscosity[END_REF][START_REF] Rothe | Global Solutions of Reaction-Diffusion Systems[END_REF][START_REF] Da Prato | Stochastic Burgers' equation[END_REF]. We now detail the proof of the third assertion, part of which can also be found in [15, Lemma A.0.6]. Let s ∈ [0, +∞), T > 0 and (v(t)) t∈[0,T ] ∈ C ([0, T ], H s 0 (T)). For any 0 ≤ t 1 ≤ t 2 ≤ T , we have

t 1 0 S t 1 -r v(r)dr - t 2 0 S t 2 -r v(r)dr H s+3/2 0 (T) ≤ t 1 0 S t 1 -r v(r) -S t 2 -r v(r) H s+3/2 0 (T) dr + t 2 t 1 S t 2 -r v(r) H s+3/2 0 (T) dr. (2.35)
Thanks to the second assertion of Proposition 2.9, we get a bound over the second term of the righthand side:

t 2 t 1 S t 2 -r v(r) H s+3/2 0 (T) dr ≤ t 2 t 1 C 4 (t 2 -r) -3/4 v(r) H s 0 (T) dr ≤ 4C 4 (t 2 -t 1 ) 1/4 sup r∈[0,T ] v(r) H s 0 (T) , (2.36
) as well as for the first term:

t 1 0 S t 1 -r v(r) -S t 2 -r v(r)dr H s+3/2 0 (T) = t 1 0 S t 1 -r (Id -S t 2 -t 1 ) v(r) H s+3/2 0 (T) dr ≤ t 1 0 C 4 (t 1 -r) -3/4 (Id -S t 2 -t 1 )v(r) H s 0 (T) dr.
(2.37)

Using (2.8) and (2.9), we write for all r ∈ [0, T ],

(Id -S t 2 -t 1 )v(r) H s 0 (T) =   m≥1 (-λ m ) s 1 -e νλm(t 2 -t 1 ) 2 v(r), e m 2 L 2 0 (T)   1/2 . ≤   m≥1 (-λ m ) s v(r), e m 2 L 2 0 (T)   1/2 = v(r) H s 0 (T) < +∞,
and thus, we can apply the dominated convergence theorem which yields

lim t 2 -t 1 →0 (Id -S t 2 -t 1 )v(r) H s 0 (T) ≤   m≥1 (-λ m ) s lim t 2 -t 1 →0
1 -e νλm(t 2 -t 1 ) 2 v(r), e m = 0.

To pass to the limit t 2 -t 1 → 0 in (2.35), we use the dominated convergence theorem once again: injecting (2.36) and (2.37) into (2.35), we get

lim t 2 -t 1 →0 t 1 0 S t 1 -r v(r)dr - t 2 0 S t 2 -r v(r)dr H s+3/2 0 (T) ≤ t 1 0 C 4 (t 1 -r) -3/4 lim t 2 -t 1 →0 (Id -S t 2 -t 1 )v(r) H s 0 (T) dr+4C 4 sup r∈[0,T ] v(r) H s 0 (T) lim t 2 -t 1 →0 (t 2 -t 1 ) 1/4 = 0,
from which we derive the wanted result.

Proof of Proposition 2.10. For any m ≥ 1, the process (w m (t)) t≥0 is an Ornstein-Uhlenbeck process.

In particular, it is the solution of the stochastic differential equation:

w m (t) = νλ m t 0 w m (s)ds + W m (t), t ≥ 0. (2.38)
As such, it satisfies the inequality

w m (t) -νλ m t 0 w m (s)ds ≤ sup s∈[0,T ] W m (s), t ∈ [0, T ],
W m (s), t ∈ [0, T ]. (2.39)
In a similar way, from the inequality

w m (t) -νλ m t 0 w m (s)ds ≥ inf s∈[0,T ] W m (s), t ∈ [0, T ],
we deduce that

t 0 w m (s)ds ≥ 1 -e νλmt -νλ m inf s∈[0,T ] W m (s), t ∈ [0, T ]. (2.40)
Combining (2.39) and (2.40), we get

t 0 w m (s)ds ≤ 1 -νλ m sup s∈[0,T ] |W m (s)|, t ∈ [0, T ].
Taking the supremum in time, the expectation and applying Doob's inequality, recalling (2.6), we have

E sup t∈[0,T ] t 0 w m (s)ds 2 ≤ 1 (νλ m ) 2 E sup t∈[0,T ] |W m (t)| 2 (2.41) ≤ 4 (νλ m ) 2 E W Q (T ), e m λ m 2
H 2 0 (T) (2.42) = 4T (νλ m ) 2 k≥1 g k , e m λ m 2
H 2 0 (T)
.

(2.43)

We deduce from (2.38) that

sup t∈[0,T ] w m (t) 2 ≤ 2 (νλ m ) 2 sup t∈[0,T ] t 0 w m (s)ds 2 + sup t∈[0,T ] |W m (t)| 2 ,
and therefore it follows from (2.41), (2.42), (2.43) that

E sup t∈[0,T ] w m (t) 2 ≤ 16T k≥1 g k , e m λ m 2
H 2 0 (T) < +∞. (2.44)
As a consequence, for any M ≥ 1, we have

M m=1 w m e m λ m ∈ L 2 Ω, C [0, T ], H 2 0 (T) .
By completeness of L 2 (Ω, C([0, T ], H 2 0 (T))), to prove the statement, it suffices to show that the sequence

( M m=1 w m e m /λ m ) M ≥1 is Cauchy in this space. That is, lim M,N →∞ E   sup t∈[0,T ] M +N m=M w m (t) e m λ m 2
H 2 0 (T)   = 0. (2.45)
We prove this last equality using (2.44):

E   sup t∈[0,T ] M +N m=M w m (t) e m λ m 2
H 2 0 (T)   = E sup t∈[0,T ] M +N m=M w m (t) 2 ≤ M +N m=M E sup t∈[0,T ] w m (t) 2 ≤ 16T M +N m=M k≥1 g k , e m λ m 2
H 2 0 (T)
.

Recall that thanks to Assumption 2.2 combined with the fact that the family (e m /λ m ) m≥1 is an orthonormal basis of H 2 0 (T),

k≥1 m≥1 g k , e m λ m 2
H 2 0 (T) = k≥1 g k 2 H 2 0 (T) < +∞.
Hence, (2.45) is proved. Moreover, (w(t)) t≥0 is (F t ) t≥0 -adapted as the limit of the sequence of (F t ) t≥0adapted processes m≥1 w m e m /λ m .

Proof of Proposition 2.12. We first show that the assumption that u 0 ∈ H 2 0 (T) ensures that t → u(t) is a continuous, H 2 0 (T)-valued mapping. The proof consists in the two first iterations of the bootstrap argument that was used in [15, Theorem A.0.7]: since the mild solution (u(t)) t∈[0,τ ] belongs to C([0, τ ], H 1 0 (T)) almost surely, then from Proposition 2.8, for all s ∈ [0, τ ], we have ∂ x A(u(s)) ∈ L 2 0 (T). As a consequence of the third assertion of Proposition 2.9, the mapping t → t 0 S t-s ∂ x A(u(s))ds is continuous from [0, τ ] to H 3/2 0 (T), and by the first assertion of Proposition 2.9 as well as Proposition 2.10, so are the mappings t → S t u 0 and t → w(t). Therefore, (2.11) shows that (u(t)) t∈[0,τ ] ∈ C([0, τ ], H Multiplying on each side by e -νλmt , we get the decomposition:

e -νλmt u(t), e m L 2 0 (T) = u 0 , e m L 2 0 (T) - t 0 e -νλms ∂ x A(u(s)), e m L 2 0 (T) ds + 1 λ m t 0 e -νλms dW m (s).
Then, the Itô formula yields

u(t), e m L 2 0 (T) = u 0 , e m L 2 0 (T) + νλ m t 0 u(s), e m L 2 0 (T) ds - t 0 ∂ x A(u(s)), e m L 2 0 (T) ds + 1 λ m W m (t).
(2.46) Since for all s ∈ [0, t], u(s) belongs to H 2 0 (T), it is possible to perform an integration by parts on the viscous term in the following way:

νλ m t 0 u(s), e m L 2 0 (T) ds = ν t 0 u(s), ∂ xx e m L 2 0 (T) ds = ν t 0 ∂ xx u(s), e m L 2 0 (T) ds.
Equation (2.46), after being injected with the above equality, multiplied by e m and summed over m, becomes the strong formulation of (2.10).

The converse statement follows from the same computations.

Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure

Invariant measure

This section is dedicated to the proof of Theorem 2.7. The existence of an invariant measure is proved in Subsection 2.3.2 using the Krylov-Bogoliubov theorem, whereas the uniqueness is addressed through a coupling argument relying on the L 1 0 (T)-contraction property established in Proposition 2.21. The proof of existence of an invariant measure we provide in the next subsection relies plainly on the presence of viscosity. Indeed, the viscous term provides the process u(t) with a dissipative -and thus a more stable -behaviour. Still, it has to be borne in mind that when the flux term is nonlinear enough, the presence of a viscous term is not a necessary condition for the stability of the underlying stochastic process. On the physical side, in his theory of turbulent flows [START_REF] Andreï | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF][START_REF] Andreï | On degeneration of isotropic turbulence in an incompressible viscous liquid[END_REF], Kolmogorov already predicted this idea: the statistical distribution of scales of intermediate size in turbulence are not determined by the viscosity coefficient. On the theoretical side, the same idea was validated theoretically by powerful results on the invariant measure for the inviscid stochastic Burgers' equation [START_REF] Weinan | Invariant Measures for Burgers Equation with Stochastic Forcing[END_REF] and, quite a few years later, for inviscid stochastic conservation laws with "non-degenerate" flux [START_REF] Debussche | Invariant measure of scalar first-order conservation laws with stochastic forcing[END_REF]. However, our framework differs substantially from the inviscid case in the sense that our stability results are driven by regularity issues which cannot be tackled without viscosity.

Preliminary results

By Definition 2.6, an invariant measure for Equation (2.1) is a Borel probability measure on H 2 0 (T). Our proofs of existence and uniqueness however involve estimates in various spaces, namely L 1 0 (T), L 2 0 (T) and H 1 0 (T). In particular, we shall manipulate and identify Borel probability measures on these spaces. We first clarify the relation between the associated Borel σ-fields thanks to the following result. For any metric space E, we respectively denote by B(E) and P(E) the Borel σ-field and the set of Borel probability measures on E. Lemma 2.20 (Borel probability measures on L q 0 (T) and H s 0 (T)). For all q ∈ [1, 2] and s ≥ 1, B(H s 0 (T)) = {B ∩ H s 0 (T) : B ∈ B(L q 0 (T))}. As a consequence:

(1) for any µ ∈ P(H s 0 (T)), the mapping µ(•) = µ(• ∩ H s 0 (T)) defines a Borel probability measure on L q 0 (T);

(2) conversely, for any µ ∈ P(L q 0 (T)) which gives full weight to H s 0 (T), there exists a unique µ ∈ P(H s 0 (T)) such that µ(B) = µ(B ∩ H s 0 (T)) for any B ∈ B(L q 0 (T)).

Proof. Let q ∈ [1, 2] and s ≥ 1. The set T defined by

T = {B ∩ H s 0 (T) : B ∈ B (L q 0 (T))} .
is a σ-field on H s 0 (T), called the trace σ-field of H s 0 (T) in B(L q 0 (T)).

(1) We denote by I the injection H s 0 (T) → L q 0 (T), so that T = {I -1 (B) : B ∈ B(L q 0 (T))}. Since I is continuous, and therefore Borel measurable, we have T ⊂ B(H s 0 (T)). Thus, for any µ ∈ P(H s 0 (T)), the pushforward measure µ defined by

µ(B) := µ • I -1 (B) = µ (B ∩ H s 0 (T)) , B ∈ B (L q 0 (T)) ,
is a Borel probability measure on L q 0 (T).

(2) Let us first notice that since H s 0 (T) is separable, the Borel σ-field B(H s 0 (T)) is the smallest σ-field on H s 0 (T) containing all closed balls. Let A ⊂ H s 0 (T) be such a ball. Since the H s 0 (T)-norm is lower semi-continuous on L q 0 (T), then A is closed in L q 0 (T) as a level set of a lower semi-continuous function, and thus A ∈ B(L q 0 (T)). It is then clear that A ∈ T , which by the minimality property of B(H s 0 (T)) entails B(H s 0 (T)) ⊂ T , and thus B(H s 0 (T)) = T . Now let µ be a Borel probability measure on L q 0 (T) which gives full weight to H s 0 (T), that is to say such that there exists B ∈ B(L q 0 (T)) such that B ⊂ H s 0 (T) and µ( B) = 1. Let us define the Borel probability measure µ on H s 0 (T) by

µ(B ∩ H s 0 (T)) := µ(B), B ∈ B (L q 0 (T)) .
Notice that this definition is not ambiguous, because the identity T = B(H s 0 (T)) ensures that any element of B(H s 0 (T)) writes under the form B ∩ H s 0 (T) for some B ∈ B(L q 0 (T)); besides, if

B 1 , B 2 ∈ B(L q 0 (T)) are such that B 1 ∩ H s 0 (T) = B 2 ∩ H s 0 (T), then µ(B 1 ) = µ(B 1 ∩ B) = µ(B 2 ∩ B) = µ(B 2 ) because the identity B 1 ∩ H s 0 (T) = B 2 ∩ H s 0 (T) implies that B 1 ∩ B = B 2 ∩ B.
Finally, the fact that any ν ∈ P(H s 0 (T)) such that µ(B) = ν(B ∩ H s 0 (T)) for any B ∈ B(L q 0 (T)) needs to coincide with µ follows again from the identity B(H s 0 (T)) = T .

To prove Theorem 2.7, we will need a standard property of scalar conservation laws, namely the L 1 0 (T)-contraction. In the stochastic setting, we mention that a similar proof of the following proposition is done in [16, Theorem 6.1], but in the case where the flux function is C ∞ . Proposition 2.21 (L 1 0 (T)-contraction). Under Assumptions 2.1 and 2.2, let (u(t)) t≥0 and (v(t)) t≥0 be two strong solutions of (2.1) starting from different initial conditions u 0 and v 0 . Then, almost surely and for every 0 ≤ s ≤ t, we have

u(t) -v(t) L 1 0 (T) ≤ u(s) -v(s) L 1 0 (T) .
Proof. We define a continuous approximation of the sign function by setting for all η > 0,

sign η (u) :=      u η , u ∈ [-η, η], 1, u ≥ η, -1, u ≤ η,
which gives rise to the following continuously differentiable approximation of the absolute value function:

|v| η := v 0 sign η (u)du, v ∈ R. Let 0 ≤ s ≤ t. We have T |u(t) -v(t)| η dx - T |u(s) -v(s)| η dx = T t s d dr |u(r) -v(r)| η drdx (2.47) = T t s d dr (u(r) -v(r)) sign η (u(r) -v(r))drdx = t s T (A(u(r)) -A(v(r)) -ν∂ x (u(r) -v(r))) ∂ x sign η (u(r) -v(r)) dxdr
(where we used the Fubini theorem and an integration by parts)

= t s T (A(u(r)) -A(v(r)) -ν∂ x (u(r) -v(r))) ∂ x (u(r) -v(r)) 1 η 1 |u(r)-v(r)|≤η dxdr ≤ t s T (A(u(r)) -A(v(r)))∂ x (u(r) -v(r)) 1 η 1 |u(r)-v(r)|≤η dxdr
We fix

M := sup r∈[s,t] u(r) L ∞ 0 (T) ∨ sup r∈[s,t] v(r) L ∞ 0 (T) , Chapter
|A(u(r)) -A(v(r))||∂ x (u(r) -v(r))| 1 η 1 |u(r)-v(r)|≤η ≤ L M |∂ x (u(r) -v(r))| , with t s T L M |∂ x (u(r) -v(r))| dxdr < +∞.
Thus, we get from the dominated convergence theorem:

lim η→0 t s T (A(u(r)) -A(v(r)))∂ x (u(r) -v(r)) 1 η 1 |u(r)-v(r)|≤η dxdr = t s T lim η→0 (A(u(r)) -A(v(r)))∂ x (u(r) -v(r)) 1 η 1 |u(r)-v(r)|≤η dxdr = 0. (2.48)
As for the left-hand side of (2.47), noticing that | • | η increases to | • | as η decreases, we have from the monotone convergence theorem

lim η→0 T |u(t) -v(t)| η dx = u(t) -v(t) L 1 0 (T) , lim η→0 T |u(s) -v(s)| η dx = u(s) -v(s) L 1 0 (T) .
Hence, (2.47) yields the wanted result.

Existence

From the semigroup (P t ) t≥0 introduced in Subsection 2.1.2, we define its time-averaged semigroup (R T ) T ≥0 by R 0 = Id, and for all T > 0,

R T ϕ(u 0 ) = 1 T T 0 P t ϕ(u 0 )dt, ϕ ∈ C b (H 2 0 (T)), u 0 ∈ H 2 0 (T), R * T α(Γ) = 1 T T 0 P * t α(Γ)dt, α ∈ P(H 2 0 (T)), Γ ∈ B(H 2 0 (T)).
Following the first part of Lemma 2.20, for any α ∈ P(H 2 0 (T)) and T ≥ 0, we denote by R * T α the Borel probability measure on

L 1 0 (T) defined by R * T α(•) = R * T α(• ∩ H 2 0 (T)).
Lemma 2.22. Under Assumptions 2.1 and 2.2, for any u 0 ∈ H 2 0 (T), there exists an increasing sequence T n n→∞ -→ +∞ and a probability measure µ ∈ P(L 1 0 (T)), such that the sequence of measures ( R * T n δ u 0 ) n≥1 converges weakly to µ in P(L 1 0 (T)).

Proof. Let u 0 ∈ H 2 0 (T). From the inequality (2.17) with p = 2, we can pass to the limit r → +∞ (which we recall implies that T r → +∞ almost surely), and we get for all T ≥ 0,

E T 0 u(t) 2 H 1 0 (T) dt ≤ 1 2ν u 0 2 L 2 0 (T) + D 0 T 2ν .
Applying now the Markov inequality when T ≥ 1, we have for all ε > 0,

1 T T 0 P u(t) 2 H 1 0 (T) > 1 ε dt ≤ ε 2ν u 0 2 L 2 0 (T) + D 0 .
(2.49)

2.3. Invariant measure 49
Setting

K ε := v ∈ H 1 0 (T) : v 2 H 1 0 (T) ≤ 1 ε ,
we know from the compact embedding H 1 0 (T) ⊂⊂ L 1 0 (T) that the set K ε is compact in L 1 0 (T). Thus, rewriting (2.49) as

R * T δ u 0 L 1 0 (T) \ K ε ≤ ε 2ν u 0 2 L 2
0 (T) + D 0 , we deduce that the family of measures { R * T δ u 0 : T ≥ 1} is tight in the space P(L 1 0 (T)). The result is then a consequence of Prokhorov's theorem [12, Theorem 5.1].

Lemma 2.23. Under the assumptions of Lemma 2.22, for all p ≥ 1, if v is a random variable in L 1 0 (T) distributed according to µ, then

E v p L p 0 (T) < +∞ and E v 2 H 2 0 (T) < +∞.
Besides, the probability measure µ ∈ P(H 2 0 (T)) associated with µ by the second part of Lemma 2.20 is invariant for the semigroup (P t ) t≥0 .

Proof. We start to show that the measure µ ∈ P(L 1 0 (T)) gives full weight to H 2 0 (T). Thanks to Lemma 2.17, since T r -→ r→∞ +∞ almost surely, we have:

∀T > 0, 1 T T 0 E u(s) 2 H 2 0 (T) ds ≤ 1 T ν u 0 2 H 1 0 (T) + C 7 1 + u 0 2p A +2 L 2p A +2 0 (T) + C 8 ν . (2.50)
Let (v n ) n≥1 be a sequence of H 2 0 (T)-valued random variables such that v n ∼ R * Tn δ u 0 and v n converges in distribution in L 1 0 (T) towards a random variable v ∼ µ. From (2.50) and the definition of (R T ) T ≥0 , we have

lim sup n→∞ E v n 2 H 2 0 (T) = lim sup n→∞ 1 T n Tn 0 E u 0 u(s) 2 H 2 0 (T) ds ≤ C 8 ν . Now, since • 2 H 2 0 (T)
is lower semi-continuous on L 1 0 (T), we get from Portemanteau's theorem:

E v 2 H 2 0 (T) ≤ lim inf n→∞ E v n 2 H 2 0 (T) ≤ C 8 ν .
In particular, v ∈ H 2 0 (T) almost surely, and thus µ gives full weight to H 2 0 (T). We now show that for any p ≥ 1, E[ v p L p 0 (T) ] < +∞. Let p ≥ 1. From Lemma 2.15, we have for all T > 0, 1

T T 0 E u 0 u(s) p L p 0 (T) ds ≤ C (p) 5 T 1 + u 0 p L p 0 (T) + C (p) 6 .
Once again, we use Portemanteau's theorem and the lower semi-continuity, this time of

• p L p 0 (T) , on L 1 0 (T): E v p L p 0 (T) ≤ lim inf n→∞ E v n p L p 0 (T) = lim inf n→∞ 1 T n Tn 0 E u 0 u(s) p L p 0 (T) ds ≤ C (p) 6 ,
and the wanted result follows.

To prove the invariance of the measure µ with respect to (P t ) t≥0 , we wish to apply the Krylov-Bogoliubov theorem [36, Theorem 3.1.1]. However, (P t ) t≥0 is a Feller semigroup on the space H 2 0 (T) (Corollary 2.5) whereas our tightness result (Lemma 2.22) holds in P(L 1 0 (T)). To overcome this inconvenience, we use Lemma 2.20 and we place ourselves at the level of the embedded probability measures in P(L It follows from the L 1 0 (T)-contraction property that the map P t ϕ : H 2 0 (T) → R is continuous with respect to the L 1 0 (T)-norm. To prove this fact, let v 0 ∈ H 2 0 (T) and let (v

(j) 0 ) j≥1 be a sequence of H 2 0 (T) such that v (j) 0 -v 0 L 1 0 (T) → 0, j → +∞. Let (v(t)
) t≥0 and (v (j) (t)) t≥0 , j ≥ 1, be the strong solutions of (2.1) respectively with initial conditions v 0 and v (j) 0 , j ≥ 1. From Proposition 2.21, we get almost surely and for all t ≥ 0,

lim j→∞ v (j) (t) -v(t) L 1 0 (T) = 0.
Since ϕ is bounded and continuous with respect to the L 1 0 (T)-norm, we have

lim j→∞ P t ϕ v (j) 0 -P t ϕ(v 0 ) ≤ lim j→∞ E ϕ v (j) (t) -ϕ(v(t)) = 0,
so that P t ϕ is continuous with respect to the L 1 0 (T)-norm. As a consequence, from Lemma 2.22, we have for all t ≥ 0 H 2 0 (T)

P t ϕdµ = L 1 0 (T) P t ϕd µ = lim n→∞ L 1 0 (T) P t ϕd R * T n δ u 0 = lim n→∞ H 2 0 (T) P t ϕdR * T n δ u 0 = lim n→∞ 1 T n T n 0 H 2 0 (T) ϕdP * s+t δ u 0 ds = lim n→∞ 1 T n T n +t t H 2 0 (T) ϕdP * s δ u 0 ds = lim n→∞ 1 T n T n 0 H 2 0 (T) ϕdP * s δ u 0 ds + 1 T n T n +t T n H 2 0 (T) ϕdP * s δ u 0 ds - 1 T n t 0 H 2 0 (T) ϕdP * s δ u 0 ds = lim n→∞ H 2 0 (T) ϕdR * T n δ u 0 = lim n→∞ L 1 0 (T) ϕd R * T n δ u 0 = L 1 0 (T) ϕd µ.
For any t ≥ 0, P * t µ gives full weight to H 2 0 (T) and therefore, following the first part of Lemma 2.20, we can define the associated Borel probability measure on L 1 0 (T) by P * t µ = P * t µ(• ∩ H 2 0 (T)). From Equation (2.51) and the above sequence of computations, it follows that for all t ≥ 0,

L 1 0 (T) ϕd P * t µ = L 1 0 (T) ϕd µ,
Given that ϕ has been chosen arbitrarily in C b (L 1 0 (T)), this last equality says that P * t µ = µ. The second part of Lemma 2.20 now ensures that P * t µ = µ.

Uniqueness

The proof of the uniqueness part of Theorem 2.7 follows the ideas of the "small-noise" coupling argument from Dirr and Souganidis [START_REF] Dirr | Large-Time Behavior for Viscous and Nonviscous Hamilton-Jacobi Equations Forced by Additive Noise[END_REF]. On one hand, due to the dissipative nature of the drift, two solutions of (2.1) perturbed by the same noise and starting from different initial conditions are driven to balls of L 2 0 (T) with small radius whenever this noise is small over sufficiently long time intervals. On the other hand, the L 1 0 (T)-contraction property ensures that when these two solutions get close to one another they stay close forever. Hence, each time the noise gets small enough, the two solutions get closer and closer and eventually, they show the same asymptotical behaviour. This idea allows to show that the law of two solutions have the same limit as the time goes to infinity. Therefore, starting from two invariant measures leads to the equality of these measures. The same kind of argument was used in [START_REF] Debussche | Invariant measure of scalar first-order conservation laws with stochastic forcing[END_REF] for the invariant measure of kinetic solutions of inviscid scalar conservation laws and in [START_REF] Debussche | Ergodicity Results for the Stochastic Navier-Stokes Equations: An Introduction[END_REF] for the stochastic Navier-Stokes equations.

Let (u(t)) t≥0 and (v(t)) t≥0 be two solutions of (2.1) driven by the same Q-Wiener process (W Q (t)) t≥0 . For all R > 0, we define the stopping time:

τ R := inf t ≥ 0 : u(t) 2 H 1 0 (T) + v(t) 2 H 1 0 (T) ≤ R .
Lemma 2.24. Under Assumptions 2.1 and 2.2, there exists R > 0 such that for any u 0 and v 0 in H 2 0 (T), the stopping time τ R is finite almost surely.

Proof. We can use here, from the statement of Lemma 2.15, the inequality (2.17) with p = 2. In this case, we get

2νE t∧τ R 0 u(s) 2 H 1 0 (T) + v(s) 2 H 1 0 (T) ds ≤ u 0 2 L 2 0 (T) + v 0 2 L 2 0 (T) + 2D 0 E[t ∧ τ R ],
from which we deduce, by definition of the stopping time τ R , that

2νRE[t ∧ τ R ] ≤ u 0 2 L 2 0 (T) + v 0 2 L 2 0 (T) + 2D 0 E[t ∧ τ R ].
Taking R > D 0 /ν yields

E[τ R ] = lim t→∞ E[τ R ∧ t] ≤ u 0 2 L 2 0 (T) + v 0 2 L 2 0 (T) 2(νR -D 0 ) < +∞,
from which we derive the wanted result.

The following result asserts that when the coupled processes (u(t)) t≥0 and (v(t)) t≥0 start from deterministic initial conditions inside some ball of L 2 0 (T), then they both attain in finite time any neighbourhood of 0 with positive probability: Lemma 2.25. Under Assumptions 2.1 and 2.2, for any M > 0 and any ε > 0, there exist a time t ε,M > 0 and a value p ε,M ∈ (0, 1) such that for all u 0 , v 0 ∈ H 2 0 (T)

satisfying u 0 2 H 1 0 (T) + v 0 2 H 1 0 (T) ≤ M , P u(t ε,M ) 2 L 2 0 (T) + v(t ε,M ) 2 L 2 0 (T) ≤ ε ≥ p ε,M .
Proof. Let u 0 , v 0 ∈ H 2 0 (T) be such that u 0 H 1 0 (T) + v 0 H 1 0 (T) ≤ M , and let us define

t ε,M = - 1 2ν log ε 4M .
Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure

To prove the lemma, we are going to compare the trajectories of (u(t)) t≥0 and (v(t)) t≥0 with the trajectories of their noiseless counterparts (u(t)) t≥0 and (v(t)) t≥0 , defined by

∂ t u(t) = -∂ x A (u(t)) + ν∂ xx u(t) u(0) = u 0 ∂ t v(t) = -∂ x A (v(t)) + ν∂ xx v(t) v(0) = v 0 .
Recall that the viscosity yields energy dissipation:

d dt u(t) 2 L 2 0 (T) + v(t) 2 L 2 0 (T) = -2ν u(t) 2 H 1 0 (T) + v(t) 2 H 1 0 (T) .
Applying (2.3) on the right-hand side, we get

d dt u(t) 2 L 2 0 (T) + v(t) 2 L 2 0 (T) ≤ -2ν u(t) 2 L 2 0 (T) + v(t) 2 L 2 0 (T) ,
and we can now apply Grönwall's lemma:

u(t) 2 L 2 0 (T) + v(t) 2 L 2 0 (T) ≤ u 0 2 L 2 0 (T) + v 0 2 L 2 0 (T) e -2νt ≤ M e -2νt .
With our choice of t ε,M , the above inequality means that as soon as t ≥ t ε,M , we have u(t) 2

L 2 0 (T) + v(t) 2 L 2 0 (T) ≤ ε/4.
Furthermore, it is a consequence of Lemma 2.17 that (u(t)) t≥0 satisfies

u(t) 2 H 1 0 (T) ≤ u 0 2 H 1 0 (T) + C 7 1 + u 0 2p A +2 L 2p A +2 0 (T) , t ≥ 0.
Indeed, when all the noise coefficients g k are equal to zero, the constant C 8 in the statement of Lemma 2.17 can also be taken equal to zero. Since the same inequality also applies to (v(t)) t≥0 , we have

u(t) 2 H 1 0 (T) + v(t) 2 H 1 0 (T) ≤ M + 2C 7 1 + M p A +1 =: C (M ) 9
.

We focus now on the trajectories of the random processes (u(t)) t≥0 and (v(t)) t≥0 . We introduce the stopping time

τ M := inf t ≥ 0 : u(t) H 1 0 (T) ∨ v(t) H 1 0 (T) ≥ 1 2 + C (M ) 9
.

Following Proposition 2.12, we may use the expressions of (u(t)) t≥0 and (u(t)) t≥0 in the mild sense. From these mild formulations, we write

u(t) -u(t) H 1 0 (T) ≤ t 0 S t-s ∂ x (A(u(s)) -A(u(s))) H 1 0 (T) ds + w(t) H 1 0 (T) , (2.52)
where (w(t)) t≥0 is the stochastic convolution associated with the Q-Wiener process (W Q (t)) t≥0 . According to Proposition 2.8, we call

L M a local Lipschitz constant of the map z ∈ H 1 0 (T) → ∂ x A(z) ∈ L 2 0 (T) over the ball {z ∈ H 1 0 (T) : z 2 H 1 0 (T) ≤ 1 2 + C (M ) 9
} , and we place ourselves in the event

sup t∈[0,t ε,M ] w(t) H 1 0 (T) ≤ δ ε,M , where δ ε,M := √ ε 2 √ 2 1 1 + 2 t ε,M C 4 L M e -C
u(t) -u(t) H 1 0 (T) ≤ t 0 C 4 √ t -s ∂ x (A(u(s)) -A(u(s))) L 2 0 (T) ds + δ ε,M ≤ t 0 C 4 L M √ t -s u(s) -u(s) H 1 0 (T) ds + δ ε,M .
Iterating this inequality and using the same arguments as in the proof of Lemma 2.19, we get for all

t ≤ t ε,M ∧ τ M , u(t) -u(t) H 1 0 (T) ≤ δ ε,M 1 + 2 t ε,M ∧ τ M C 4 L M + C 2 4 L 2 M π t 0 u(s) -u(s) H 1 0 (T) ds.
Using now Grönwall's lemma, we deduce

u(t) -u(t) H 1 0 (T) ≤ δ ε,M 1 + 2 t ε,M ∧ τ M C 4 L M e C 2 4 L 2 M πt ≤ √ ε 2 √ 2 .
Since the same arguments apply for the processes (v(t)) t≥0 and (v(t)) t≥0 , and given Equation (2.3), we have shown that for all t ≤ τ M ∧ t ε,M ,

u(t) 2 L 2 0 (T) + v(t) 2 L 2 0 (T) ≤ 2 u(t) 2 L 2 0 (T) + v(t) 2 L 2 0 (T) + 2 u(t) -u(t) 2 L 2 0 (T) + v(t) -v(t) 2 L 2 0 (T) ≤ ε 2 + ε 2 = ε.
We shall prove now that the event τ M < t ε,M is impossible. Indeed, assume for instance that

u ( τ M ) H 1 0 (T) ≥ 1 2 + C (M ) 9
, then we would have

u ( τ M ) -u ( τ M ) H 1 0 (T) ≤ √ ε 2 and u ( τ M ) 2 H 1 0 (T) ≤ C (M ) 9
, and thus,

√ ε 2 ≥ u ( τ M ) -u ( τ M ) H 1 0 (T) ≥ u ( τ M ) H 1 0 (T) -u ( τ M ) H 1 0 (T) ≥ 1 2 + C (M ) 9 -C (M ) 9 = 1 2 ,
which is false for too small values of ε.

We just have proved that for M > 0 arbitrarily chosen and for all u 0 , v 0 ∈ H 2 0 (T) such that

u 0 2 H 1 0 (T) + v 0 2 H 1 0 (T) ≤ M , we have P u(t ε,M ) 2 L 2 0 (T) + v(t ε,M ) 2 L 2 0 (T) ≤ ε ≥ P sup t∈[0,t ε,M ] w(t) H 1 0 (T) ≤ δ ε,M .
To conclude the proof, it remains to check that Proof.

p ε,M := P sup t∈[0,t ε,M ] w(t) H 1 0 (T) ≤ δ ε,M > 0. ( 2
Step 1. Almost sure confluence. We start by fixing ε > 0 small to which we associate the value t ε,R defined at Lemma 2.25, where R has been defined at Lemma 2.24. We define the increasing stopping time sequence [START_REF] Burgers | Hydrodynamics. -Application of a model system to illustrate some points of the statistical theory of free turbulence[END_REF] and the strong Markov property (Corollary 2.5) ensure that every T j is finite almost surely. We claim that

T 1 := τ R T 2 := inf t ≥ T 1 + t ε,R : u(t) 2 H 1 0 (T) + v(t) 2 H 1 0 (T) ≤ R T 3 := inf t ≥ T 2 + t ε,R : u(t) 2 H 1 0 (T) + v(t) 2 H 1 0 (T) ≤ R . . . Lemma 2.
∀J ∈ N * , P ∀j = 1, . . . , J, u(T j + t ε,R ) 2 L 2 0 (T) + v(T j + t ε,R ) 2 L 2 0 (T) > ε ≤ (1 -p ε,R ) J .
(2.54) Indeed, it is true for J = 1 thanks to the strong Markov property and Lemma 2.25:

P (u 0 ,v 0 ) u(τ R + t ε,R ) 2 L 2 0 (T) + v(τ R + t ε,R ) 2 L 2 0 (T) > ε = E (u 0 ,v 0 ) P (u 0 ,v 0 ) u(τ R + t ε,R ) 2 L 2 0 (T) + v(τ R + t ε,R ) 2 L 2 0 (T) > ε|F τ R = E (u 0 ,v 0 ) P (u(τ R ),v(τ R )) u(t ε,R ) 2 L 2 0 (T) + v(t ε,R ) 2 L 2 0 (T) > ε ≤ 1 -p ε,R ,
and the general case follows by induction: assuming that inequality (2.54) is true for some J ∈ N * , we have

P (u 0 ,v 0 ) ∀j = 1, . . . , J + 1, u(T j + t ε,R ) 2 L 2 0 (T) + v(T j + t ε,R ) 2 L 2 0 (T) > ε = E (u 0 ,v 0 ) P (u 0 ,v 0 ) ∀j = 1, . . . , J + 1, u(T j + t ε,R ) 2 L 2 0 (T) + v(T j + t ε,R ) 2 L 2 0 (T) > ε|F T J+1 = E (u 0 ,v 0 )     J j=1 1 u(T j +t ε,R ) 2 L 2 0 (T) + v(T j +t ε,R ) 2 L 2 0 (T) >ε   P (u(T J+1 ),v(T J+1 )) u(t ε,R ) 2 L 2 0 (T) + v(t ε,R ) 2 L 2 0 (T) > ε   ≤ (1 -p ε,R ) J × (1 -p ε,R ) = (1 -p ε,R ) J+1 .
Taking the limit when J goes to infinity, we get

P ∀j ∈ N * , u(T j + t ε,R ) 2 L 2 0 (T) + v(T j + t ε,R ) 2 L 2 0 (T) > ε = lim J→∞ P ∀j = 1, . . . , J, u(T j + t ε,R ) 2 L 2 0 (T) + v(T j + t ε,R ) 2 L 2 0 (T) > ε ≤ lim J→∞ (1 -p ε,R ) J = 0,
and consequently,

P ∃t ≥ 0, u(t) 2 L 2 0 (T) + v(t) 2 L 2 0 (T) ≤ ε = 1.
(2.55)

2.3. Invariant measure 55 Since u(t) -v(t) 2 L 1 0 (T) ≤ u(t) -v(t) 2 L 2 0 (T) ≤ 2( u(t) 2 L 2 0 (T) + v(t) 2 L 2 0 (T)
) and since the value ε > 0 has been chosen arbitrarily at the beginning of this proof, then Equality (2.55) means that almost surely, ∀ε > 0, ∃t ≥ 0, u(t) -v(t) 2 L 1 0 (T) ≤ 2ε. Recall however that Proposition 2.21 states that almost surely, the mapping t → u(t) -v(t) L 1 0 (T) is non-decreasing. It follows that almost surely,

lim t→∞ u(t) -v(t) L 1 0 (T) = 0.
(2.56)

Step 2. Uniqueness. Let us now assume that there exist two invariant measures µ 1 , µ 2 for the solution of (2.1), and let us take initial conditions u 0 and v 0 with distributions µ 1 and µ 2 respectively. For any test function φ : L 1 0 (T) → R bounded and Lipschitz continuous, we have for all t ≥ 0,

|E [φ(u 0 )] -E [φ(v 0 )]| = |E [φ(u(t))] -E [φ(v(t))]| ≤ E [|φ(u(t)) -φ(v(t))|] .
Since φ is Lipschitz continuous, from (2.56), we have almost surely

lim t→∞ |φ(u(t)) -φ(v(t))| = 0.
Moreover, for any t ≥ 0, we have almost surely |φ(u(t)) -φ(v(t))| ≤ 2 sup |φ|. Thus, we may apply the dominated convergence theorem, which yields

|E [φ(u 0 )] -E [φ(v 0 )]| ≤ lim t→∞ E [|φ(u(t)) -φ(v(t))|] = 0, so that E[φ(u 0 )] = E[φ(v 0 )],
or in other words,

H 2 0 (T) φdµ 1 = H 2 0 (T) φdµ 2 .
(2.57)

According to Lemma 2.20, let µ 1 and µ 2 be the probability measures on P(L 1 0 (T)) associated to µ 1 and µ 2 respectively. Equation (2.57) rewrites Abstract. This chapter corresponds to the first four sections of the preprint [START_REF] Boyaval | Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law[END_REF], written in collaboration with S. Boyaval and J. Reygner. We aim to give a numerical approximation of the invariant measure of the solution of the equation studied in Chapter 2, that is, a viscous scalar conservation law, one-dimensional and periodic in the space variable, and stochastically forced with a white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz and to have at most polynomial growth. The numerical scheme we employ discretises the SPDE according to a finite volume method in space, and a split-step backward Euler method in time. As a first result, we prove the well-posedness as well as the existence and uniqueness of an invariant measure for both the spatial semi-discretisation and the fully discrete scheme. Our main result is then the convergence of the invariant measures of the discrete approximations, as the space and time steps go to zero, towards the invariant measure of the SPDE, with respect to the second-order Wasserstein distance.

L 1 0 (T) φd µ 1 = L 1 0 (T) φd µ 2 , ∀φ ∈ C b L 1 0 (T) , so that µ 1 = µ 2 and

Introduction

Viscous scalar conservation law with random forcing

We consider the following viscous scalar conservation law with stochastic forcing

du = -∂ x A(u)dt + ν∂ xx udt + k≥1 g k dW k (t), x ∈ T, t ≥ 0. (3.1)
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law Periodic boundary conditions are assigned over the space variable x as T = R/Z denotes the one-dimensional torus, and (W k ) k≥1 is a family of independent real Brownian motions. The viscosity coefficient ν is assumed to be positive. In Chapter 2, we have shown the well-posedness in a strong sense of Equation (3.1), as well as the existence and uniqueness of an invariant measure for its solution. These results are recalled in Proposition 3.2 below. In this work, we aim to provide a numerical scheme, based on the finite-volume method, that allows to approximate this invariant measure. In this perspective, we place ourselves in the setting of Chapter 2 and recall our main notations and assumptions.

Notations. For all p ∈ [1, +∞], we denote by L p 0 (T) the set of functions f ∈ L p (T) such that T f (x)dx = 0. We write • L p 0 (T) the L p -norm induced on L p 0 (T) and •, • L 2 0 (T) the L 2 -scalar product induced on L 2 0 (T). In a similar manner, for any integer m ≥ 0 and any p ∈ [1, +∞], we introduce the Sobolev space W m,p 0 (T) := L p 0 (T) ∩ W m,p (T) which we equip with the norm

• W m,p 0 (T) := ∂ m x • L p 0 (T)
. Incidentally, we will denote by H m 0 (T) the space W m,2 0 (T), which we recall is separable and Hilbert when endowed with the norm

• H m 0 (T) := • W m,2 0
(T) and the associated scalar product •, • H m 0 (T) . We recall the following inequalities: for all 1 ≤ p ≤ q ≤ +∞,

u L p 0 (T) ≤ u L q 0 (T) , ∀u ∈ L q 0 (T), (3.2)
and

u L ∞ 0 (T) ≤ u W 1,1 0 (T) , ∀u ∈ W 1,1 0 (T). (3.3)
In the sequel, we denote by N the set of non-negative integers, and by N * the set of positive integers.

Assumption 3.1. The function A : R → R is of class C 2 , its first derivative has at most polynomial growth:

∃C A > 0, ∃p A ∈ N * , ∀v ∈ R, |A (v)| ≤ C A (1 + |v| p A ) , (3.4)
and its second derivative A is locally Lipschitz continuous on R. Furthermore, for all k ≥ 1, g k ∈ H 2 0 (T) and

D 0 := k≥1 g k 2 H 2 0 (T) < +∞.
(3.5)

The assumptions (3.4) and (3.5) will be needed in the arguments contained in this chapter while the local Lipschitz continuity of A is only necessary for Proposition 3.2.

Let (Ω, F, P) be a probability space, equipped with a normal filtration (F t ) t≥0 in the sense of [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF]Section 3.3], on which (W k ) k≥1 is a family of independent Brownian motions. Under Assumption 3.1, the series k g k W k converges in L 2 (Ω, C([0, T ], H 2 0 (T))), for any T > 0, towards an H 2 0 (T)-valued Wiener process (W Q (t)) t∈[0,T ] with respect to the filtration (F t ) t≥0 , defined in the sense of [35, Section 4.2], with the trace class covariance operator Q defined by

Q : H 2 0 (T) -→ H 2 0 (T) v -→ k≥1 g k v, g k H 2 0 (T)
. Given a normed vector space E, B(E) denotes the Borel sets of E, P(E) denotes the set of Borel probability measures over E, and for p ∈ [1, +∞), P p (E) denotes the subset of P(E) of probability measures with finite p-th order moment. The well-posedness of (3.1) as well as the existence and uniqueness of an invariant measure for its solution is proved in Chapter 2 (Theorems 2.4 and 2.7): Proposition 3.2. Let u 0 ∈ H 2 0 (T). Under Assumption 3.1, there exists a unique strong solution (u(t)) t≥0 to Equation (3.1) with initial condition u 0 . That is, an (F t ) t≥0 -adapted process (u(t)) t≥0 with values in H 2 0 (T) such that, almost surely:

1. the mapping t → u(t) is continuous from [0, +∞) to H 2 0 (T);

2. for all t ≥ 0, the following equality holds:

u(t) = u 0 + t 0 (-∂ x A (u(s)) + ν∂ xx u(s)) ds + W Q (t). (3.6)
Furthermore, the process (u(t)) t≥0 admits a unique invariant measure µ ∈ P(H 2 0 (T)). Besides, if v is a random variable with distribution µ, then E[ v 2

H 2 0 (T) ] < +∞ and for all p ∈ [1, +∞), E[ v p L p 0 (T) ] < +∞.
Let us precise that for any t ≥ 0, u(t) will always refer to an element of the space H 2 0 (T). The scalar values taken by this function are denoted by u(t, x), for x ∈ T.

Space discretisation

In order to discretise (3.1) with respect to the space variable, we first define a regular mesh T on the torus:

T := i -1 N , i N , i ∈ Z/N Z .
Averaging in (3.1) over each cell of T , we get

d N i N i-1 N u(t, x)dx = -N A u t, i N -A u t, i -1 N dt + νN ∂ x u t, i N -∂ x u t, i -1 N dt + k≥1 N i N i-1 N g k (x)dxdW k (t), i ∈ Z/N Z. (3.7)
Finite-volume schemes aim to approximate the dynamics of the average value of the solution over each cell of the mesh. This leads to the introduction of a numerical flux function A(u, v) approximating the flux of the conserved quantity at the interface between two adjacent cells. As regards the viscous term in (3.7), we replace the space derivatives by their finite difference approximations. As for the noise coefficients, we introduce the shorthand notation

σ k i := N i N i-1 N g k (x)dx, k ≥ 1, i ∈ Z/N Z.
These operations result in the following stochastic differential equation

dU i (t) = -N A (U i (t), U i+1 (t)) -A (U i-1 (t), U i (t)) dt + νN 2 (U i+1 (t) -2U i (t) + U i-1 (t)) dt + k≥1 σ k i dW k (t), i ∈ Z/N Z, t ≥ 0, (3.8)
as a semi-discrete finite-volume approximation of (3.1) in the sense that U i (t) is meant to be an approximation of the spatial average N Let us notice that the R N -valued stochastic process (W Q,N (t)) t≥0 whose components are defined by

W Q,N i (t) := k≥1 σ k i W k (t), i ∈ Z/N Z, t ≥ 0,
is a Wiener process with the covariance

E W Q,N i (t)W Q,N j (t) = t k≥1 σ k i σ k j , 60
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law which is finite as the Jensen inequality, Assumption 3.1 and (3.3) ensure that for all i ∈ Z/N Z,

k≥1 σ k i 2 2 = k≥1 N i N i-1 N g k (x)dx 2 ≤ k≥1 N i N i-1 N g k 2 L ∞ 0 (T) dx ≤ k≥1 g k 2 H 2 0 (T) ≤ D 0 . (3.9)
Furthermore, each vector σ k = (σ k 1 , . . . , σ k N ) satisfies a discrete cancellation condition:

N i=1 σ k i = N T g k (x)dx = 0.
Thus, denoting

R N 0 := {u = (u 1 , . . . , u N ) ∈ R N : u 1 + • • • + u N = 0}, we get that (W Q,N (t)) t≥0 is an R N 0 -valued process.
We equip the space R N 0 with the renormalised L p norm • p and scalar product •, • : for any u, v ∈ R N 0 and any p ∈ [1, +∞),

u p p := 1 N N i=1 |u i | p , u, v := 1 N N i=1 u i v i .
Furthermore, for any u ∈ R N 0 , we set by convention u 0 0 = 1. Besides, notice that for any 1 ≤ p ≤ q < +∞, we have

u p ≤ u q , ∀u ∈ R N 0 . (3.10)
The drift function in (3.8) is the function b defined on R N 0 by the components b

i (v) := -N A(v i , v i+1 ) -A(v i-1 , v i ) + νN 2 (v i+1 -2v i + v i-1 ), i ∈ Z/N Z.
These notations being set, we can write the SDE (3.8) in the vectorised form

dU(t) = b(U(t))dt + dW Q,N (t), t ≥ 0. (3.11)
It appears that b takes values in R N 0 . As a consequence, Equation (3.11) is conservative in the following sense: if U 0 ∈ R N 0 , then for all t ≥ 0, U(t) ∈ R N 0 . We may now state our assumptions on the numerical flux: Assumption 3.3. The function A belongs to C 1 (R 2 , R), its first derivatives ∂ 1 A and ∂ 2 A are locally Lipschitz continuous on R 2 , and it satisfies the following properties:

(i) Consistency: ∀u ∈ R, A(u, u) = A(u); (3.12) (ii) Monotonicity: ∀u, v ∈ R, ∂ 1 A(u, v) ≥ 0, ∂ 2 A(u, v) ≤ 0; (3.13)
(iii) Polynomial growth:

∃C A > 0, ∃p A ∈ N * , ∀u, v ∈ R, |∂ 1 A(u, v)| ≤ C A (1 + |u| p A ), |∂ 2 A(u, v)| ≤ C A (1 + |v| p A ). ( 3
A EO (u, v) := A(u) + A(v) 2 - 1 2 v u |A (z)|dz.

Space and time discretisation

The second stage in constructing a numerical scheme for (3.1) is the time discretisation of the SDE (3.11). Considering a time step ∆t > 0 and a positive integer n, we introduce the notation

∆W Q,N n := W Q,N (n∆t) -W Q,N ((n -1)∆t).
As it was already noticed in [START_REF] Mattingly | Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise[END_REF], explicit numerical schemes for SDEs with non-globally Lipschitz continuous coefficients do not preserve in general the large time stability, whereas implicit schemes are more robust. Therefore, since our main focus in this chapter is to approximate invariant measures, we propose the following split-step stochastic backward Euler method :

   U n+ 1 2 = U n + ∆tb U n+ 1 2 , U n+1 = U n+ 1 2 + ∆W Q,N n+1 .
(3.15)

The well-posedness of the scheme, i.e. the existence and uniqueness of the value U n+ 1 2 in the first line of (3.15), is ensured by Proposition 3.23.

Main results

Our first focus is on the large-time behaviour of the processes (U(t)) t≥0 and (U n ) n∈N . In this context, we state our first result: Theorem 3.5. Under Assumptions 3.1 and 3.3, the following two statements hold:

(i) for any N ≥ 1, the process (U(t)) t≥0 solution of the SDE (3.11) admits a unique invariant measure ν N ∈ P(R N 0 );

(ii) for any N ≥ 1 and any ∆t > 0, the process (U n ) n∈N defined by (3.15) admits a unique invariant measure ν N,∆t ∈ P(R N 0 ).

Moreover, for any N ≥ 1 and any ∆t > 0, the measures ν N and ν N,∆t belong to P 2 (R N 0 ).

The proofs for these two statements are given separately in Section 3.2. The structure of the proof is the same as for Theorem 2.7 where we derived the existence and uniqueness of an invariant measure for the solution of (3.1) from two important properties: respectively the dissipativity of the solution and an L 1 -contraction property. In Lemma 3.13 below, we show that both of these properties are preserved at the discrete level. Therefore, we will address the existence part with a tightness argument (the Krylov-Bogoliubov theorem) and the uniqueness with a coupling argument. To compare two different probability measures, we will make use of the following metric: Definition 3.6 (Wasserstein distance). Let (E, • E ) be a normed vector space and let α, β ∈ P 2 (E). The second order Wasserstein distance between α and β is defined by

W 2 (α, β) := inf π∈Π(α,β) E×E u -v 2 E dπ(u, v) 1/2
, where Π(α, β) is the set of probability measures on E × E with marginals α and β:

Π(α, β) := {π ∈ P 2 (E × E) : ∀B ∈ B (E) , π(B × E) = α(B) and π(E × B) = β(B)} .
The reader is referred to [START_REF] Villani | Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften[END_REF]Chapter 6] for further details on the Wasserstein distance, and in particular for the proof that it actually defines a distance on P 2 (E). From now on, the space P 2 (E) will be endowed with the distance W 2 : convergence of elements of P 2 (E) will always be meant in the Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law sense of the Wasserstein distance. The only cases we will address in this chapter are E = L 2 0 (T) and E = R N 0 . As a first step to approximate numerically the measure µ, we start to embed the measures ν N and ν N,∆t into P(L 2 0 (T)). For m = 0, 1, 2, let Ψ (m)

N : R N 0 → H m 0 (T) denote embedding functions in such a way that for any u ∈ R N 0 , Ψ

N u, Ψ

N u and Ψ

N u correspond respectively to piecewise constant, piecewise affine, and piecewise quadratic interpolations of the vector u on the mesh T . The functions Ψ (m) N will be precisely defined at the beginning of Section 3.3.

For m = 0, 1, 2, we define the pushforward measures µ and moreover, for any N ≥ 1, lim

∆t→0 ν N,∆t = ν N in P 2 (R N 0 ). (3.17)
In short, we have the following approximation result for all m = 0, 1, 2:

lim

N →∞ lim ∆t→0 µ (m) N,∆t = µ in P 2 (L 2 0 (T)).
Remark 3.8. In Theorem 3.7, µ is seen as a probability measure of P 2 (L 2 0 (T)) giving full weight to H 2 0 (T), as opposed to Proposition 3.2 where µ was directly seen as a measure of the space P 2 (H 2 0 (T)).

Remark 3.9 (Ergodicity). As the invariant measure µ of the process (u(t)) t≥0 is unique from Proposition 3.2, it is ergodic. In particular, a consequence of Birkhoff 's ergodic theorem (see for instance [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF]Theorem 1.2.3]) is that for any ϕ ∈ L 1 (µ) and for µ-almost every initial condition u 0 ∈ H 2 0 (T), almost surely,

lim t→∞ 1 t t 0 ϕ(u(s))ds = E [ϕ(v)] , where v ∼ µ.
By virtue of Theorem 3.5, this property also holds at the discrete level: the process (U n ) n∈N satisfies for any ϕ ∈ L 1 (ν N,∆t ) and for ν N,∆t -almost every initial condition U 0 ∈ R N 0 , almost surely,

lim n→∞ 1 n n-1 l=0 ϕ(U l ) = E [ϕ(V)] , where V ∼ ν N,∆t .
Thanks to this property, it is possible to approximate numerically expectations of functionals under the invariant measure by averaging in time the simulated process. We used this method to perform the numerical experiments presented in Chapter 4.

Review of literature

Many results are found concerning the numerical approximation in finite time of stochastic conservation laws. A particular case of interest is the stochastic Burgers equation which corresponds to the case of the flux function A(u) = u 2 /2. Finite difference schemes are presented in [START_REF] Alabert | On Numerical Approximation of Stochastic Burgers' Equation[END_REF][START_REF] Hairer | Approximations to the stochastic Burgers equation[END_REF] to approximate its solution. When the viscosity coefficient is equal to zero, the SPDE falls into a different framework. Convergence of finite-volume schemes in this hyperbolic case have been established both under the kinetic [START_REF] Dotti | Convergence of approximations to stochastic scalar conservation laws[END_REF][START_REF] Dotti | Convergence of the Finite Volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation[END_REF][START_REF] Sylvain | Numerical approximation of hyperbolic stochastic scalar conservation laws[END_REF] and the entropic formulations [START_REF] Bauzet | Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation[END_REF][START_REF] Bauzet | Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise[END_REF].

3.2. Semi-discrete and split-step schemes: well-posedness and invariant measure 63

As regards the numerical approximation of the invariant measure of an SPDE, we may start by mentioning [START_REF] Chen | Approximation of invariant measure for damped stochastic nonlinear Schrödinger equation via an ergodic numerical scheme[END_REF] concerning the damped stochastic non-linear Schrödinger equation, where a spectral Galerkin method is used for the space discretisation and a modified implicit Euler scheme for the temporal discretisation. Several works of Bréhier are devoted to the numerical approximation of the invariant measures of semi-linear SPDEs in Hilbert spaces perturbed with white noise [START_REF] Bréhier | Approximation of the invariant measure with an Euler scheme for Stochastic PDE's driven by Space-Time White Noise[END_REF][START_REF] Bréhier | Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme[END_REF][START_REF] Bréhier | High order integrator for sampling the invariant distribution of a class of parabolic stochastic PDEs with additive space-time noise[END_REF], where spectral Galerkin and semi-implicit Euler methods are used. Those results hold under a global Lipschitz assumption on the nonlinearity. In the more recent works [START_REF] Chen | A full-discrete exponential Euler approximation of the invariant measure for parabolic stochastic partial differential equations[END_REF][START_REF] Cui | Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE[END_REF], non-Lipschitz nonlinearities are considered, but they still need to satisfy a one-sided Lipschitz condition.

In the present work, our assumptions on the flux function do not imply that the non-linear term is globally Lipschitz in L 2 0 (T) nor even one-sided Lipschitz. In particular, the case of the Burgers' equation is covered. However, Equation (3.1) satisfies an L 1 -contraction property (Proposition 2.21) which may be viewed as a one-sided Lipschitz condition in the Banach space L 1 0 (T).

Outline of the chapter

The existence and uniqueness of an invariant measure for the solution of (3.11), i.e. the first part of Theorem 3.5, is proved in Section 3.2.2, and for the split-step scheme (3.15), i.e. the second part of Theorem 3.5, it is proved in Section 3.2.3. The proof of Theorem 3.7 is also split in two separate parts. The convergence in space (3.16) is proved in Section 3.3 and then, in Section 3.4, we prove the convergence with respect to the time step, i.e. Equation (3.17).

We performed numerical experiments to test the stationarity and convergence results in the Burgers case. The results of these experiments are exposed in the next chapter, where we furthermore illustrate some properties regarding the turbulent behaviour of the process in its stationary regime.

Semi-discrete and split-step schemes: well-posedness and invariant measure

Preliminary results are given in Subsection 3.2.1. In Subsection 3.2.2, we prove the well-posedness of (3.11), and after establishing some properties for the solution, we prove the existence of an invariant measure at Proposition 3.19. Then, Lemmas 3.21 and 3.22 lead to the proof of uniqueness of this invariant measure, i.e. to the proof of the first assertion of Theorem 3.5.

Notations and properties

All the lemmas stated in this section will be proved in the appendix. We define the discrete differential operators D (m)

N : R N → R N , m = 0, 1, 2, by D (0) N u i := u i , D (1)
N u i := N (u i+1 -u i ), D (2)
N u i := N 2 (u i+1 -2u i + u i-1 ), i ∈ Z/N Z.
We will often make use, in this chapter, of the summation by parts identity

D (1) N u, D (1)
N v = -D (2) N u, v , u, v ∈ R N 0 . (3.18)
These operators satisfy furthermore the following properties: then for all M ≥ 0, we have τ M ≤ T * almost surely.

Lemma 3.10 (Discrete Poincaré inequality). Let u ∈ R N . If there exist i -, i + ∈ Z/N Z such that u i -≤ 0 ≤ u i + ,
3.2. Semi-discrete and split-step schemes: well-posedness and invariant measure 65 From Dynkin's formula applied to Equation (3.11), we get for all t ≥ 0,

E U(t ∧ τ M ) 2 2 = u 0 2 2 + 2E t∧τ M 0 U(s), b(U(s)) ds + E[t ∧ τ M ] k≥1 σ k 2 2 ≤ u 0 2 2 -2νE t∧τ M 0 D (1)
N U(s)

2 2 ds + E[t ∧ τ M ] k≥1 σ k 2 2 ,
where the inequality comes from Lemma 3.13.(ii). As a consequence, using (3.9),

E U(t ∧ τ M ) 2 2 ≤ u 0 2 2 + tD 0 .
From Markov's inequality, we now derive

P(τ M ≤ t) = P U(t ∧ τ M ) 2 2 ≥ M ≤ E U(t ∧ τ M ) 2 2 M ≤ u 0 2 2 + tD 0 M -→ M →∞ 0.
As the random variable 1 τ M ≤t is almost surely non-decreasing as M increases, it admits an almost sure limit as M → +∞. From the dominated convergence theorem, this limit is actually zero:

E lim M →∞ 1 τ M ≤t = lim M →∞ P(τ M ≤ t) = 0.
As a consequence, almost surely, lim M →∞ τ M = +∞, and then T * = +∞ almost surely, meaning that (U(t)) t≥0 is a global solution of (3.11). Now, if the initial condition of (U(t)) t≥0 is an F 0 -measurable random variable U 0 distributed under some probability measure α on R N 0 , then we have

P(T * = +∞) = R N 0 P u 0 (T * = +∞)dα(u 0 ) = 1, (3.20)
where P u 0 is the conditional probability given the event U(0) = u 0 .

We now turn to the proof of existence of an invariant measure. The following lemma, Proposition 3.17, and Corollary 3.18 are preliminary results. Lemma 3.16 (Moment estimates on the semi-discrete approximation). Let p ∈ 2N * and let U 0 be an F 0 -measurable random variable such that E[ U 0 p p ] < +∞. Then, under Assumptions 3.1 and 3.3, the strong solution (U(t)) t≥0 of (3.11) with initial condition U 0 satisfies:

(i) For all t ≥ 0, E U(t) p p + νpE t 0 D (1)
N U(s) p-1 , D (1)
N U(s) ds ≤ E U 0 p p + D 0 p(p -1) 2 E t 0 U(s) p-2 p-2 ds (3.21)
where U(s) p denotes the vector (U 1 (s) p , . . . , U N (s) p ) and when p = 2, we recall the convention

• p-2 p-2 = 1.
(ii) There exist six positive constants c

(p) 0 , c (p) 1 , c (p) 2 , θ (p) 0 , θ (p) 1 and θ (p)
2 depending only on D 0 , ν and p such that we have

∀t > 0, E t 0 U(s) p p ds ≤ θ (p) 0 + θ (p) 1 E U 0 p p + θ (p) 2 t, (3.22)
and ∀T > 0, sup

t∈[0,T ] E U(t) p p ≤ c (p) 0 + c (p) 1 E U 0 p p + c (p) 2 T. (3.23)
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law Proof. Let τ M be the stopping time defined at (3.19). Applying Dynkin's formula to Equation (3.11), we get the following dynamics for the p-th order moment: for all t ≥ 0 and all M ≥ 0,

E U(t ∧ τ M ) p p = E U 0 p p -pE t∧τ M 0 N i=1 U i (s) p-1 A (U i (s), U i+1 (s)) -A (U i-1 (s), U i (s)) ds + νN pE t∧τ M 0 U(s) p-1 , D (2)
N U(s) ds + p(p -1) 2N E   t∧τ M 0 N i=1 U i (s) p-2 k≥1 σ k i 2 ds   . (3.24)
From (3.9), we have for all i = 1, . . . , N ,

k≥1 σ k i 2 ≤ D 0 .
On the other hand, the second term of the right-hand side in (3.24) is non-positive thanks to Lemma 3.12. Hence, using (3.18) in the viscous term, we get

E U(t ∧ τ M ) p p ≤ E U 0 p p -νN pE t∧τ M 0 D (1)
N U(s) p-1 , D (1)
N U(s) ds + p(p -1) 2 D 0 E t∧τ M 0 U(s) p-2 p-2 ds . (3.25)
Letting M go to +∞, applying the monotone convergence theorem on the right-hand side and Fatou's lemma on the left-hand side yields the first assertion of the lemma.

From the first assertion and Lemma 3.11, we have

4ν(p -1) p E t 0 U(s) p p ds ≤ E U 0 p p + D 0 p(p -1) 2 E t 0 U(s) p-2 p-2 ds . (3.26)
Noticing that • p-2 p-2 ≤ 1 + • p p , by (3.26) and an induction argument, we can show that for all p ∈ 2N * , (3.22) holds. Now, from the first assertion once again and (3.22), we have for all p ∈ 2N * and all 0 ≤ t ≤ T ,

E U(t) p p ≤ E U 0 p p + D 0 p(p -1) 2 E T 0 U(s) p-2 p-2 ds ≤ E U 0 p p + D 0 p(p -1) 2 θ (p-2) 0 + θ (p-2) 1 E U 0 p-2 p-2 + θ (p-2) 2 T ≤ E U 0 p p + D 0 p(p -1) 2 θ (p-2) 0 + θ (p-2) 1 1 + E U 0 p p + θ (p-2) 2 T =: c (p) 0 + c (p) 1 E U 0 p p + c (p) 2 T.
Since the right-hand side does not depend on t, we get (3.23).

Proposition 3.17 (L 1 -contraction). Under Assumptions 3.1 and 3.3, two strong solutions (U(t)) t≥0 and (V(t)) t≥0 of (3.11) (driven by the same Wiener process W Q,N ) with possibly different initial conditions satisfy almost surely:

U(t) -V(t) 1 ≤ U(s) -V(s) 1 , 0 ≤ s ≤ t.
3.2. Semi-discrete and split-step schemes: well-posedness and invariant measure 67

Proof. Since (U(t)) t≥0 and (V(t)) t≥0 are driven by the same Wiener process, then (U(t) -V(t)) t≥0 is an absolutely continuous process:

d(U(t) -V(t)) = (b(U(t)) -b(V(t))) dt.
In particular, we can write for all t ≥ 0,

d dt U(t) -V(t) 1 = sign (U(t) -V(t)) , b(U(t)) -b(V(t)) ≤ 0,
where the inequality comes from Lemma 3.13.(i), and the result follows by integrating in time.

This last property ensures the following result which we state without a proof:

Corollary 3.18 (Feller property). Under Assumption 3.3, the strong solution (U(t)) t≥0 of Equation (3.11) satisfies the Feller property, i.e. for any continuous and bounded function ϕ : R N 0 → R and any t ≥ 0, the mapping

u 0 ∈ R N 0 -→ E u 0 [ϕ(U(t))
] ∈ R is continuous and bounded, where E u 0 is the conditional expectation given the event U(0) = u 0 . Proposition 3.19 (Existence of an invariant measure for the semi-discrete scheme). Under Assumptions 3.1 and 3.3, the strong solution (U(t)) t≥0 of (3.11) admits an invariant measure ν N ∈ P(R N 0 ). Moreover, for all p ∈ [1, +∞), there exists a constant C 0,p not depending on N such that if V is a random variable with distribution ν N , then

E V p p ≤ C 0,p .
Proof. Let (U(t)) t≥0 be the solution of (3.11) with a deterministic initial condition u 0 ∈ R N 0 . From Lemma 3.16.(ii), we have for all t > 0 and all p ∈ 2N * ,

1 t t 0 E u 0 U(s) p p ds ≤ 1 t θ (p) 0 + 1 t θ (p) 1 u 0 2 2 + θ (p) 2 . (3.27)
Let us take p = 2. Applying the Markov inequality and taking the limit superior in t, we get for all ε > 0, lim sup

t→∞ 1 t t 0 P u 0 U(s) 2 2 > 1 ε ds ≤ εθ (2) 2 .
Since from Corollary 3.18, (U(t)) t≥0 is Feller, the existence of an invariant measure ν

N ∈ P(R N 0 ) for (U(t)) t≥0 is now a consequence of the Krylov-Bogoliubov theorem [36, Corollary 3.1.2].
Let V be a random variable with distribution ν N . We will derive now from Equation (3.27) that V has finite moments. A computation of the same kind as the one below is found for instance in the proof of [START_REF] Hairer | Ergodic Properties of Markov Processes[END_REF]Proposition 4.24]. For any M > 0 and any p ∈ 2N * ,

E V p p ∧ M = 1 t t 0 R N 0 E u 0 U(s) p p ∧ M dν N (u 0 )ds = R N 0 1 t t 0 E u 0 U(s) p p ∧ M dsdν N (u 0 ) ≤ R N 0 1 t t 0 E u 0 U(s) p p ds ∧ M dν N (u 0 ) ≤ R N 0 1 t θ (p) 0 + θ (p) 1 u 0 p p t + θ (p) 2 ∧ M dν N (u 0 ).
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law Now, letting t → +∞, we get from the dominated convergence theorem,

E V p p ∧ M ≤ R N 0 lim t→∞ 1 t θ (p) 0 + θ (p) 1 u 0 p p t + θ (p) 2 ∧ M dν N (u 0 ) = θ (p) 2 ∧ M ≤ θ (p) 2 ,
and the result for p ∈ 2N * follows by letting M → +∞ and using the monotone convergence theorem. This result extends readily to the general case p ∈ [1, +∞) by using for instance the Jensen inequality.

Corollary 3.20. Under Assumptions 3.1 and 3.3, let ν N be an invariant measure for the solution (U(t)) t≥0 of (3.11) and let V be a random variable with distribution ν N . Then, for all p ∈ 2N * , V satisfies

E D (1)
N V p-1 , D (1)
N V ≤ D 0 (p -1) 2ν C 0,p-2 ,
where we set C 0,0 = 1.

Proof. Let (U(t)) t≥0 be a solution of (3.11)

N V p-1 , D (1)
N V ≤ D 0 p(p -1) 2 E V p-2 p-2 ,
and it remains to apply Proposition 3.19 to conclude.

We now turn to the proof of uniqueness of the invariant measure ν N . We will first need some preliminary results: Lemma 3.21. Let Assumptions 3.1 and 3.3 hold and let (U(t)) t≥0 and (V(t)) t≥0 be two strong solutions of (3.11) driven by the same Wiener process. Then, for all M > 0 and all ε > 0, there exists t ε,M > 0 such that

p ε,M := inf u 0 2 ∨ v 0 2 ≤M P (u 0 ,v 0 ) U(t ε,M ) 1 + V(t ε,M ) 1 ≤ ε > 0.
Proof. We recall that b : R N 0 → R N 0 is locally Lipschitz continuous (for every norm over R N 0 ). Let M > 0 and ε > 0. Let us also fix the deterministic values u 0 , v 0 ∈ R N 0 satisfying u 0 2 ∨ v 0 2 ≤ M , along with the following constants:

t ε,M := - 1 2ν log ε 2 16M 2 ; L M +ε := Lipschitz constant of b over the ball u ∈ R N 0 : u 1 ≤ M + ε ; δ ε := ε 4 e -L M +ε t ε,M .
Let (U(t)) t≥0 and (V(t)) t≥0 denote two solutions of (3.11) with the initial conditions u 0 and v 0 . We introduce the stopping times

τ U := inf {t ≥ 0 : U(t) 1 ≥ M + ε} ; τ V := inf {t ≥ 0 : V(t) 1 ≥ M + ε} .
Furthermore, we denote by (u(t)) t≥0 and (v(t)) t≥0 the noiseless counterparts of (U(t)) t≥0 and (V(t)) t≥0 :

d dt u(t) = b (u(t)) , d dt v(t) = b (v(t)) , 3.2.
Semi-discrete and split-step schemes: well-posedness and invariant measure 69 with respective initial conditions u 0 and v 0 . By the dissipativity property (Lemma 3.13.(ii)) and Lemma 3.10, we have

d dt u(t) 2 2 + v(t) 2 2 = 2 ( u(t), b (u(t)) + v(t), b (v(t)) ) ≤ -2ν D (1)
N u(t)

2 2 + D (1)
N v(t)

2 2 ≤ -2ν u(t) 2 2 + v(t) 2 2 ,
so that Grönwall's lemma yields the upper bound

u(t) 2 2 + v(t) 2 2 ≤ u 0 2 2 + v 0 2 2 e -2νt ,
meaning that for t ≥ t ε,M , we have

u(t) 2 2 + v(t) 2 2 ≤ ε 2 8 ,
and consequently, by (3.10),

u(t) 1 + v(t) 1 ≤ u(t) 2 + v(t) 2 ≤ ε 2 .
We now restrict ourselves to the situation where

ω ∈ sup t∈[0,t ε,M ] W Q,N (t) 1 ≤ δ ε .
For any t ≤ τ U ∧ τ V ∧t ε,M , the four vectors U(t), V(t), u(t) and v(t) stay in the ball { • 1 ≤ M +ε}, and thanks to the local Lipschitz continuity assumption on b we have

U(t) -u(t) 1 + V(t) -v(t) 1 = t 0 (b (U(s)) -b (u(s))) ds + W Q,N (t) 1 + t 0 (b (V(s)) -b (v(s))) ds + W Q,N (t) 1 ≤ t 0 ( b (U(s)) -b (u(s)) 1 + b (V(s)) -b (v(s)) 1 ) ds + 2 W Q,N (t) 1 ≤ L M +ε t 0 ( U(s) -u(s) 1 + V(s) -v(s) 1 ) ds + 2δ ε ,
so by Grönwall's lemma, we have

U(t) -u(t) 1 + V(t) -v(t) 1 ≤ 2δ ε e L M +ε t ≤ 2δ ε e L M +ε t ε,M = ε 2 , (3.28)
for every t ∈ [0, τ U ∧ τ V ∧ t ε,M ]. But it appears that the case τ U ∧ τ V < t ε,M is impossible for small values of ε. Indeed, it would either imply (U -u)( τ U ) 1 ≤ ε/2 or (V -v)( τ V ) 1 ≤ ε/2
which is impossible because we have on the one hand

U τ U 1 ≥ M + ε or V τ V 1 ≥ M + ε ,
and on the other hand

u τ U 1 ≤ u τ U 2 ≤ u 0 2 ≤ M or v τ V 1 ≤ M .
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law Therefore, Inequality (3.28) holds for all t ∈ [0, t ε,M ]. Thus,

U(t ε,M ) 1 + V(t ε,M ) 1 ≤ U(t ε,M ) -u(t ε,M ) 1 + V(t ε,M ) -v(t ε,M ) 1 + u(t ε,M ) 1 + v(t ε,M ) 1 ≤ ε,
and we have just shown that

sup t∈[0,t ε,M ] W Q,N (t) 1 ≤ δ ε ⊂ U(t ε,M ) 1 + V(t ε,M ) 1 ≤ ε .
and therefore,

P (u 0 ,v 0 ) U(t ε,M ) 1 + V(t ε,M ) 1 ≤ ε ≥ P sup t∈[0,t ε,M ] W Q,N (t) 1 ≤ δ ε .
Notice that the right-hand side does not depend on u 0 nor v 0 . Furthermore, it is positive since W Q,N is an R N -valued Wiener process. Hence, taking the infimum over u 0 and v 0 on the left-hand side yields the wanted result.

For two solutions (U(t)) t≥0 and (V(t)) t≥0 of (3.11) driven by the same Wiener process W Q,N , we define the following entrance time for all M ≥ 0:

τ M := inf {t ≥ 0 : U(t) 2 ∨ V(t) 2 ≤ M } . (3.29)
Lemma 3.22. Under Assumptions 3.1 and 3.3, there exists M > 0 such that for any deterministic initial conditions u 0 , v 0 ∈ R N 0 , τ M < +∞ almost surely.

Proof. From Itô's formula, we have for all t ≥ 0,

U(τ M ∧ t) 2 2 + V(τ M ∧ t) 2 2 = U 0 2 2 + V 0 2 2 + τ M ∧t 0 b(U(s)), U(s) ds+ τ M ∧t 0 b(V(s)), V(s) ds + k≥1 τ M ∧t 0 U(s) + V(s), σ k dW k (s) + 2 k≥1 τ M ∧t 0 σ k 2 2 ds. (3.30)
The fifth term of the right-hand side is a martingale. Indeed, by the Cauchy-Schwarz inequality, Inequality (3.9), and the bound (3.22), we have

E   k≥1 τ M ∧t 0 U(s) + V(s), σ k 2 ds   ≤   k≥1 σ k 2 2   E t 0 U(s) + V(s) 2 2 ds ≤ 2D 0 E t 0 U(s) 2 2 ds + E t 0 V(s) 2 2 ds ≤ 2D 0 2θ (2) 0 + θ (2) 1 u 0 2 2 + v 0 2 2 + 2θ (2)
2 t < +∞.

Thus, taking the expectation in (3.30), applying Lemma 3.13.(ii), Inequality (3.9), Lemma 3.10 3.2. Semi-discrete and split-step schemes: well-posedness and invariant measure 71 and (3.29), we get

E U(τ M ∧ t) 2 2 + V(τ M ∧ t) 2 2 = u 0 2 2 + v 0 2 2 + 2E τ M ∧t 0 ( b (U(s)) , U(s) + b (V(s)) , V(s) ) ds + 2E   τ M ∧t 0 k≥1 σ k 2 2 ds   ≤ u 0 2 2 + v 0 2 2 -2νE τ M ∧t 0 D (1)
N U(s)

2 2 + D (1)
N U(s)

2 2 ds + 2E [τ M ∧ t] D 0 ≤ u 0 2 2 + v 0 2 2 -2νE τ M ∧t 0 U(s) 2 2 + V(s) 2 2 ds + 2E [τ M ∧ t] D 0 ≤ u 0 2 2 + v 0 2 2 + 2 D 0 -νM 2 E [τ M ∧ t] So if we choose M > D 0 /ν, we get E[τ M ∧ t] ≤ u 0 2 2 + v 0 2 2
2 (νM 2 -D 0 ) , so that we can apply the monotone convergence theorem:

E[τ M ] = lim t→∞ E[τ M ∧ t] < +∞,
which concludes the proof.

Proof of Theorem 3.5, Assertion (i). We start to fix ε > 0 to which we associate the quantities t ε,M and p ε,M defined at Lemma 3.21, where M has been defined at Lemma 3.22. Let (U(t)) t≥0 and (V(t)) t≥0 start respectively from arbitrary deterministic initial conditions u 0 and v 0 and be driven by the same Wiener process. We define the increasing stopping time sequence

T 1 := τ M T 2 := inf {t ≥ T 1 + t ε,M : U(t) 2 ∨ V(t) 2 ≤ M } T 3 := inf {t ≥ T 2 + t ε,M : U(t) 2 ∨ V(t) 2 ≤ M } . . .
By the strong Markov property and Lemma 3.22, each term of this sequence is finite almost surely. We claim that

∀J ∈ N * , P ∀j = 1, . . . , J, U(T j + t ε,M ) 1 + V(T j + t ε,M ) 1 > ε ≤ (1 -p ε,M ) J . (3.31)
Indeed, it is true for J = 1 thanks to the strong Markov property and Lemma 3.21:

P U(τ M + t ε,M ) 1 + V(τ M + t ε,M ) 1 > ε = E P U(τ M + t ε,M ) 1 + V(τ M + t ε,M ) 1 > ε|F τ M ≤ 1-p ε,M ,
and the general case follows by induction: assuming that Inequality (3.31) is true for some J ∈ N * , we have

P ∀j = 1, . . . , J + 1, U(T j + t ε,M ) 1 + V(T j + t ε,M ) 1 > ε = E P ∀j = 1, . . . , J + 1, U(T j + t ε,M ) 1 + V(T j + t ε,M ) 1 > ε|F T J+1 = E     J j=1 1 U(T j +t ε,M ) 1 + V(T j +t ε,M ) 1 >ε   P U(T J+1 + t ε,M ) 1 + V(T J+1 + t ε,M ) 1 > ε|F T J+1   ≤ (1 -p ε,M ) J × (1 -p ε,M ) = (1 -p ε,M ) J+1 .
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law

Letting J → +∞, we get

P ∀j ∈ N * , U(T j + t ε,M ) 1 + V(T j + t ε,M ) 1 > ε = lim J→∞ P ∀j = 1, . . . , J, U(T j + t ε,M ) 1 + V(T j + t ε,M ) 1 > ε ≤ lim J→∞ (1 -p ε,M ) J = 0,
and consequently, P (∃t ≥ 0,

U(t) 1 + V(t) 1 ≤ ε) = 1, meaning that almost surely, ∃t ≥ 0, U(t) -V(t) 1 ≤ ε.
Now recall that thanks to Proposition 3.17, U(t) -V(t) 1 is non-increasing in time almost surely. Since ε has been chosen arbitrarily, the above inequality actually indicates that U(t) -V(t) 1 converges almost surely to 0 as t → +∞ when the initial conditions are deterministic. However, this almost sure consergence extends naturally to random and F 0 -measurable initial conditions using the same argument as for (3.20). Let φ : R N 0 → R be a Lipschitz continuous and bounded test function, with Lipschitz constant L φ . We have in particular, almost surely,

lim t→∞ |φ(U(t)) -φ(V(t))| ≤ L φ lim t→∞ U(t) -V(t) 1 = 0. (3.32)
To conclude the proof, assume that there exist two invariant measures ν

N and ν

N for the solution of (3.11), and take random initial conditions U 0 and V 0 with distributions ν

N and ν

N respectively. We have for all t ≥ 0,

|E [φ (U 0 )] -E [φ (V 0 )]| = |E [φ (U(t))] -E [φ (V(t))]| ≤ E [|φ (U(t)) -φ (V(t))|] .
Letting t go to +∞, by (3.32) and the dominated convergence theorem, we have

|E [φ (U 0 )] -E [φ (V 0 )]| ≤ lim t→∞ E [|φ (U(t)) -φ (V(t))|] = 0.
As a consequence, U 0 and V 0 have the same distribution, meaning that ν

(1) N = ν (2) N .

Invariant measure for the split-step scheme

In this subsection, we aim to prove the existence and uniqueness of an invariant measure for the discrete time process (U n ) n∈N defined by (3.15). The general argument is the same as the one used in Subsection 3.2.2 for the semi-discrete case and the intermediary results are stated in the same order. Therefore, the proofs which are not affected by the time discretisation are omitted.

As the time step ∆t is meant to converge towards 0, we may consider that it will always lie in an interval (0, ∆t max ] for some arbitrarily chosen ∆t max > 0.

The following preliminary result ensures that the scheme (3.15) is well-posed.

Proposition 3.23. Under Assumption 3.3, given ∆t > 0 and v ∈ R N 0 , there exists a unique w ∈ R N 0 such that w = v + ∆tb(w).

Proof. Uniqueness. It is a straightforward consequence of Lemma 3.13.(i): if w 1 and w 2 are two solutions, then

w 1 -w 2 1 = sign(w 1 -w 2 ), w 1 -w 2 = ∆t sign(w 1 -w 2 ), b(w 1 ) -b(w 2 ) ≤ 0.
3.2. Semi-discrete and split-step schemes: well-posedness and invariant measure 73

Existence. The mapping Id -∆tb : R N 0 → R N 0 is continuous. Furthermore, by Lemmas 3.13.(ii) and 3.10, we have for all w ∈ R N 0 ,

(Id -∆tb)(w), w w 2 = w 2 -∆t b(w), w w 2 ≥ w 2 + ν∆t D (1) N w 2 2 w 2 ≥ (1 + ν∆t) w 2 .
Thus, as a consequence of [42, Theorem 3.3], Id -∆tb is surjective in R N 0 and, for any v ∈ R N 0 , there exists w ∈ R N 0 such that w = v + ∆tb(w).

Lemma 3.24 (L 1 -contraction). Let Assumptions 3.1 and 3.3 hold and let (U n ) n∈N and (V n ) n∈N be two solutions of (3.15) (driven by the same Wiener process W Q,N ). Then, almost surely and for any

n ∈ N, U n+1 -V n+1 1 ≤ U n -V n 1 .
Proof. From Equations (3.15) and Lemma 3.13.(ii), we write

U n+1 -V n+1 1 = U n+ 1 2 -V n+ 1 2 1 = sign U n+ 1 2 -V n+ 1 2 , U n+ 1 2 -V n+ 1 2 = sign U n+ 1 2 -V n+ 1 2 , U n -V n + ∆t sign U n+ 1 2 -V n+ 1 2 , b U n+ 1 2 -b V n+ 1 2 ≤ sign U n+ 1 2 -V n+ 1 2 , U n -V n ≤ U n -V n 1 .
Remark 3.25. Note that the choice of the split-step backward Euler scheme is essential for the L 1contraction property to hold. Indeed, consider for instance two processes ( U n ) n∈N and ( V n ) n∈N built via a explicit Euler method, that is,

U n+1 = U n + ∆tb U n + ∆W Q,N n+1
(and naturally, the same construction for ( V n ) n∈N ), then the expansion of the L 1 distance gives

U n+1 -V n+1 1 = sign U n+1 -V n+1 , U n -V n +∆t sign U n+1 -V n+1 , b U n -b V n .
Thus, we would need to control the second term of the right-hand side in the above equation, which is delicate given that b is not globally Lipschitz.

As for the semi-discrete scheme, Lemma 3.24 induce the following property: (1)

N V 1 2 2 2 ≤ D 0 2ν , (3.34)
where

V 1 2 denotes the solution of V 1 2 = V + ∆tb(V 1 2
).

Proof. Let u 0 ∈ R N 0 be the deterministic initial condition of the process (U n ) n∈N . Starting from the first equation in (3.15), we have

U n+ 1 2 -∆tb U n+ 1 2 2 2 = U n 2 2 ,
by expanding the left-hand side, we derive the inequality

U n+ 1 2 2 2 ≤ U n 2 2 + 2∆t b U n+ 1 2 , U n+ 1 2 .
Using the dissipativity inequality (Lemma 3.13.(ii)), we get

U n+ 1 2 2 2 ≤ U n 2 2 -2ν∆t D (1) N U n+ 1 2 2 2 .
(3.35)

Now, from the second equation in (3.15), we have

U n+1 2 2 = U n+ 1 2 2 2 + 2 U n+ 1 2 , ∆W Q,N n+1 + ∆W Q,N n+1 2 2 . (3.36)
Injecting Inequality (3.35) into Equation (3.36), we get

U n+1 2 2 -U n 2 2 ≤ -2ν∆t D (1)
N U n+ 1 2 2 2 + 2 U n+ 1 2 , ∆W Q,N n+1 + ∆W Q,N n+1 2 2
.

(3.37)

By definition of W Q,N and from (3.9), we have

E ∆W Q,N n+1 2 2 = 1 N ∆t N i=1 k≥1 σ k i 2 ≤ D 0 ∆t. (3.38)
On the other hand, the variables U n+ 1 2 and ∆W Q,N n+1 are independent, so that taking the expectation in (3.37) yields

E U n+1 2 2 -E U n 2 2 ≤ -2ν∆tE D (1) N U n+ 1 2 2 2 + D 0 ∆t,
which is valid for any n ∈ N, so that we can get a telescopic sum:

E U n 2 2 -u 0 2 2 = n-1 l=0 E U l+1 2 2 -E U l 2 2 ≤ -2ν∆t n-1 l=0 E D (1) N U l+
N ∆W Q,N l+1 2 2 = N E   N i=1   k≥1 σ k i+1 -σ k i W k ((l + 1)∆t) -W k (l∆t)   2   = N N i=1 k≥1 σ k i+1 -σ k i 2 E W k ((l + 1)∆t) -W k (l∆t) 2 = ∆t k≥1 D (1)
N σ k 2 2 . (1)
Now, from the definition of σ k , the Jensen inequality and (3.5), we have

D (1) N σ k 2 2 = N N i=1 k≥1 N i N i-1 N g k x + 1 N -g k (x) dx 2 ≤ N 2 N i=1 k≥1 i N i-1 N g k x + 1 N -g k (x) 2 dx = N 2 N i=1 k≥1 i N i-1 N x+ 1 N x ∂ x g k (y)dy 2 dx ≤ N 2 N i=1 i N i-1 N 1 N x+ 1 N x k≥1 ∂ x g k (y) 2 dydx ≤ D 0 . ( 3
N U l+1 2 2 ≤ 1 2nν∆t u 0 2 2 + D 0 2ν + ∆tD 0 . (3.44) Since D (1)
N • 2 defines a norm on R N 0 and since from Corollary 3.26, the process (U n ) n∈N is Feller, the result follows from Markov's inequality and the Krylov-Bogoliubov theorem [START_REF] Da | Ergodicity for Infinite Dimensional Systems[END_REF]Theorem 3.1.1].

Using the same arguments as for the end of the proof of Proposition 3.19, Inequalities (3.44) and (3.40) yield respectively (3.33) and (3.34).

We now proceed to the proof of uniqueness of the invariant measure ν N,∆t . Lemma 3.28 (Hitting any neighbourhood of 0 with positive probability). Let Assumptions 3.1 and 3.3 hold. Let (U n ) n∈N and (V n ) n∈N be two solutions of (3.15) driven by the same Wiener process W Q,N . For any ε > 0 and any M > 0, there exists n ε,M ∈ N such that p ε,M := inf

u 0 2 ∨ v 0 2 ≤M P (u 0 ,v 0 ) U n ε,M 1 + V n ε,M 1 ≤ ε > 0.
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law Proof. First, let ε > 0 and let us fix u 0 , v 0 ∈ R N 0 such that u 0 2 ≤ M and v 0 2 ≤ M . Let (u n ) n∈N and (v n ) n∈N denote the noiseless counterparts of the processes (U n ) n∈N and (V n ) n∈N , i.e.

u n+1 = u n + ∆tb (u n+1 ) v n+1 = v n + ∆tb (v n+1 ) , (3.45)
with initial conditions u 0 and v 0 . Then (u n ) n∈N and (v n ) n∈N are subject to non-perturbed dissipativity, and consequently the sum of their energies decreases to 0 over time. Indeed, we have

u n 2 2 + v n 2 2 = u n+1 -∆tb (u n+1 ) 2 2 + v n+1 -∆tb (v n+1 ) 2 2 = u n+1 2 2 + v n+1 2 2 + (∆t) 2 b (u n+1 ) 2 2 + b (v n+1 ) 2 2 -2∆t ( u n+1 , b (u n+1 ) + v n+1 , b (v n+1 ) )
therefore, using successively Lemma 3.13.(ii) and Lemma 3.10,

u n+1 2 2 + v n+1 2 2 -u n 2 2 + v n 2 2 ≤ 2∆t ( u n+1 , b (u n+1 ) + v n+1 , b (v n+1 ) ) ≤ -2∆tν D (1) N u n+1 2 2 + D (1)
N v n+1 2 2 ≤ -2∆tν u n+1 2 2 + v n+1 2 2 so that u n+1 2 2 + v n+1 2 2 ≤ 1 1 + 2∆tν u n 2 2 + v n 2 2
, by induction, we get for all n ∈ N,

u n 2 2 + v n 2 2 ≤ 1 1 + 2∆tν n u 0 2 2 + v 0 2 2 .
It appears now that if we fix the value

n ε,M := -1 log(1 + 2∆tν) log ε 2 16M 2 ,
we get for all n ≥ n ε,M ,

u n 1 + v n 1 ≤ u n 2 + v n 2 ≤ ε 2 .
Now, we fix δ ε := ε/(4n ε,M ) and we restrict ourselves to the event

sup n=1,...,n ε,M ∆W Q,N n 1 ≤ δ ε . (3.46)
Let (U n ) n∈N and (V n ) n∈N be two solutions of (3.15) with the deterministic initial conditions u 0 and v 0 respectively. With similar arguments as for the proof of Proposition 3.17, we get from (3.15), (3.45) and Lemma 3.13.(ii), for all n ∈ N,

U n+1 -u n+1 1 + V n+1 -v n+1 1 ≤ U n+ 1 2 -u n+1 1 + V n+ 1 2 -v n+1 1 + 2 ∆W Q,N n+1 1 = sign U n+ 1 2 -u n+1 , U n -u n + ∆t sign U n+ 1 2 -u n+1 , b U n+ 1 2 -b (u n+1 ) + sign V n+ 1 2 -v n+1 , V n -v n + ∆t sign V n+ 1 2 -v n+1 , b V n+ 1 2 -b (v n+1 ) + 2 ∆W Q,N n+1 1 ≤ U n -u n 1 + V n -v n 1 + 2 ∆W Q,N n+1 1 .
On the event (3.46), we have for all n = 1, . . . , n ε,M ,

U n+1 -u n+1 1 + V n+1 -v n+1 1 ≤ U n -u n 1 + V n -v n 1 + 2δ ε .
In particular, by induction, we have

U n ε,M -u n ε,M 1 + V n ε,M -v n ε,M 1 ≤ 2n ε,M δ ε = ε 2 .
Thus,

U n ε,M 1 + V n ε,M 1 ≤ U n ε,M -u n ε,M 1 + V n ε,M -v n ε,M 1 + u n ε,M 1 + v n ε,M 1 ≤ ε 2 + ε 2 = ε.
We just have shown that

P (u 0 ,v 0 ) U n ε,M 1 + V n ε,M 1 ≤ ε ≥ P sup n=1,...,n ε,M ∆W Q,N n 1 ≤ δ ε > 0.
Since the event (3.46) does not depend on u 0 nor v 0 , we get the result.

For two solutions (U n ) n∈N and (V n ) n∈N of (3.15) driven by the same Wiener process W Q,N , we define the entrance time

η M := inf {n ∈ N : U n+1 2 ∨ V n+1 2 ≤ M } .
The following lemma is the time-discrete version of Lemma 3.22. The proof is omitted as it is very similar to its time-continuous counterpart. Lemma 3.29 (Almost sure entrance in some ball). Under Assumptions 3.1 and 3.3, there exists M > 0 such that for any initial conditions u 0 , v 0 ∈ R N 0 for the processes (U n ) n∈N and (V n ) n∈N , η M < +∞ almost surely.

Proof of Theorem 1, Assertion (ii). Given Lemmas 3.24, 3.28 and 3.29, the proof is done in exactly the same way as for Assertion (i).

Finally, the fact that ν N and ν N,∆t belong to P 2 (R N 0 ) come from Propositions 3.19 and 3.27 respectively (for ν N,∆t , we use in particular the fact that D

Convergence of invariant measures: semi-discrete scheme towards SPDE

The purpose of this section is to prove that W 2 (µ

(m)
N , µ) -→ 0, N → +∞, m = 0, 1, 2, which will be the first part of the proof of Theorem 3.7. In Subsection 3.3.1, we provide a result ensuring that it is sufficient that the convergence holds for only one m ∈ {0, 1, 2}, in which case it will hold for the three of them. Then, we show that (µ (m) N ) N ≥1 is relatively compact in P 2 (L 2 0 (T)), and in Subsection 3.3.2, we present a procedure to identify any subsequential limit of (µ (m) N ) N ≥1 as the invariant measure µ for the solution of (3.1), which leads to the proof the first assertion of Theorem 3.7. Subsection 3.3.3 contain the proofs of the lemmas from Subsections 3.3.1 and 3.3.2.

Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law

Notations and preliminary results

For m = 0, 1, 2, we define the interpolation operators Ψ (m)

N : R N 0 → W m,∞ 0 (T) by Ψ (m) N v(x) = N i=1 v i φ (m) N x - i N , v = (v 1 , . . . , v N ) ∈ R N 0 ,
where φ (0)

N (x) = 1 (-1 N ,0] (x), φ (1)
N (x) = N x + 1 N 1 (-1 N ,0] (x) + N 1 N -x 1 (0, 1 N ] (x), φ (2)
N (x) = N 2 2 x + 1 N x + 2 N 1 (-2 N ,-1 N ] (x) -N 2 x + 1 N x - 1 N 1 (-1 N ,0] (x) + N 2 2 x - 1 N x - 2 N 1 (0, 1 N ] (x), so that Ψ (0) N v, Ψ (1)
N v and Ψ

N v are the respectively piecewise constant, linear and quadratic interpolations of the values v i at the points i/N . In this regard, note that for v ∈ R N 0 , i ∈ Z/N Z and m = 0, 1, 2, we have

Ψ (m) N v( i N ) = v i .
We recall that these operators allowed to define the sequences of embedded invariant measures µ

(m) N = ν N • (Ψ (m) N ) -1 , where µ (m)
N is here considered as an element of P(L 2 0 (T)). We prove the following lemma in the appendix:

N v W m,p 0 (T) = D (m) N v p , (ii) for any v ∈ R N 0 , Ψ (m)
N v -Ψ (0) N v 2 L 2 0 (T) = 1 3N 2 D (1) N v 2 2 (1)
and Ψ

(2)

N v -Ψ (0) N v 2 L 2 0 (T) ≤ 3 20N 4 D (2) N v 2 2 + 1 2N 2 D (1) N v 2 2 .
The proof of the following Lemma is given below in Subsection 3.3.3. Lemma 3.31 (Discrete H 1 0 and H 2 0 bounds). Let Assumptions 3.1 and 3.3 hold and let V be a random variable in R N 0 with distribution ν N , then V satisfies

E D (1) N V 2 2 ≤ D 0 2ν =: C 1,2 . (3.47)
Furthermore, there exists a positive constant C 2,2 not depending on N such that,

E D (2) N V 2 2 ≤ C 2,2 .
(3.48)

3.3. Convergence of invariant measures: semi-discrete scheme towards SPDE

79

The following result ensures in particular that the estimates obtained at Lemma 3.31 and Proposition 3.19 remain true when passing to the limit N → +∞. Lemma 3.32 (Relative compactness). Under Assumptions 3.1 and 3.3, the three families of probability measures (µ (m) N ) N ≥1 , m = 0, 1, 2, are relatively compact in the space P 2 (L 2 0 (T)). Moreover, for any m = 0, 1, 2 and for any subsequential limit µ * ∈ P 2 (L 2 0 (T)) of (µ

(m) N ) N ≥1 , a random variable v ∼ µ * satisfies E v 2 H 1 0 (T) ≤ C 1,2 , E v 2 H 2 0 (T) ≤ C 2,2 and E v p L p 0 (T) ≤ C 0,p , p ∈ [1, +∞). (3.49) Proof.
Step 1. Relative compactness of (µ

(1) N ) N ≥1 in P(L 2 0 (T)). Let V be an R N 0 -valued random variable with distribution ν N . Then, Ψ (1)
N V has distribution µ (1)
N . Thanks to Lemmas 3.30 and 3.31, we have

E Ψ (1) N V 2 H 1 0 (T) = E D (1) N V 2 2 ≤ C 1,2 .
Thus, Markov's inequality implies ∀ε > 0, P Ψ

(1)

N V 2 H 1 0 (T) > 1 ε ≤ εC 1,2 .
The space H 1 0 (T) is compactly embedded in L 2 0 (T), so this last inequality means that the sequence (µ

N ) N ∈N * is tight in the space P(L 2 0 (T)). As a consequence of Prokhorov's theorem [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 5.1], any subsequence of (µ

N ) N ≥1 admits itself a weakly converging subsequence in P(L 2 0 (T)). In this respect, let µ * ∈ P(L 2 0 (T)) be a subsequential limit of (µ

N ) N ≥1 and let (µ

N j ) j∈N be the associated subsequence.

Step 2. Relative compactness of (µ

(0) N ) N ≥1 in P 2 (L 2 0 (T)).
Let v be a random variable with distribution µ * . On the one hand, the sequence of random variables (Ψ (1) N j V) j∈N converges in distribution towards v. On the other hand, by Lemmas 3.30.(ii) and 3.31, we have

E Ψ (0) N j V -Ψ (1) N j V 2 L 2 0 (T) ≤ 1 3N 2 j E D (1) N j V 2 2 ≤ C 1,2 3N 2 j -→ j→∞ 0, so that Ψ (0) N j V -Ψ (1)
N j V converges in probability towards 0 as j → +∞. As a consequence, by Slutsky's theorem [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 3.9], the couple (Ψ (1)

N j V, Ψ (0) N j V -Ψ (1)
N j V) converges in distribution towards (v, 0) as j → +∞. In particular, Ψ (0)

N j V converges in distribution towards v, which means that µ (0) N j converges weakly in P(L 2 0 (T)) towards µ * . Moreover, µ (0)
N j has uniform moment bounds with respect to j thanks to (3.2), Lemma 3.30.(i) and Proposition 3.19: N j ) j∈N also converge towards µ * in P 2 (L 2 0 (T)) by Lemmas 3.30 and 3.31. Indeed, we have

E Ψ (0) N j V p L 2 0 (T) ≤ E Ψ (0) N j V p L p 0 (T) = E V p p ≤ C 0,p , ∀p ≥ 2. ( 3
W 2 µ (1) N j , µ (0) N j 2 ≤ E Ψ (1) N j V -Ψ (0) N j V 2 L 2 0 (T) ≤ C 1,2 3N 2 j , (3.51)
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law and W 2 µ

(2)

N j , µ (0) N j 2 ≤ E Ψ (2) N j V -Ψ (0) N j V 2 L 2 0 (T) ≤ E 3 20N 4 j D (2) N j V 2 2 + 1 2N 2 j D (1) N j V 2 2 ≤ 3C 2,2 20N 4 j + C 1,2 2N 2 j • (3.52)
Step 4. Moment estimates. Finally, the estimates (3.49) follow from Portemanteau's theorem: since • 2 H 1 0 (T) is lower semi-continuous on the space L 2 0 (T), we have

E v 2 H 1 0 (T) ≤ lim inf j→∞ E Ψ (1)
N j V 2 H 1 0 (T) ≤ C 1,2 ,
and the same argument applies for • 2 H 2 0 (T) and • p L p 0 (T) using respectively the sequences of random variables (Ψ (2)

N j V) j∈N and (Ψ (0) N j V) j∈N .
The three following lemmas will be useful for the proof of finite time convergence stated in the next subsection, namely Proposition 3.36. The proofs are given in Subsection 3.3.3. T not depending on N such that

E sup t∈[0,T ] D (1)
N U(t) T such that the solution (u(t)) t≥0 of (3.1) with initial condition u 0 ∼ µ * satisfies for all t ∈ [0, T ]:

sup t∈[0,T ] E u(t) p L p 0 (T) ≤ C 0,p T and sup t∈[0,T ] E u(t) 2 H 1 0 (T) ≤ C 1,2 T .

Characterisation of the limit

In Lemma 3.32 we proved the existence of subsequential limits µ * for the sequences of embedded invariant measures (µ (m) N ) N ≥1 . Our convergence argument now consists in identifying any such limit µ * with the unique invariant measure µ of the solution (u(t)) t≥0 of (3.1) (see Proposition 3.2). We proved in Lemma 3.32 that µ * gives full weight to H 2 0 (T). As a consequence of this result, the measure µ * can be considered as an initial distribution for (u(t)) t≥0 . The weak convergence of the subsequence µ

(1) N j towards µ * can be represented, by virtue of the Skorokhod theorem, by the almost sure L 2 0 (T)-convergence, on some particular probability space, of a sequence of random variables u

(1) j,0

Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law using Itô's formula: for all t ≥ 0, D

N U(t)

2 2 = D (1)
N U 0 2 2 + 2 t 0 D (1)
N b(U(s)), D

N U(s) ds

+ 2 t 0 D (1)
N U(s), d D

N W Q,N (s) + t k≥1 D (1)
N σ k 2 2 . (3.55) (1)
It appears that the third term of the right-hand side is a martingale since

k≥1 E t 0 D (1)
N U(s), D

N σ k 2 ds ≤ t   k≥1 D (1)
N σ k 2 2   E D (1) N V 2 2 ≤ tD 0 C 1,2 < +∞, (1)
where we used the stationarity of (U(t)) t≥0 , Inequality (3.42), and the first inequality from this lemma. Thus, taking the expectation and expanding the drift term, we get

E D (1)
N U(t)

2 2 = E D (1)
N U 0 2 2 + 2 t 0 E D (1)
N U(s), D

N b(U(s)) ds + t k≥1 D (1)
N σ k 2 2 . (1)
Since the process starts from its invariant measure, the left-hand side cancels with the first term of the right-hand side. Besides, we may drop the time index. Using the decomposition b = b 1 + b 2 , we may sum by parts both the viscous and the flux term, and after dividing by t, it remains 2νE D

(2)

N V 2 2 = -2E D (2) N V, b 1 (V) + k≥1 D (1)
N σ k 2 2 ≤ 2 E D (2) N V 2 2 E b 1 (V) 2 2 +D 0 , (3.57
) where we used in particular the Cauchy-Schwarz inequality. We can bound the term in the second square root thanks to Assumption 3.3:

E b 1 (V) 2 2 = E N N i=1 A(V i , V i+1 ) -A(V i-1 , V i ) 2 = E   N N i=1 V i+1 V i ∂ 2 A(V i , z)dz + V i V i-1 ∂ 1 A(z, V i )dz 2   (by (3.13)) ≤ 2E N N i=1 (V i+1 -V i ) V i+1 V i ∂ 2 A(V i , z) 2 dz + 2E N N i=1 (V i -V i-1 ) V i V i-1 ∂ 1 A(z, V i ) 2 dz (by Jensen) ≤ 4C 2 A E N N i=1 (V i -V i-1 ) V i V i-1 (1 + |z| p A ) 2 dz (by (3.14)) ≤ 8C 2 A E N N i=1 (V i -V i-1 ) 2 + E N N i=1 (V i -V i-1 ) V i V i-1 |z| 2p A dz = 8C 2 A E D (1) N V 2 2 + 1 2p A + 1 E D (1) N V 2p A +1 , D (1)
N V ≤ 8C 2 A C 1,2 + D 0 2ν C 0,2p
N U 2 2 ≤ 2 E D (2) N U 2 2 4C 2 A D 0 ν 1 + C 0,2p A + D 0 . (2)
Applying Young's inequality on the right-hand side, we get 2νE D

(2)

N U 2 2 ≤ νE D (2) N U 2 2 + 4C 2 A D 0 ν 2 1 + C 0,2p A + D 0 , which rewrites E D (2) N U 2 2 ≤ 4C 2 A D 0 ν 3 1 + C 0,2p A + D 0 ν .
Since the right-hand side does not depend on N , we get the result.

Proof of Lemma 3.33. From summations by parts and Hölder's inequality, we establish a discrete Gagliardo-Nirenberg inequality in the following way (similar inequalities in the multi-dimensional case are given for instance in [START_REF] Bouchut | Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data[END_REF]Lemma 6] or [START_REF] Bessemoulin-Chatard | On discrete functional inequalities for some finite volume schemes[END_REF]Theorem 4]):

E N i=1 |V i+1 -V i | 3 = E N i=1 (V i+1 -V i ) 2 |V i+1 -V i | = -E N i=1 V i ((V i+1 -V i )|V i+1 -V i | -(V i -V i-1 )|V i -V i-1 |) = -E N i=1 V i ((V i+1 -2V i + V i-1 )|V i+1 -V i | + (V i -V i-1 ) (|V i+1 -V i | -|V i -V i-1 |)) ≤ E N i=1 |V i ||V i+1 -V i ||V i+1 -2V i + V i-1 | + E N i=1 |V i ||V i -V i-1 ||V i+1 -2V i + V i-1 | ≤ 2E N i=1 |V i | 6 1 6 E N i=1 |V i+1 -V i | 3 1 3 E N i=1 |V i+1 -2V i + V i-1 | 2 1 2
.

Dividing on both sides by E[

i |V i+1 -V i | 3 ] 1/3
and then passing to the power 3/2, we obtain

E N i=1 |V i+1 -V i | 3 ≤ 2 3/2 E N i=1 |V i | 6 1 4 E N i=1 |V i+1 -2V i + V i-1 | 2 3 4
.

Multiplying on both sides by N 2 , we derive the inequality E D

(1)

N V 3 3 ≤ 2 √ 2E V 6 6 1/4 E D (2) N V 2 2 3/4
, and we conclude thanks to Proposition 3.19 and Lemma 3.31:

E D (1) N V 3 3 ≤ 2 √ 2 C 0,6 1/4 C 2,2 3/4 =: C 1,3 .
N U(t)

2 2 = D (1) N U 0 2 2 +2 t 0 D (1)
N b 1 (U(s)), D

N U(s) ds+2ν t 0 D (1)
N D

N U(s), D

N U(s) ds

+ 2 t 0 D (1)
N U(s), d D

N W Q,N (s) + t D (1) N σ k 2 2 . (3.59) (1)
We shall address the second term of the right-hand side by applying (3.18) and Young's inequality:

2 t 0 D (1)
N b 1 (U(s)), D (1)
N U(s) ds = -2 t 0 b 1 (U(s)), D (2)
N U(s) ds ≤ 1 2ν t 0 b 1 (U(s)) 2 2 ds + 2ν t 0 D (2)
N U(s)

N D

N U(s), D

N U(s) ds = -2ν t 0 D (1)
N U(s)

N U(t)

N W Q,N (s) + tD 0 .

(3.62) Taking the supremum in time and the expectation over the second term of the right-hand side, by stationarity of (U(t)) t≥0 , we get the bound

E sup t∈[0,T ] 1 2ν t 0 b 1 (U(s)) 2 2 ds ≤ 1 2ν E T 0 b 1 (U(s)) 2 2 ds = 1 2ν T 0 E b 1 (U(s)) 2 2 ds = T 2ν E b 1 (U 0 ) 2 2
.

Applying now inequality (3.58), we get

E sup t∈[0,T ] 1 2ν t 0 b 1 (U(s)) 2 2 ds ≤ 2C 2 A T D 0 ν 2 1 + C 0,2p A . ( 3

.63)

We now turn our attention to the third term of the right-hand side in (3.62). Recall that by (3.56), the process

( t 0 D (1) N U(s), d D (1)
N W Q,N (s) ) t≥0 is a martingale. Therefore, applying successively the Jensen and the Doob inequalities, the Itô isometry, the Cauchy-Schwarz inequality and Lemma 3.31, we get

E sup t∈[0,T ] t 0 D (1) N U(s), d D (1) N W Q,N (s) ≤ E sup t∈[0,T ] t 0 D (1) N U(s), d D (1)
N W Q,N (s) 2 1/2 ≤ 2E T 0 D (1)
N U(s), d D

N W Q,N (s)

2 1/2 = 2E   k≥1 T 0 D (1)
N U(s), D

N σ k 2 ds   1/2 ≤ 2 √ T E D (1)
N U 0 2 2 1/2   k≥1 D (1)
N σ k 2 2   1/2 ≤ 2 T C 1,2 D 0 = D 0 2T ν . ( (1)
E sup t∈[0,T ] D (1) N U(t) 2 2 ≤ C 1,2 + 2C 2 A T D 0 ν 2 1 + C 0,2p A + D 0 2T ν + T D 0 .
Since the right-hand side of the above inequality does not depend on N , the result follows.

Proof of Lemma 3.35. Let p ∈ 2N * and let us repeat the proof of Lemma 2.15 up to (2.23). When the initial condition u 0 is random and has distribution µ * , this equation writes

E u (t ∧ T r ) p L p 0 (T) = E u 0 p L p 0 (T) -pE t∧Tr 0 T ∂ x A(u(s))u(s) p-1 dxds -νp(p -1)E t∧Tr 0 T ∂ x u(s) 2 u(s) p-2 dxds + p(p -1) 2 k≥1 E t∧Tr 0 T u(s) p-2 g 2 k dxds ,
for all t ∈ [0, T ] and r ≥ 0, where T r is a stopping time converging almost surely towards +∞ as r → +∞ (by Corollary 2.18). Using (2.24), the non-positivity of the third term of the right-hand side, and bounding the g k 's by their supremum, we get the inequality

E u (t ∧ T r ) p L p 0 (T) ≤ E u 0 p L p 0 (T) + p(p -1) 2   k≥1 g k 2 L ∞ 0 (T)   E t∧Tr 0 u(s) p-2 L p-2 0 (T)
ds .

Using now Lemma 3.32, (3.3), (3.5), and Inequality (2.18), we get

E u (t ∧ T r ) p L p 0 (T) ≤ C 0,p + p(p -1) 2 D 0 C (p-2) 5 1 + E u 0 p-2 L p-2 0 (T) + C (p-2) 6 t ,
where the constants C (p-2) 5

and

C (p-2) 6
, defined in Chapter 2, depend only on ν, p and D 0 . Using once again Lemma 3.32, letting r → +∞ and bouding t by T , we obtain

lim sup r→∞ E u (t ∧ T r ) p L p 0 (T) ≤ C 0,p + p(p -1) 2 D 0 C (p-2) 5 1 + C 0,p-2 + C (p-2) 6 T =: C 0,p T .
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law Applying Fatou's lemma on the left-hand side, we get

E u(t) p L p 0 (T) ≤ C 0,p T ,
and since the right-hand side does not depend on t, we get the first wanted inequality in the case p ∈ 2N * . The general case p ∈ [2, +∞) then follows from the Jensen inequality.

To prove the second inequality, we start from Lemma 2.17 which, when u 0 is random, gives the estimate

E u(t ∧ T r ) 2 H 1 0 (T) ≤ E u 0 2 H 1 0 (T) + C 7 1 + E u 0 2p A +2 L 2p A +2 0 (T) + C 8 t,
from which we deduce, by applying Fatou's lemma on the left-hand side and Lemma 3.32 on the right-hand side:

E u(t) 2 H 1 0 (T) ≤ C 1,2 + C 7 1 + C 0,2p A +2 + C 8 T =: C 1,2
T .

We now turn to the proof of the main lemma in this section:

Proof of Proposition 3.36. In all this proof, for notational convenience, the subsequence (u

N j ) j∈N will be denoted by (u

N ) N ≥1 .

Step 0. Decomposition of the error. Let us fix a time horizon T > 0. We introduce the stopping time

τ M,N := inf t ≥ 0 : u(t) H 1 0 (T) ∨ u (1)
N (t)

H 1 0 (T) ≥ M ,
and we split the expectation in two parts: for all t ∈ [0, T ],

E u

N (t) -u(t) 2 L 2 0 (T) = E u (1)
N (t) -u(t) 2 L 2 0 (T) 1 t≤τ M,N + E u (1)
N (t) -u(t) 2 L 2 0 (T) (1)
1 t>τ M,N .

(3.65) We will address the first term of the RHS in the steps 1 to 6, and the second one in the step 7.

We have

E u (1) N (t) -u(t) 2 L 2 0 (T) 1 t≤τ M,N ≤ E u (1) N (τ M,N ∧ t) -u(τ M,N ∧ t) 2 L 2 0 (T) , (3.66)
and we will use this localization argument to take benefit from the local Lipschitz continuity of the non-linear term which, by use of the Grönwall lemma, will lead us to show that for any fixed M > 0, E u

N (t) -u(t) 2 L 2 0 (T) 1 t≤τ M,N -→ N →∞ 0. (1)
3.3. Convergence of invariant measures: semi-discrete scheme towards SPDE 87 Applying Itô's formula [START_REF] Da | Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications[END_REF]Theorem 4.32] and taking the expectation, we have E u

(1)

N (τ M,N ∧ t) -u(τ M,N ∧ t) 2 L 2 0 (T) = E u (1)
N (0) -u(0) 2 L 2 0 (T) -2E τ M,N ∧t 0 T u (1) N (s, x) -u(s, x) N i=1 N A (U i (s), U i+1 (s)) -A (U i-1 (s), U i (s)) φ (1) N x - i N -∂ x A u (1)
N (s, x) dxds

-2E τ M,N ∧t 0 T u (1) N (s, x) -u(s, x) ∂ x A u (1)
N (s, x) -∂ x A(u(s, x)) dxds + 2νE τ M,N ∧t 0 T u (1) N (s, x) -u(s, x) N 2 N i=1 (U i+1 (s) -2U i (s) + U i-1 (s)) φ (1) N x - i N -∂ xx u(s, x) dxds + k≥1 E   τ M,N ∧t 0 T N i=1 σ k i φ (1) N x - i N -g k (x) 2 dxds   =: I N 1 + I N 2 (t) + I N 3 (t) + I N 4 (t) + I N 5 (t),
where the local martingale term vanished thanks to the localisation. From Step 1 to Step 5, we will get an upper bound over each of the terms I N l , l = 1, . . . , 5. More precisely, for all l = 1, 2, 4, 5, we will show that there exists a sequence (ε N l ) N ∈N of non-negative real numbers not depending on t such that lim N →∞ ε N l = 0 and such that the following inequalities are satisfied for all N ∈ N and all t ∈ [0, T ]:

I N 1 ≤ ε N 1 , (3.67)
I N 2 (t) ≤ E τ M,N ∧t 0 u (1)
N (s) -u(s) 2 L 2 0 (T) ds + ε N 2 , (3.68)
I N 4 (t) ≤ -2νE τ M,N ∧t 0 u (1)
N (s) -u(s) 2 H 1 0 (T) ds + ε N 4 , (3.69)
I N 5 (t) ≤ ε N 5 . (3.70)
In the case l = 3, we will show that there exists a constant γ M > 0 not depending on N nor t such that

I N 3 (t) ≤ 2νE τ M,N ∧t 0 u (1) N (s) -u(s) 2 H 1 0 (T) ds + γ M E τ M,N ∧t 0 u (1) N (s) -u(s) 2 L 2 0 (T)
ds .

Step 1. The initial condition. By the construction of the sequence (u

N (0) -u(0) 4 L 2 0 (T) ≤ 8E u (1) N (0) 4 L 2 0 (T) + 8E u(0) 4 L 2 0 (T) ≤ 8E u (1)
N (0) and the convergence of I N 1 towards 0 follows (see for instance [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 3.5]). Thus, since I N 1 does not depend on t, we may take ε N 1 := I N 1 .

Step 2. The flux-numerical flux approximation. Using Young's inequality, we have

I N 2 (t) ≤ E τ M,N ∧t 0 u (1)
N (s) -u(s) 2 L 2 0 (T) ds +E   τ M,N ∧t 0 T N i=1 N A (U i (s), U i+1 (s)) -A (U i-1 (s), U i (s)) φ (1) N x - i N -∂ x A u (1)
N (s, x)

2 dxds   . (3.73)
We focus on the second term of the right-hand side which we can rewrite by

E τ M,N ∧t 0 N i=1 i N i-1 N N A(U i-1 (s), U i (s)) -A(U i-2 (s), U i-1 (s)) -∂ x A u (1)
N (s, x) φ (1) N x - i -1 N + N A (U i (s), U i+1 (s)) -A (U i-1 (s), U i (s)) -∂ x A u (1)
N (s, x) φ

(1) N

x -i N 2 dxds , which we control by the following upper bound

2E τ M,N ∧t 0 N i=1 i N i-1 N N A(U i-1 (s), U i (s)) -A(U i-2 (s), U i-1 (s)) -∂ x A u (1)
N (s, x)

2 + N A(U i (s), U i+1 (s)) -A(U i-1 (s), U i (s)) -∂ x A u (1)
N (s, x) 2 dxds .

By definition of u

N , we have for all s ≥ 0 and all x ∈

( i-1 N , i N ], ∂ x u (1)
N (s, x) = N (U i (s) -U i-1 (s)).
Let us now focus on the second term of the above integrand and observe that by symmetry, the left one may be treated in exactly the same way. We have thanks to (3.12):

N A(U i (s), U i+1 (s)) -A(U i-1 (s), U i (s)) -N (U i (s) -U i-1 (s))A u (1) N (s, x) 2 = N A(U i (s), U i+1 (s)) -A(U i-1 (s), U i (s)) -N (U i (s) -U i-1 (s)) ∂ 1 A u (1) N (s, x), u (1)
N (s, x) + ∂ 2 A u (1) N (s, x), u (1)
N (s, x) 2 ≤ 2 N U i (s) U i-1 (s) ∂ 1 A(z, U i (s)) -∂ 1 A u (1) N (s, x), u (1)
N (s, x) dz 2 + 2N 2 U i+1 (s) U i (s) ∂ 2 A(U i (s), z)dz -(U i (s) -U i-1 (s)) ∂ 2 A u (1) N (s, x), u (1)

N (s, x)

In the following computations, we will get upper bounds on the terms

I N 2.1 (t) := E   τ M,N ∧t 0 N i=1 i N i-1 N N U i (s) U i-1 (s) ∂ 1 A(z, U i (s)) -∂ 1 A u (1)
N (s, x), u

N (s, x) dz 2 dxds   , I N 2.2 (t) := E τ M,N ∧t 0 N i=1 i N i-1 N N 2 U i+1 (s) U i (s) ∂ 2 A(U i (s), z)dz -(U i (s) -U i-1 (s)) ∂ 2 A u (1)
N (s, x), u

N (s, x) 2 dxds .

We first look at the term

I N 2.1 (t). The function ∂ 1 A is uniformly continuous on [-M, M ] 2 . In particular, ∀ε > 0, ∃δ M,ε > 0, max (|w -x|, |y -z|) ≤ δ M,ε , max {|w|, |x|, |y|, |z|} ≤ M ⇒ |∂ 1 A(w, y)-∂ 1 A(x, z)| ≤ ε.
Furthermore, for every s ≤ τ M,N , by Lemma 3.30 and (3.3), we have sup

i=1,...,N |U i (s)| = u (1) N (s) L ∞ 0 (T) ≤ u (1) N (s) H 1 0 (T) ≤ M . Let C (M ) ∂ be a local bound of ∂ 1 A and ∂ 2 A over the square [-M, M ] 2 . Let ε > 0. We have I N 2.1 (t) = E τ M,N ∧t 0 N i=1 i N i-1 N N U i (s) U i-1 (s) ∂ 1 A(z, U i (s)) -∂ 1 A u (1)
N (s, x), u

N (s, x) dz1 |U i (s)-U i-1 (s)|≤δ M,ε 2 dxds + E   τ M,N ∧t 0 N i=1 i N i-1 N N U i (s) U i-1 (s) ∂ 1 A(z, U i (s)) -∂ 1 A u (1)
N (s, x), u

N (s, x) dz1 |U i (s)-U i-1 (s)|>δ M,ε 2 dxds   ≤ E   τ M,N ∧t 0 N i=1 i N i-1 N N U i (s) U i-1 (s) εdz1 |U i (s)-U i-1 (s)|≤δ M,ε 2 dxds   + E   τ M,N ∧t 0 N i=1 i N i-1 N   N U i (s) U i-1 (s) 2C (M ) ∂ dz |U i (s) -U i-1 (s)| 1/2 δ 1/2 M,ε   2 dxds   ≤ ε 2 E t 0 N N i=1 |U i (s) -U i-1 (s)| 2 ds + 4 C (M ) ∂ 2 δ M,ε t 0 E N N i=1 |U i (s) -U i-1 (s)| 3 ds = ε 2 tE D (1) N U 0 2 2 + 4 C (M ) ∂ 2 δ M,ε N tE D (1) N U 0 3 3 ≤ ε 2 T C 1,2 + 4 C (M ) ∂ 2 T C 1,3 (1)
δ M,ε N (by Lemmas 3.31 and 3.33).

Since ε was chosen arbitrarily, it follows that ε

N 2.1 := sup t∈[0,T ] I N 2.1 (t) satisfies lim N →∞ ε N 2.1 = 0.
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law

As for the term I N 2.2 (t) we have

I N 2.2 (t) = E τ M,N ∧t 0 N i=1 i N i-1 N N 2 U i+1 (s) U i (s) ∂ 2 A(U i (s), z) -∂ 2 A u (1)
N (s, x), u

N (s, x) dz + (U i+1 (s) -2U i (s) + U i-1 (s))∂ 2 A u (1)
N (s, x), u

N (s, x)

2 dxds ≤ 2E   τ M,N ∧t 0 N i=1 i N i-1 N N 2 U i+1 (s) U i (s) ∂ 2 A(U i (s), z) -∂ 2 A u (1)
N (s, x), u

N (s, x) dz

2 dxds   + 2E τ M,N ∧t 0 N i=1 i N i-1 N N 2 (U i+1 (s) -2U i (s) + U i-1 (s)) 2 ∂ 2 A u (1)
N (s, x), u

N (s, x)

I N 2.2.2 (t) ≤ C (M ) ∂ 2 E τ M,N ∧t 0 N N i=1 (U i+1 (s) -2U i (s) + U i-1 (s)) 2 ds ≤ 2 C (M ) ∂ 2 t 0 E N N i=1 (U i+1 (s) -2U i (s) + U i-1 (s)) 2 ds = 2 C (M ) ∂ 2 t N 2 E D (2) N U 0 2 2 ≤ 2 C (M ) ∂ 2 T C 2,2 N 2 (by Lemma 3.31) =: ε N 2.2.2 .
At last, the sum of all these error terms amounts to an error term ε N 2 satisfying the requirements of Step 0, so that the inequality (3.73) reduces to (3.68).

Step 3. The flux term. Integrating by parts and applying Young's inequality, we get

I N 3 (t) = 2E τ M,N ∧t 0 T ∂ x u (1)
N (s, x) -u(s, x) A u

N (s, x) -A(u(s, x)) dxds ≤ 2νE τ M,N ∧t 0 T ∂ x u (1)
N (s, x) -∂ x u(s, x) 2 dxds + 1 2ν E τ M,N ∧t 0 T A u (1)
N (s, x) -A(u(s, x)) 2 dxds . (1)
Denoting by L M a local Lipschitz constant of A over the interval [-M, M ], we get

I N 3 (t) ≤ 2νE τ M,N ∧t 0 u (1) N (s) -u(s) 2 H 1 0 (T) ds + L 2 M 2ν E τ M,N ∧t 0 u (1) N (s) -u(s) 2 L 2 0 (T)
ds , and we set therefore γ M := L 2 M /(2ν).

Step 4. The viscous term. We shall compare the term

J N (s) = T u (1) N (s, x) -u(s, x) N i=1 N 2 (U i+1 (s) -2U i (s) + U i-1 (s))φ (1) N x - i N -∂ xx u(s, x) dx,
Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law As a consequence,

J N 1 (s) -J N 1 (s) = N i=1 N 2 (U i+1 (s) -2U i (s) + U i-1 (s)) 1 6N U i-1 (s) + 2 3N U i (s) + 1 6N U i+1 (s) + N i=1 U i (s) (-N U i-1 (s) + 2N U i (s) -N U i+1 (s)) = N 6 N i=1 (U i+1 (s) -2U i (s) + U i-1 (s)) 2 = 1 6N 2 D (2) N U(s) 2 2
. By Lemma 3.31, we deduce that

E J N 1 (s) -J N 1 (s) ≤ C 2,2 6N 2 .
In order to compute J N 3 (s) -J N 3 (s), we first rewrite

J N 3 (s) = T ∂ x u(s, x)∂ x u (1)
N (s, x)dx

= N i=1 U i (s) T ∂ x u(s, x)∂ x φ (1) N x - i N dx = N i=1 N U i (s) i N i-1 N ∂ x u(s, x)dx - i+1 N i N ∂ x u(s, x)dx = - N i=1 N U i (s) u s, i + 1 N -2u s, i N + u s, i -1 N = - N i=1 N u s, i N (U i+1 (s) -2U i (s) + U i-1 (s)) .
As a consequence,

J N 3 (s) -J N 3 (s) = - N i=1 N (U i+1 (s) -2U i (s) + U i-1 (s)) N T u(s, x)φ (1) N x - i N dx -u s, i N = - N i=1 N 2 (U i+1 (s) -2U i (s) + U i-1 (s)) T u(s, x) -u s, i N φ (1) N x - i N dx = - N i=1 N 2 (U i+1 (s) -2U i (s) + U i-1 (s)) T x i N ∂ x u(s, y)dyφ (1) N x - i N dx.
Using the rough bound |φ As a conclusion,

E J N 3 (s) -J N 3 (s) ≤ 4 C 2,2 C 1,2 T N .
Coming back to the expression of I N 4 (t), we have

I N 4 (t) ≤ 2νE
N (s) -u(s)

N (s) -u(s) ds + ε N 4 .

Step 5. The noise term. We have

I N 5 (t) = E   τ M,N ∧t 0 T k≥1 N i=1 σ k i φ (1) N x - i N -g k (x) 2 dxds   ≤ t T k≥1 N i=1 σ k i φ (1) N x - i N -g k (x) 2 dx.
Using the fact that φ

(1)

N x -i-1 N + φ (1)
N x -i N = 1, for all x ∈ [(i -1)/N, i/N ], we get

I N 5 (t) ≤ t N i=1 i N i-1 N k≥1 σ k i-1 -g k (x) φ (1) N x - i -1 N + σ k i -g k (x) φ (1) N x - i N 2 dx,
and by periodicity of the indexes, we can say that

I N 5 (t) ≤ 2t N i=1 i N i-1 N k≥1 N i N i-1 N g k (y)dy -g k (x) 2 φ (1) N x - i N 2 dx ≤ 2t N i=1 i N i-1 N k≥1 N i N i-1 N (g k (y) -g k (x))dy 2 φ (1) N x - i N 2 dx ≤ 2t N i=1 i N i-1 N N i N i-1 N k≥1
(g k (y) -g k (x)) 2 dydx (since |φ

N | ≤ 1 and by Jensen)

≤ 2tN N i=1 i N i-1 N i N i-1 N D 0 N 2 dydx = 2tD 0 N 2 ≤ 2T D 0 N 2 =: ε N 5 .

94

Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law

Step 6. Conclusion for the "bounded" event. Summing all the I N i terms, we get for all t ∈ [0, T ] E u We set ε N := ε N 1 + ε N 2 + ε N 4 + ε N 5 . By (3.66), we have E u

(1)

N (t) -u(t) 2 L 2 0 (T) 1 t≤τ M,N ≤ (1 + γ M ) t 0 E u (1)
N (s) -u(s) Thus, Grönwall's lemma applies and gives E u

(1)

N (t) -u(t) 2 L 2 0 (T)
1 t≤τ M,N ≤ ε N e (1+γ M )t .

Step 7. Conclusion of the proof. We want now to take care of the second term of the righthand side in (3.65). Using in particular the Cauchy-Schwarz inequality, Proposition 3.19, the Markov inequality, Lemma 3.35 and Lemma 3.34, we have E u

1 t>τ M,N = E u (1)
N (t) -u(t) 2 L 2 0 (T)
1 sup s∈[0,t] u(s) H 1 0 (T) ∨sup s∈[0,t] u

N (s)

H 1 0 (T) >M ≤ E u (1)
N (t) -u(t)

N (s) Combining this inequality with the conclusion of the step 6 yields the wanted result.

H 1 0 (T) > M 1/2 ≤ √ 8 E u
3.4 Convergence of invariant measures: split-step scheme towards semi-discrete scheme

In this section, we aim to prove the second part of Theorem 3.7, namely Equation (3.17). The structure of the proof is the same as for the first part of Theorem 3.7. In Subsection 3.4.1, we show that the 3.4. Convergence of invariant measures: split-step scheme towards semi-discrete scheme 95 family of probability measures {ν N,∆t : ∆t ∈ (0, ∆t max ]} is tight in P(R N 0 ) and then relatively compact in P 2 (R N 0 ). In Subsection 3.4.2, in a similar manner as in Subsection 3.3.2 for the semi-discrete case, we identify each subsequential limit of the family {ν N,∆t : ∆t ∈ (0, ∆t max ]}, when ∆t → 0, as the invariant measure of the process (U n ) n∈N , which leads to the final part of the proof of Theorem 3.7. Subsection 3.4.3 contain the proofs of the lemmas stated in Subsections 3.4.1 and 3.4.2.

3.4.1 Tightness, relative compactness and some estimates Lemma 3.38 (Tightness). Under Assumptions 3.1 and 3.3, for any N ≥ 1, the family of probability measures {ν N,∆t : ∆t ∈ (0, ∆t max ]} is tight in the space P(R N 0 ).

Proof. We established at Proposition 3.27 that a random variable V ∼ ν N,∆t satisfies the discrete H 1 0 estimate:

E D (1) N V 2 2 ≤ C 1,2 .
Since D

N • 2 defines a norm on R N 0 , the result follows from the Markov inequality.

Lemma 3.39 (Fourth-order moment). Under Assumptions 3.1 and 3.3, there exists a constant C 0,4 > 0, depending only on D 0 , ν and ∆t max , such that for any time step ∆t ∈ (0, ∆t max ] and any random variable V ∼ ν N,∆t , we have

E V 4 4 ≤ C 0,4 .
Corollary 3.40 (Relative compactness). Let N ≥ 1. Under Assumptions 3.1 and 3.3, the family {ν N,∆t : ∆t ∈ (0, ∆t max ]} is relatively compact in P 2 (R N 0 ).

Proof. By virtue of the Prokhorov theorem [12, Theorem 5.1] and Lemma 3.38, any sequence extracted from {ν N,∆t : ∆t ∈ (0, ∆t max ]} admits a weakly converging subsequence in P(R N 0 ). Let ν * be a subsequential weak limit and let (ν N,∆t j ) j∈N be a sequence weakly converging towards ν * . Let (V j ) j∈N be a sequence of R N 0 -valued random variables such that V j ∼ ν N,∆t j . By virtue of the Portemanteau theorem, since • 4 4 is continuous (and thus lower semi-continuous) on R N 0 , we have

E V 4 4 ≤ lim inf j→∞ E V j 4 4 ≤ C 0,4 ,
so that ν * admits a fourth-order moment and thus belongs to P 4 (R N 0 ). Moreover, it follows also from Lemma 3.38 that the sequence (ν N,∆t j ) j∈N satisfies a uniform integrability condition in the sense of [START_REF] Villani | Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften[END_REF]Definition 6.8], and the result is now a consequence of [102, Theorem 6.9].

Characterisation of the limit

As in Subsection 3.3.2 for the semi-discrete scheme, we want to use a result of convergence in finite time of the numerical scheme (3.15) in order to identify each subsequential limit of the family {ν N,∆t : ∆t ∈ (0, ∆t max ]}, when ∆t → 0, as the invariant measure ν N for the solution of Equation (3.11). By virtue of Corollary 3.40, let ν * ∈ P 2 (R N 0 ) and let (∆t j ) j∈N be a sequence of time steps decreasing to and ∆W Q,N i,n+1 are independent. Since ∆W Q,N i,n+1 ∼ N (0, ∆t k≥1 (σ k i ) 2 ) and k≥1 (σ k i ) 2 ≤ D 0 (by (3.9)), we write To conclude, we use Inequality (3.75) once again: (3.83)

E U n+1
E V 4 4 ≤ E V 1 2 4 4 + 6D 0 ∆tE V 1
Step 2. Bound over the term (3.83). Let L M be a Lipschitz constant of b over the ball E U (j) (t) -U(t) E U (j) (t) -U(t) Here, x ∼ y means that there exists a constant C > 0 such that C -1 y ≤ x ≤ Cy. The inertial range I is defined with more details in [START_REF] Boritchev | Generalised Burgers equation with random force and small viscosity[END_REF]Section 4.6]. To give a physical interpretation of this interval, it corresponds to the range of scales in which the energy of the system is transported from large scales to smaller ones. If we write I = [α, β], the inertial range is positioned between the energy range [β, 1] containing the large scales, which in our case are generated by the stochastic forcing, and the dissipation range [0, α] containing the small scales dissipated by the viscous term. In particular, α depends linearly on ν.

E U (j) (t) -U(t) 2 2 1 ρ (j) M ≤t ≤ E U (j) (t) -U(t)
The second universal property of interest concerns the flatness, that is the function F defined by

D

  u(t, x)ϕ(x)dx = D u 0 (x)ϕ(x)dx + ν t 0 D u(s, x)∆ϕ(x)dxds + t 0 D A(u(s, x)) • ∇ϕ(x)dxds + t 0 D

1. 2 .

 2 Viscous scalar conservation laws with stochastic forcingpositive measure m on D × [0, T ] × R, such that for all test functions ϕ ∈ C ∞ c ([0, T ] × D × R), the function f satisfies almost surely T 0 f (t), ∂ t ϕ(t) dt + f 0 , ϕ(0) + t 0 f (t), A • ∇ϕ(t) dt = -T 0 D ϕ(t, x, u(t, x))Φ(t, x, u(t, x))dxdW (t) , x, u(t, x))| 2 ∂ ξ ϕ(t,x, u(t, x))dxdt + m(∂ ξ ϕ).

Figure 1 . 2 :

 12 Figure 1.2: Numerical approximation results for invariant measures of SPDEs

Figure 1 . 3 :

 13 Figure 1.3: Approximation of the invariant measure µ

6 >

 6 0 depending only on ν, p and D 0 such that E t∧Tr 0 u(s) p L p 0 (T) ds ≤ C

Chapter 2 .

 2 Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure where we fixed a time horizon T > 0. Thus, for all t ∈ [0, T ], we get d dt e -νλmt t 0 w m (s)ds = e -νλmt -νλ m t 0 w m (s)ds + w m (t) ≤ e -νλmt sup s∈[0,T ] W m (s), which leads, after integrating in time and dividing by e -νλmt on each side, to the inequality t 0 w m (s)ds ≤ 1 -e νλmt -νλ m sup s∈[0,T ]

3 / 2 0

 32 (T)). Iterating this argument, we get (u(t)) t∈[0,τ ] ∈ C([0, τ ], H 2 0 (T)). We now show that (u(t)) t∈[0,τ ] satisfies the strong formulation of (2.10) on [0, τ ]. Along each coordinate of the basis (e m ) m≥1 , the mild formulation (2.11) writes for all t ∈ [0, τ ] u(t), e m L 2 0 (T) = e νλmt u 0 , e m L 2 0 (T) -t 0 e νλm(t-s) ∂ x A(u(s)), e m L 2 0 (T) ds + 1 λ m w m (t).

  x)dx. We may interpret the noise term (i.e. the last term in (3.8)) as a discrete version of the Q-Wiener process (W Q (t)) t≥0 introduced in Section 3.1.1.

. 14 )

 14 Note in particular that the flux function, and therefore the non-linearity of Equation (3.1), is not subject to a global Lipschitz continuity assumption. Nevertheless, we will prove in Proposition 3.15 below that (3.11) is well-posed under Assumption 3.3.Remark 3.4 (Engquist-Osher numerical flux). A notable class of numerical fluxes satisfying the monotonicity and polynomial growth conditions (under Assumption 3.1) are the flux-splitting schemes [57, Example 5.2], among which a commonly employed example is the Engquist-Osher flux[START_REF] Engquist | Stable and entropy satisfying approximations for transonic flow calculations[END_REF] defined by

N ) - 1 .

 1 In particular, the measures µ (m) N and µ (m) N,∆t give full weight to H m 0 (T). Sections 3.3 and 3.4 are devoted to the proof of our main result: Theorem 3.7. Under Assumptions 3.1 and 3.3, we have for all m = 0, 1, 2, lim N →∞ µ (m) N = µ in P 2 (L 2 0 (T)), (3.16)

  then for any m = 0, 1, any p ∈ [1, +∞), Let u 0 ∈ R N 0 . Since the function b is locally Lipschitz continuous, there exists a unique strong solution (U(t)) t∈[0,T * ) to Equation (3.11) with initial condition u 0 defined up to an explosion time T * , i.e. a stopping time taking values in (0, +∞] such that almost surely, if T * < +∞, then lim t→T * U(t) 2 = +∞ (see for instance [75, Theorem 2.3, Lemma 2.1, Theorem 3.1]). In particular, if we define the stopping times τ M := inf t ≥ 0 : U(t) 2 2 ≥ M , (3.19)

Corollary 3 . 26 .Proposition 3 . 27 . 74 Chapter 3 .

 326327743 Under Assumptions 3.1 and 3.3, the solution (U n ) n∈N of (3.15) is a Feller process. Under Assumptions 3.1 and 3.3, for any time step ∆t > 0, the process (U n ) n∈N solution of the split-step backward Euler method (3.15) admits an invariant measure ν N,∆t . Moreover, if V is a random variable with distribution ν N,∆t , then E D Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law and E D

N U l+ 1 2 2 2 ≤ u 0 2 2

 222 + n∆tD 0 . (3.40) 3.2. Semi-discrete and split-step schemes: well-posedness and invariant measure 75 Besides,

2 ≤

 2 ∆tD 0 . (3.43) Injecting (3.43) into (3.41), and (3.41) into (3.40), we get

N • 2

 2 defines a norm on R N 0 ).

Lemma 3 . 30 .

 330 The following properties hold:(i) for v ∈ R N 0 , any p ∈ [1, +∞] and any m = 0, 1, 2, Ψ

. 50 )

 50 As a consequence, (µ (0) N j ) j∈N satisfies a uniform integrability condition in the sense of [102, Definition 6.8.(iii)] and thus, is converging for the Wasserstein distance in P 2 (L 2 0 (T)) towards µ * . Step 3. Relative compactness of (µ(1) N j ) j∈N and (µ (2) N j ) j∈N in P 2 (L 2 0 (T)). The sequences (µ(1)N j ) j∈N and (µ[START_REF] Albeverio | Stratified structure of the Universe and Burgers' equation-a probabilistic approach[END_REF]

Lemma 3 . 33 ( 3 ≤ C 1 , 3 .

 333313 Discrete W 1,3 0 bound). Let Assumptions 3.1 and 3.3 hold and let V be an R N 0 -valued random variable distributed according to ν N . Then, there exists a constant C 1,3 depending only on ν, p A and D 0 such that Lemma 3.34 (Discrete H 1 0 bound in finite time). Let Assumptions 3.1 and 3.3 hold and let (U(t)) t≥0 be the solution of(3.11) with an initial condition U 0 ∼ ν N . For every T > 0, there exists a constant C1,2

Lemma 3 . 35 (

 335 Moments on the solution of (3.1)). Under Assumption 3.1, for all p ∈ [2, +∞) and T > 0, there are constants C 0,p T and C 1,2

84 Chapter 3 .

 843 Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law Proof of Lemma 3.34. Let U 0 ∼ ν N . Using the decomposition b = b 1 + b 2 introduced at (3.54), we may expand Equation (3.55): D

0 D

 0 As for the viscous term in (3.59), Equation (3.18) leads to 2ν t

  (3.61) and (3.60) into (3.59) and using the bound (3.42) results in: D

( 1 )

 1 N ) N ≥1 in Section 3.3.2, we have almost surely lim N →∞ u (1) N (0) -u(0) L 2 0 (T) = 0. (3.71) Moreover, Proposition 3.19 and Lemma 3.32 ensure the uniform bound with respect to N over the Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law following fourth order moment:

≤

  16C 0,4 (by Proposition 3.19 and Lemma 3.32),(3.72)

1 NN

 1 |∂ x u(s, y)| dy, whenceJ N 3 (s) -J N 3 (s) ≤ 2 N i=1 |U i+1 (s) -2U i (s) + U i-1 (s)| i+1 N i-1 N|∂ x u(s, y)| dy.

3. 3 .

 3 Convergence of invariant measures: semi-discrete scheme towards SPDE 93 By the Cauchy-Schwarz inequality,E J N 3 (s) -J N 3 (s) ≤ 2N E N i=1 |U i+1 (s) -2U i (s) + U i-1 (s)| 2 E

τ M,N ∧t 0 J

 0 N (s)ds ≤ 2νE τ M,N ∧t 0 J N (s)ds + 2ν t 0 E J N 1 (s) -J N 1 (s) ds + 2ν t 0 E J N 3 (s) -J N 3 (s) ds ≤ -2νE τ M,N ∧t 0 u

( 1 )

 1 N (τ M,N ∧ t) -u(τ M,N ∧ t)

1

  s≤τ M,N ds + ε N .

Furthermore

  , as (u(t)) t≥0 is continuous from [0, +∞) to H 1 0 (T) (see Proposition 3.2), the random variable sup s∈[0,t] u(s) H 1 0 (T) is finite almost surely. As a consequence,

2 L 2 0

 22 (T) 1 t>τ M,N = 0.

Lemma 3 . 41 ( 2 T

 3412 Finite time bound). Let Assumptions 3.1 and 3.3 hold, let T > 0 be a time horizon and let (U n ) n∈N be a solution of (3.15) with an initial condition U 0 ∼ ν N,∆t . There exists a constant C 0,depending only on D 0 , T , ν and ∆t max , such that for any time step ∆t ∈ (0, ∆t max ], we have

4 4 = E U n+ 1 2 + 2 + 4 ≤ E U n+ 1 2 4 4 + 6D 0 ∆tE U n+ 1 2 2 2 + 3D 2 0 ∆t 2 ( 3 . 75 ) 3 . 4 .U n 4 4 ≥ E U n+1 4 4-

 4224242223753444 E ∆W Q,N n+1 4 Convergence of invariant measures: split-step scheme towards semi-discrete scheme 97 Combining Inequalities (3.74) and (3.75), we get E 6D 0 ∆tE U n+ 1 Using now the same arguments as for the end of the proof of Proposition 3.19, letting V ∼ ν N,∆t , we get

Proof of Lemma 3 . 41 . 2 . 98 Chapter 3 . 2 .U l+ 1 2 , 2 . 2 , 2 = 0 .MM 2 -

 34129832222202 Let us repeat the proof of Proposition 3.27 up to Equation(3.37). For all n = 0, 1, . . . ,T ∆t , we writeThe viscous term may be removed from the inequality. Taking the supremum in time and the expectation, we getE   sup n=0,1,..., T∆t Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law First, by Lemma 3.10 and Proposition 3.27, we have Noticing that the process ( n-1 l=0 ∆W Q,N l+1 ) n≥1 is a martingale, we get by applying successively Jensen's and Doob's inequalities to the second term of the right-hand side, From (3.15), we may observe that each increment ∆W Q,N l+1 is independent of the family (U m+ 1 ∆W Q,N m ) m=0,...,l . Therefore, letting V ∼ ν N,∆t and letting V 1 2 be the random variable satisfying V 1 Proof of Lemma 3.42. Step 0. Decomposition of the error in two events. Let T > 0. We start by introducing the exit time of some ball for the time-continuous and the time-discretised processes:Then, we decompose the approximation error according to whether or not the discrete and continuous processes stay in the ball of radius M : for all t ∈ [0, T ],E U (j) (t) -U(t) -n [t,M,j] ∆t j b U (j) n [t,M,j] -n [t,M,j] ) b U (j) n [t,M,j] + 1 b U (j) n [t,M,j]

Chapter 3 .nnnStep 3 .M 2 - 2 ≤ 1 2 2 2 ≤ U l 2 2 3 . 4 . 101 Thus 2 ] 2 +

 33221223410122 Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law{ • 2 2 ≤ M }. From Jensen's inequality, we have [t,M,j] ∆t j b U (j) n [t,M,j] -b(U(s)) ds [t,M,j] ∆t j b U (j) n [t,M,j] -b(U(s)) [t,M,j] ∆t j U (j) n [t,M,j] -U(s) Decomposition of the term (3.82). Let C M be a supremum of b 2 2 over the ball { • 2 2 ≤ M }.Using the Jensen inequality and the locally Lipschitz continuity of b, we write -n [t,M,j] ) b U (j) n [t,M,j] + 1 4T (T + ∆t j ) ∆t 2 j L 2 M C M ,where at the last line, we have used the fact that for any l = 0, . . . , n [t,M,j] , by (3.35), we have U l+ ≤ M .Step 4. Conclusion for the bounded event. Summing the estimates obtained from Step 1 to Step 3, we getE U (j) (t) -U(t) T + ∆t j ) ∆t 2 j L 2 M C M .By construction, the random variable U (j) 0 -U 0 2 tends to 0 almost surely as j → +∞. Furthermore, it has a fourth order moment uniform in j thanks to Lemma 3.39 and the Portemanteau theorem: Convergence of invariant measures: split-step scheme towards semi-discrete scheme tends to 0 as j → +∞. Now, by Grönwall's lemma, we have E U (j) (t) -U(t) 4T (T + ∆t j ) ∆t 2 j L 2 M C M e 4T 2 L 2 M .

Step 5 .

 5 The exiting trajectories. We want to bound the second term of the RHS in (3.79) uniformly in j. Observing thatρ (j) M ≤ t = sup s∈[0,t]we get from the Cauchy-Schwarz inequality, Lemma 3.39 and Lemma 3.16.(ii):

2 .

 2 As for the second term, we have thanks to the Markov inequality and Lemma 3.41,

Step 6 . 2 ≤ 2 2 1 ρ

 6221 Conclusion. In the end, from (3.79) and (3.84), we have for every M > 0, lim sup j→∞ sup t∈[0,T ]E U (j) (t) -U(t) 2 lim sup j→∞ sup t∈[0,T ] E U (j) (t) -U(t) (j) M ≤t .Thus, letting M → +∞ and applying (3.85) yields the wanted result.

Figure 4 . 3 :Proposition 4 . 1 .

 4341 Figure 4.3: Strong error convergence at a large time with respect to ∆t

F

  (l) := E T |u(x + l) -u(x)| 4 dx E T |u(x + l) -u(x)| 2 dx

  .2. Well-posedness and regularity

	and				t∧Tr		
	lim M →∞	p 2 E		k≥1	0	T	g 2 k u(s) 2(p-1) φ M (u(s) p )dxds  = 0.
	Letting M go to +∞ in (2.19), (2.20), (2.21) and (2.22), we get

  ) ) be the maximal solution to Equation (2.1) given by Lemma 2.14. By Corollary 2.18, T * = +∞ almost surely. Therefore, (u(t)) t≥0 is the unique (global) mild solution to Equation (2.1), and by Proposition 2.12, it is also the unique (global) strong solution to this equation. It remains to check that this solution depends continuously on u 0 .

2.2.4 Proof of Theorem 2.4 Under Assumptions 2.1 and 2.2, let u 0 ∈ H 2 0 (T), and (T * , (u(t)) t∈[0,T * Lemma 2.19 (Continuous dependence on initial conditions). If (u

  2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure and we denote by L M a Lipschitz constant of A over the interval [-M, M ]. Since (u(r)) r∈[s,t] and (v(r)) r∈[s,t] belong to C([s, t], H 2 0 (T)) almost surely, then M is finite almost surely and for all r ∈ [s, t]

  be associated with µ by the second part of Lemma 2.20, and let ϕ ∈ C b (L 1 0 (T)).In particular, the restriction ϕ |H 2 0 (T) is bounded and continuous on H 2 0 (T) and we can write

	Chapter 2. Viscous scalar conservation law with stochastic forcing: strong
				solution and invariant measure
	Let µ ∈ P(H 2 0 (T)) H 2 0 (T)	ϕdP * t µ =	0 (T) H 2	P t ϕdµ.	(2.51)

1 0 (T)), where we can adapt, thanks to Proposition 2.21, the proof of

[START_REF] Da | Ergodicity for Infinite Dimensional Systems[END_REF] Theorem 3.1.1]

.

  has been defined at Proposition 2.8. Taking t ≤ τ M ∧ t ε,M , applying the second part of Proposition 2.9 and Proposition 2.8 to (2.52), we get

	2.3. Invariant measure	53
	where C 4
	2 4 L 2 M t ε,M ,

  Chapter 2. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure Lemma 2.26. Under Assumptions 2.1 and 2.2, any invariant measure µ for the process (u(t)) t≥0 solution to (2.1) is unique.

.

[START_REF] Weinan | Probability Distribution Functions for the Random Forced Burgers Equation[END_REF]

We can write {sup t∈[0,t ε,M ] w(t)

H 1 0 (T) ≤ δ ε,M } = {(w(t)) t∈[0,t ε,M ] ∈ B} where B is the closed ball of C([0, t ε,M ], H 1 0 (T))

with radius δ ε,M . Since the process (w(t)) t∈[0,t ε,M ] is the mild solution to the stochastic heat equation (i.e. Equation (2.7) with initial condition w(0) ≡ 0 and flux A ≡ 0), we can apply the support theorem from [95, Theorem 1.1] which implies P((w(t)) t∈[0,t ε,M ] ∈ B) > 0, so that (2.53) is satisfied.

  thus, by Lemma 2.20, µ 1 = µ 2 . Proof of Theorem 2.7. It follows from Lemmas 2.23 and 2.26.

	Chapter 3
	Finite-volume approximation of the
	invariant measure of a viscous stochastic
	scalar conservation law

Résumé. Ce chapitre correspond aux quatre premières sections de la pré-publication

[START_REF] Boyaval | Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law[END_REF]

, écrite en collaboration avec S. Boyaval et J. Reygner. On se propose d'établir une approximation numérique de la mesure invariante de la solution de l'équation étudiée dans le chapitre 2, c'est-à-dire d'une loi de conservation scalaire avec viscosité, uni-dimensionnelle et périodique en espace, et forcée aléatoirement avec un bruit blanc en temps mais spatialement correlé. La fonction de flux est supposée localement lipschitzienne et à croissance polynomiale. Le schéma numérique utilisé discrétise l'EDPS en espace selon la méthode des volumes finis, et en temps selon une méthode d'Euler à pas fractionné. En premier lieu, pour la semi-discrétisation spatiale puis pour le schéma totalement discrétisé, on prouve l'existence et l'unicité d'une solution puis d'une mesure invariante pour cette solution. Le résultat principal est alors la convergence des mesures invariantes de ces approximations, lorsque les pas de temps et d'espace tendent vers zéro, vers l'unique mesure invariante de l'EDPS par rapport à la distance de Wasserstein d'ordre deux.

  whose initial condition U 0 has distribution ν N . According to Proposition 3.19, one has E[ U 0

		p p ] < +∞. Thus, one can apply Lemma 3.16.(i) which in the
	stationary case, writes
	νpE D	(1)

  A ,(3.58) where we used Lemma 3.31 and Corollary 3.20 with p = 2p A + 2 at the last line.

	Injecting (3.58) into (3.57), we get
	2νE D

in which case Equation (1.2) is usually complemented with boundary conditions, e.g. the Dirichlet condition u |∂D = 0.

depending on the chosen assumptions.

For semi-linear equations, the semi-implicit scheme is implicit with respect to the linear term and explicit with respect to the non-linear term.

These properties constitute what is now called the K41 theory.

, l ∈ T,

Remerciements

Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law

It should be noted that this discrete Poincaré inequality holds in particular for u ∈ R N 0 . Several times in this chapter, we will establish estimates uniformly in N (resp. in ∆t) over the moment of the discrete Sobolev norm E[ D (m) N V p p ] where V is an invariant measure for the semidiscrete scheme (resp. the fully discrete scheme). Whenever this situation appears, we will denote by C m,p (resp. C m,p ) the uniform upper bound.

Lemma 3.11. For any u ∈ R N 0 and p ∈ 2N * , we have

N u ≥

where u p-1 := (u p-1 1 , . . . , u p-1 N ).

Lemma 3.12. Under Assumption 3.3, for any u ∈ R N 0 and any q ∈ 2N * , we have

For any z ∈ R, we write sign(z) := 1 z≥0 -1 z<0 . By extension, for u ∈ R N 0 , sign(u) denotes the vector of {-1, +1} N defined by (sign(u)) i = sign(u i ).

The discretised drift b preserves some nice properties of Equation (3.1) that we will use repeatedly throughout this chapter:

N u 2 2 (dissipativity).

Remark 3.14. The dissipativity property actually holds for the family of E-fluxes [START_REF] Makridakis | Sharp CFL, discrete kinetic formulation, and entropic schemes for scalar conservation laws[END_REF], a larger family than the class of monotone numerical fluxes. The monotonicity assumption (3.13) seems however necessary as regards the L 1 -contraction property.

The semi-discrete scheme

Before addressing the invariant measure of the solution of (3.11), we first ensure the existence and uniqueness of this solution:

Proposition 3.15. Let U 0 be an R N 0 -valued, F 0 -measurable random variable. Under Assumptions 3.1 and 3.3, the stochastic differential equation (3.11) admits a unique strong solution (U(t)) t≥0 taking values in R N 0 and with initial condition U 0 .

towards u 0 , where u

(1)

N j , ∀j ∈ N, and u 0 ∼ µ * . We may define on this probability space, up to enlarging it, a Q-Wiener process (W Q (t)) t≥0 defined as in Section 3.1.1, independent of u 0 and u (1) j,0 , along with a normal filtration. In such a way, we may consider u 0 and u (1) j,0 as initial conditions for the solution of (3.1) and the embedded solutions of (3.11) respectively. More precisely, if we denote by U 0 = (U 1,0 , . . . , U N j ,0 ) the R N j 0 -valued random variable such that U i,0 = u j,0 (i/N ), and if we define (U(t)) t≥0 the solution of (3.11) starting at U 0 , then we define the process (u

N j (t)) t≥0 by u

(1)

N j U(t), for all t ≥ 0. Given that (u

N j (t)) t≥0 is a numerical approximation of (u(t)) t≥0 , convergence at time 0 shall lead to convergence at every finite time t: Proposition 3.36. Under Assumptions 3.1 and 3.3, for every t ≥ 0, we have

This result is proved in Section 3.3.3 below. Let us explain how this finite time result leads to the convergence of (µ

N j is invariant for the process (u

N j (t)) t≥0 . For all t ≥ 0, let µ * t ∈ P(L 2 0 (T)) denote the probability distribution of u(t). By Definition 3.6 and Proposition 3.36, we have

By continuity of the Wasserstein distance [START_REF] Villani | Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften[END_REF]Corollary 6.11] and Lemma 3.32, this leads to

From Lemma 2.20, there exists a unique probability measure in P(H 2 0 (T)) coinciding with µ * on the Borel sets of H 2 0 (T) (for convenience, we still call this measure µ * ). The meaning of (3.53) is that this measure µ * ∈ P(H 2 0 (T)) is invariant for the process (u(t)) t≥0 . However, this process already has a unique invariant measure µ ∈ P(H 2 0 (T)), so that necessarily µ * = µ. As a consequence, µ is the only subsequential limit in P 2 (L 2 0 (T)) of the sequence (µ

N ) N ≥1 , and since from Lemma 3.32, (µ

N ) N ≥1 is relatively compact in P 2 (L 2 0 (T)), we just proved (3.16) (see for instance [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 2.6]) in the case m = 1. The cases m = 0 and m = 2 follow from the bounds (3.51) and (3.52).

Remark 3.37. This last proof shows in particular that µ * is invariant for (u(t)) t≥0 . Therefore it provides a second proof for the existence part in Theorem 2.7.

Proofs

N .

(3.54)

To prove the first inequality of the lemma, apply Corollary 3.20 with p = 2 and recall that we may take C 0,0 = 1.

We focus now on the discrete H 2 0 estimate. Let V ∼ ν N and let (U(t)) t≥0 be the solution of (3.11) with initial distribution ν N . We may compute the dynamics of the discrete H 1 0 -norm of (U(t)) t≥0 by with J N (s) := -

Expanding the product in the definition of J N , we first write

, where

, where

N (s, x)∂ x u(s, x)dx,

N (s, x)dx,

Integration by parts shows that J N 2 (s) = J N 2 (s) and J N 4 (s) = J N 4 (s). We now focus on the computation of J N 1 (s) -J N 1 (s) and J N 3 (s) -J N 3 (s). Using the definition of u (1) N , we get

Direct computation yields

otherwise, and ) n∈N be the solution of (3.15) with initial condition U (j) 0 . For any j ∈ N, we define the piecewise constant approximation (U

Then, for all T > 0,

Proof of Theorem 3.7: Part 2/2. The arguments are identical to the proof of the first part.

Proofs

Proof of Lemma 3.39. Let (U n ) n∈N = (U 1,n , . . . , U N,n ) n∈N be a solution of (3.15) with a deterministic initial condition u 0 . By convexity of the function v → v 4 , for any α, β ∈ R, we have

Hence, expanding the drift function and summing over i, we get

We know thanks to Lemma 3.12 that the second term of the right-hand side is non-negative. Summing by parts the third term, we get

From Lemma 3.11, we get

On the other hand, let us look at the second step of the scheme (3.15). By the construction of the split-step scheme, the random variables U i,n+ 1 Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law

3.A Proofs

Proof of Lemma 3.10. Let m = 0. Let p ∈ [1, +∞) and let u ∈ R N be as in the statement. We have

where we used the Jensen inequality passing from the third to the fourth line. We just have proved the wanted inequality when m = 0 but the case m = 1 is proved in the same way.

Proof of Lemma 3.11. For u ∈ R N 0 and p ∈ 2N * , we have

We can now apply Lemma 3.10 to the vector (sign (u i ) |u i | p/2 ) N i=1 and eventually, we obtain

Proof of Lemma 3.12. Let u ∈ R N 0 and q ∈ 2N * . Summing by parts and using (3.12) and (3.13),

where A q denotes a function defined on R such that A q (z) = A(z 1/(q-1) ).

Proof of Lemma 3.13.

From the definition of b, we write

By periodicity, both terms of the right-hand side can be summed by parts, which leads to

Observe that since the function sign : R → R is non-decreasing, each term of the second sum is non-negative. As for the first sum, it follows from the monotonicity property of A that each term is non-positive. Let us address for instance the case where u i+1 ≥ v i+1 and u i ≤ v i . Then, on the one hand, we have sign(

On the other hand, we have

The case where

Lemma 3.12 with q = 2 shows that the first term of the above decomposition is non-positive. Summing by parts the second term yields the result.

Chapter 3. Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law Proof of Lemma 3.30. (i) The wanted equality follows from standard computations.

(ii) Let us start with the first inequality. We have

As for the second inequality, we have

Chapter 4

Numerical experiments

Résumé. Ce chapitre correspond à la section 5 de [START_REF] Boyaval | Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law[END_REF]. On présente des tests numériques illustrant les théorèmes 3.5 et 3.7 dans le cas de l'équation de Burgers. La stationnarité du schéma totalement discrétisé est observée via un calcul de la moyenne ergodique. Des bornes supérieures sur les taux de convergence en temps et en espace, pour l'approximation en distance de Wasserstein de la mesure invariante, sont calculées empiriquement. Enfin, des simulations du spectre d'énergie et du facteur d'aplatissement (flatness) de la solution stationnaire sont comparées à des résultats théoriques relatifs à la théorie de la turbulence.

Abstract. This chapter corresponds to the fifth section of [START_REF] Boyaval | Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law[END_REF]. We provide numerical experiments to illustrate Theorems 3.5 and 3.7 in the case of the Burgers equation. The stationarity of the fully discrete scheme is observed through the computation of the ergodic mean. Upper bounds on the convergence rates in space and in time, for the approximation of the invariant measure with respect to the Wasserstein distance, are derived empirically. Finally, numerical simulations of the energy spectrum and of the flatness of the stationary solution are compared to theoretical results in turbulence theory.

All the experiments in this section are performed on the Burgers equation, i.e. the flux function is set to be A(u) = u 2 /2. Moreover, we will also fix the following set of parameters: ν = 10 -5 , u 0 ≡ 0, g k (x) = cos(2πkx) for k = 1, . . . , 4 and g k ≡ 0 for k ≥ 5. The implicit equation in (3.15) is solved numerically by use of the Newton-Raphson method.

Stationarity

We seek here to give a numerical illustration of the stationarity of the Markov chain (U N,∆t n ) n∈N defined by (3.15) (in all this section, the number of cells and the time step will always appear as a superscript in the solutions). As already mentioned in Remark 3.9, by virtue of Birkhoff's ergodic theorem, for any test function ϕ : R N 0 → R such that ϕ ∈ L 1 (ν N,∆t ) and any random variable V ∼ ν N,∆t , the process (U N,∆t n ) n∈N shall satisfy

In

Convergence in space

In the following experiment, we aim to retrieve numerically the convergence result of Theorem 3.7 as N tends to infinity and for a fixed time step. Instead of computing directly the Wasserstein distance, we compute the strong L 2 error with respect to a reference solution computed with N ref = 2 11 . More precisely, we record in Figure 4.2 the values of

as N takes values in {2 3 , 2 4 , . . . , 2 10 }. For n sufficiently large, the discrete processes aim to be close to their stationary state and thus, the value (4.1) is meant to be an upper bound of the Wasserstein error approximation of the invariant measure µ:

Here, the other parameters are set to: ∆t = 10 -3 , n = 10 4 .

The result shows that the convergence in space happens at an order of at least 1/2.

Convergence in time

We apply the same procedure to study the convergence with respect to the time step ∆t. A reference solution is computed for the time step ∆t ref = 2 -11 , and for a number n = 10 × 2 11 of iterations, we compute the following L 2 error

, which is supposed to be an upper bound of the Wasserstein distance error between the respective invariant measures for (3.11) and for (3.15):

This error is evaluated when ∆t takes values in {2 -4 , 2 -5 , . . . , 2 -10 } and for N = 256.

A rate of convergence of 1/2 also stems from this experiment.

Burgulence estimates

Endowed with the Burgers flux function, Equation (3.1) may be interpreted as a one-dimensional and simplified version of the Navier-Stokes system, and as such, it is considered a toy model for turbulence (the so-called burgulence, see for instance [15, Chapter 1] or [START_REF] Weinan | Stochastic hydrodynamics[END_REF][START_REF] Weinan | Stochastic PDEs in turbulence theory[END_REF] in this prospect). According to the turbulence theory dating back to Kolmogorov, universal properties emerge as a turbulent dynamical 4.4. Burgulence estimates

where u is a random variable with distribution µ. The flatness aims to be an indicator of the spatial intermittency in the turbulent system described by (3.1). A decay rate for F in the inertial range is provided in [15, Corollary 4.6.9]: Proposition 4.2. Let I be the inertial range from Proposition 4.1. Then,

From (3.15), we computed the numerical approximations of the energy spectrum and the flatness. These computations are plotted in Figure 4.4. More precisely, we used the following respective approximations:

, k = 1, . . . , N 2 ;

Here, ÛN,∆t n is the discrete Fourier transform of U N,∆t n , which we computed using an FFT algorithm from the Python library numpy.fft (v1.17). In both experiments, we have taken N = 256, ∆t = 10 -3 , and n = 10000.

The result of Proposition 4.1 seems recovered as the slope of the energy spectrum tends to behave like k -2 in some sub-interval of [0, 1]. As regards the flatness, for intermediate scales, a decay rate varying from the order -3/2 to the order -1 is observed.

Appendix A

Invariant measures for the numerical schemes in the inviscid multi-dimensional case

Résumé. On se propose dans cette partie d'étudier la stationnarité des schémas semi-discret et complètement discret introduits au chapitre 3 lorsque la viscosité est nulle. Il s'agit de généraliser le résultat du théorème 3.5 au cas non-visqueux. Dans la première section, on expose comment les schémas à flux strictement monotone ont un caractère diffusif exploitable pour assurer l'existence et l'unicité d'une mesure invariante. Dans la seconde section, ce résultat est généralisé à un domaine spatial périodique de dimension quelconque.

Abstract. In this part, we study the stationarity of the semi and fully discrete schemes introduced in Chapter 3 when the viscosity coefficient is equal to zero. We aim to generalise Theorem 3.5 to the inviscid case. In the first section, we expose how strongly monotone schemes have a diffusive nature that can be exploited to ensure the existence and uniqueness of an invariant measure. In the second section, this result is generalised to a periodic spatial domain of arbitrary dimension.

A.1 A remark on strongly monotone numerical fluxes

Let us place ourselves in the setting of Chapter 3 and let us complement Assumption 3.3 with the fact that the numerical flux A is strongly monotone, that is, there exist λ > 0 and a numerical flux function A satisfying Assumption 3.3 such that

This notion was used for instance in [82, Definition 5]. We may notice that it is possible to construct a strongly monotone numerical flux from any numerical flux function satisfying Assumption 3.3 just by adding a term λ(u -v). In particular, there is no restriction on the class of flux functions A that are covered here. We consider here the function b used to establish the semi and fully discrete approximations, respectively (3.11) and (3.15), to Equation (2.1) with such a numerical flux. Recall that this function is defined from R N 0 to R N 0 by its components

Using (A.1), we may write

Appendix A. Invariant measures for the numerical schemes in the inviscid multi-dimensional case

Observe that the numerical scheme we obtain here is the one we would get with the numerical flux A, but with some additional amount of viscosity (actually, with the new viscosity coefficient ν + λ N ). In this respect, strongly monotone numerical fluxes provide a good example of schemes inducing numerical diffusion.

In this context, if we take ν = 0, the function b still satisfies the L 1 -contraction property (Lemma 3.13.(i)), whereas the dissipativity property also holds but with a dependence on N which is not of the same order as Lemma 3.13.(ii) as we have

As a consequence, all the qualitative properties of the processes (U(t)) t≥0 and (U n ) n∈N obtained in Chapter 3 hold, and we have notably Proposition A.1. With a numerical flux of the form (A.1) and a viscosity coefficient ν ≥ 0, the respective solutions (U(t)) t≥0 and (U n ) n∈N to (3.11) and (3.15) both admit a unique invariant measure in P(R N 0 ), denoted respectively ν N and ν N,∆t . However, in the quantitative estimates, all of the upper bounds that relied on the viscosity have been multiplied by N . For instance, Inequality (3.47) has been replaced by

from which we cannot derive directly the tightness of the family (µ N ) N ≥1 , which is the first step of our proof of convergence with respect to N .

A.2 Inviscid and multi-dimensional version of Theorem 3.5

The purpose of this section is to generalise, in the inviscid case, Proposition A.1 to the case where the space variable is multi-dimensional. That is, we prove uniqueness and existence of invariant measures for semi-discrete and fully discrete numerical schemes that aim to approximate conservation laws of the form du = -div (A(u)) dt +

where

• (W k ) k≥1 is a family of independent real Brownian motions.

We consider a mesh T on T d constructed as in [START_REF] Dotti | Convergence of the Finite Volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation[END_REF], i.e. T is a finite family of disjoint open connected sets which form a partition of T d up to a negligible set. For all distinct K, L ∈ T , the intersection of closures K ∩ L is assumed to be contained in a hyperplane. For all K ∈ T ,

defines the set of neighbours of K.

To discretise the flux in Equation (A.2), we will not use an single function A as in the onedimensional case, but a family of functions (A K→L ) K∈T ,L∈N (K) , where each function A K→L : R 2 → R represents the flux of the conserved quantity at the interface K ∩ L.

A.2. Inviscid and multi-dimensional version of Theorem 3.5 113 Definition A.2 (Monotone family of numerical fluxes). A family of continuous functions ( A K→L ) K∈T ,L∈N (K) from R 2 to R is said to be a monotone family of numerical fluxes if for all K ∈ T and all L ∈ N (K), it satisfies the following conditions :

• monotonicity: for all v ∈ R, A K→L (•, v) is non-decreasing and for all u ∈ R, A K→L (u, •) is non-increasing;

• consistency:

where |K ∩ L| denotes the (d -1)-dimensional Hausdorff measure of K ∩ L and n K,L is the outward unit normal to K on K ∩ L;

In this section, our family (A K→L ) K∈T ,L∈N (K) is assumed to be monotone and, as in Section A.1 for the numerical flux A, we make furthermore the following Assumption A.3 (Strong monotonicity). There exist λ > 0 and a monotone family of numerical fluxes ( A K→L ) K∈T ,L∈N (K) such that for all K ∈ T and all L ∈ N (K),

In order to dicretise the noise coefficients, we define for any K ∈ T and any k ≥ 1,

where |K| is the Lebesgue measure of K. We can now define our semi-discrete approximation of (A.2) as the following SDE:

Setting N := #T , Equation (A.3) turns out to be an SDE on the state space

Denoting σ k := (σ k K ) K∈T for all k ≥ 1, the vectorised form of (A.3) writes

We now introduce the split-step time discretisation of (A.4) in exactly the same way as in Chapter 3. Given a time step ∆t > 0, and using the notation

(A.5)

Appendix A. Invariant measures for the numerical schemes in the inviscid multi-dimensional case

Let us now endow the space R N 0 with the renormalised L p norms and L 2 scalar product: for all u = (u K ) K∈T and v = (v K ) K∈T in R N 0 , we define

In this setting, we have Proposition A.4. Under Assumption A.3, for all u, v ∈ R N 0 , we have

where K∼L means the sum over each pair of neighbouring cells K and L.

Before proceeding to the proof of Proposition A.4, we will first establish a preliminary result:

Lemma A.5. For any u ∈ R N 0 , we have

Proof. First of all, we have thanks to the Green formula:

A L→K (z, z) = 0 (A.8)

(see [START_REF] Vovelle | La méthode des volumes finis appliquée aux lois de conservation d'ordre un. Notes de cours[END_REF]Equation (22)]). Let m ∈ R d such that for all neighbouring cells K and L of T , we have n K,L • m = 0. Since the union of all the orthogonal hyperplanes of the normal units n K,L cannot cover R d , existence of such a vector m is forthright.

For every cell K ∈ T , we split its family of neighbours N (K) in the two subsets

Splitting the sum in (A.8), we have by symmetry of the numerical fluxes: for all K ∈ T and all z ∈ R,

Let ψ L→K denote an antiderivative of z → A L→K (z, z). Integrating in (A.9) up to u K and summing over K ∈ T , we get

A.2. Inviscid and multi-dimensional version of Theorem 3.5

115

Observe by the way that each of the double sums in (A.10) amounts to a sum over exactly all the interfaces. Let us now rewrite the left-hand side in (A.10):

Replacing this last term back in (A.10), we obtain

Proof of Proposition A.4 Notice first that for a mapping a : T 2 → R, we have

By monotonicity of A L→K , each term of this sum is non-positive (see the proof of Lemma 3.13).

Appendix A. Invariant measures for the numerical schemes in the inviscid multi-dimensional case

Let us now prove (A.7). Using Assumption A.3, we write

Let us show that S 1 ≤ 0. To finish the proof, we will show that S 2 = -λ K∼L (u L -u K ) 2 .

As in the one-dimensional case, the existence and uniqueness of a solution (U(t)) t≥0 to (A.4) is ensured by (A.7), while the existence and uniqueness of a solution (U n ) n∈N to (A.4) is ensured by (A.7) and (A.6) respectively. Furthermore, the uniqueness of an invariant measure is proved exactly the same way as the one-dimensional case by the use of the L 1 -contraction property for the processes (U(t)) t≥0 and (U n ) n∈N , which follows itself from (A.6). The existence is not as straightforward though. Indeed, Inequality (A.7) leads, in the semi-discrete case, to an estimate of the form

In order for the Krylov-Bogoliubov theorem to apply, one needs the level sets of the mapping u ∈ R N 0 → λ K∼L (u K -u L ) 2 to be compact, which is actually the case as we have Lemma A.6 (Coercivity of the discrete gradient). For every u ∈ R N 0 , we have

A.2. Inviscid and multi-dimensional version of Theorem 3.5

117

Proof. First of all, the result is obvious when u = 0. We shall therefore consider the case u = 0. Let K max be a cell of T such that |u Kmax | = max K∈T |u K |. Without loss of generality, we assume that u Kmax > 0. Since u ∈ R N 0 , there exists K -∈ T such that u K -≤ 0. Now, let (K 1 , . . . , K l ) ∈ T l be such that:

• l ≤ N ;

• K 1 = K -and K l = K max ;

• K i ∈ N (K i+1 ), for all i ∈ {1, . . . , l -1};

• K i = K j , for all i = j.

In other words, (K 1 , . . . , K l ) is a simple path of length l between K -and K max . Since there are no repeating cells in this path, we have

(by Cauchy-Schwarz)

We can now state:

Theorem A.7. Under Assumption A.3, the following two statements hold:

(i) for any N ≥ 1, the process (U(t)) t≥0 solution of the SDE (A.4) admits a unique invariant measure ν N ∈ P(R N 0 );

(ii) for any N ≥ 1 and any ∆t > 0, the process (U n ) n∈N defined by (A.5) admits a unique invariant measure ν N,∆t ∈ P(R N 0 ).