Victor Marx
email: victor.marx@unice.fr

A new approach for the construction of a Wasserstein diffusion

Keywords: Wasserstein diffusion, interacting particle system, coalescing particles, modified Arratia flow, Brownian sheet, differential calculus on Wasserstein space, Itô formula for measure-valued processes AMS MSC 2010: Primary 60K35, 60J60, 60B12, Secondary 60G44, 82B21

. We obtain that process by the construction of a system of particles having shortrange interactions and by letting the range of interactions tend to zero. This construction can be seen as an approximation of the singular process of Konarovskyi by a sequence of smoother processes.

Introduction

This paper introduces a new approach to construct the stochastic diffusion process studied by Konarovskyi ([10], [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF], [START_REF]A system of coalescing heavy diffusion particles on the real line[END_REF], [START_REF] Konarovskyi | Modified Massive Arratia Flow and Wasserstein Diffusion[END_REF]). It is a close relative to the Wasserstein diffusion, introduced by von Renesse and Sturm [START_REF] Von Renesse | Entropic measure and Wasserstein diffusion[END_REF]. Our interest is to construct an analogous process to the Euclidean Brownian motion taking values on the Wasserstein space P 2 (R), defined as the set of probability measures on R having a second-order moment.

In [START_REF] Von Renesse | Entropic measure and Wasserstein diffusion[END_REF], von Renesse and Sturm construct a strong Markov process called Wasserstein diffusion on P 2 (M ), for M equal either to the interval [0, 1] or to the circle S 1 . Two major features of that process illustrate the analogy with the standard Brownian motion on a Euclidean space. First, the energy of the martingale part of the Wasserstein diffusion has the same form as that of a k-dimensional standard Brownian motion, up to replacing the Euclidean norm on R k by the L 2 -Wasserstein distance:

d W (µ, ν) = inf E |X -Y | 2 1/2 ,
where the infimum is taken over all couplings of two random variables X and Y such that X (resp. Y ) has law µ (resp. ν). It should be noticed that the geometry of P 2 (M ), equipped with the Wasserstein distance, for M a Euclidean space, was the subject of fundamental studies conducted by Ambrosio, Gigli, Savare, Villani, Lions and many others ( [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], [START_REF] Cardaliaguet | Notes on Mean Field Games (from P.-L. Lions' lectures at collège de France)[END_REF], [15], [START_REF] Villani | Topics in optimal transportation[END_REF], [START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF]), which led to important improvements in optimal transport theory. Second, the transition costs of the Wasserstein diffusion are given by a Varadhan formula (see [START_REF] Von Renesse | Entropic measure and Wasserstein diffusion[END_REF], Corollary 7. [START_REF] Sturm | A monotone approximation to the Wasserstein diffusion, Singular phenomena and scaling in mathematical models[END_REF]). The formula is identical to the Euclidean case, up to the replacement of the Euclidean norm by d W .

Although the existence of a Wasserstein diffusion was initially proven by von Renesse and Sturm using Dirichlet processes and the theory of Dirichlet forms (see [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]), it can also be obtained as a limit of finite-dimensional systems of interacting particles, see [START_REF] Andres | Particle approximation of the Wasserstein diffusion[END_REF] and [START_REF] Sturm | A monotone approximation to the Wasserstein diffusion, Singular phenomena and scaling in mathematical models[END_REF]. Nevertheless, we will focus in this paper on a construction of a system of particles which seems more natural and simpler and which is due to Konarovskyi in [START_REF] Konarovskii | On an infinite system of diffusing particles with coalescing[END_REF] and [START_REF]A system of coalescing heavy diffusion particles on the real line[END_REF].

Konarovskyi's model

In [START_REF]A system of coalescing heavy diffusion particles on the real line[END_REF], Konarovskyi studies a simple system of N interacting and coalescing particles and proves its convergence to an infinite-dimensional process which has the features of a diffusion on the L 2 -Wasserstein space of probability measures (see also [START_REF] Konarovskii | On an infinite system of diffusing particles with coalescing[END_REF], [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF], [START_REF] Konarovskyi | Modified Massive Arratia Flow and Wasserstein Diffusion[END_REF]). However, even if it has common properties with the diffusion of von Renesse and Sturm, there are also important differences between the two processes. An outstanding property of Konarovskyi's process is the fact that, for a large family of initial measures, it takes values in the set of measures with finite support for each time t > 0 (see [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF]), whereas the values of the Wasserstein diffusion of von Renesse and Sturm are probability measures on [0, 1] with no absolutely continuous part and no discrete part.

The model introduced by Konarovskyi is a modification of the Arratia flow, also called Coalescing Brownian flow, introduced by Arratia [START_REF] Alejandro | Coalescing brownian motions on the line[END_REF] and subject of many interest, among others in [START_REF] Dorogovtsev | One Brownian stochastic flow[END_REF], [START_REF] Le | Flows, coalescence and noise[END_REF], [START_REF] Norris | Weak convergence of the localized disturbance flow to the coalescing Brownian flow[END_REF], [START_REF] Piterbarg | Expansions and contractions of isotropic stochastic flows of homeomorphisms[END_REF]. It consists of Brownian particles starting at discrete points of the real line and moving independently until they meet another particle: when they meet, they stick together to form a single Brownian particle.

In his model (see [START_REF]A system of coalescing heavy diffusion particles on the real line[END_REF]), Konarovskyi adds a mass to every particle: at time t = 0, N particles, denoted by (x k (t)) k∈{1,...,N } , start from N points regularly distributed on the unit interval [0, 1], and each particle has a mass equal to 1 N . When two particles stick together, they form as in the standard Arratia flow a unique particle, but with a mass equal to the sum of the two incident particles. Furthermore, the quadratic variation process of each particle is assumed to be inversely proportional to its mass. In other words, the heavier a particle is, the smaller its fluctuations are.

Konarovskyi constructs an associated process (y N (u, t)) u∈[0,1],t∈[0,T ] in the set D([0, 1], C[0, T ]) of càdlàg functions on [0, 1] taking values in C[0, T ] by setting:

y N (u, t) := N k=1 x k (t)1 {u∈[ k-1 N , k N )} + x N (t)1 {u=1} .
In other words, y N (•, t) is the quantile function associated to the empirical measure 1 N N k=1 δ x k (t) . Konarovskyi showed in [START_REF]A system of coalescing heavy diffusion particles on the real line[END_REF] that the sequence (y N ) N 1 is tight in D([0, 1], C[0, T ]). Hence, by passing to the limit upon a subsequence, there exists a process (y(u, t)) u∈[0,1],t∈[0,T ] belonging to D([0, 1], C[0, T ]) and satisfying the following four properties: (i 0 ) for all u ∈ [0, 1], y(u, 0) = u;

(ii) for all u v, for all t ∈ [0, T ], y(u, t) y(v, t);

(iii) for all u ∈ [0, 1], y(u, •) is a square integrable continuous martingale relatively to the filtration (F t ) t∈[0,T ] := (σ(y(v, s), v ∈ [0, 1], s t)) t∈[0,T ] ;

1 {y g (u,t)=y g (v,t)} dv.

Moreover, we can compare the diffusive properties of the process (µ t ) t∈[0,T ] in the Wasserstein space P 2 (R) with the Wasserstein diffusion of von Renesse and Sturm. To that extent and thanks to Lions' differential calculus on P 2 (R) ( [15], [START_REF] Cardaliaguet | Notes on Mean Field Games (from P.-L. Lions' lectures at collège de France)[END_REF]), we give in Appendix A an Itô formula on P 2 (R) for the process (µ t ) t∈[0,T ] in order to describe the energy of the martingale part of this diffusion. Appendix A also contains a small introduction to the differentiability on P 2 (R) in the sense of Lions.

Approximation of a Wasserstein diffusion

In this paper, we propose a new method to construct a process y satisfying properties (i)-(iv), by approaching y by a sequence of smooth processes. Finding smooth approximations of processes having singularities has already led to interesting results, typically in the case of the Arratia flow. Piterbarg [START_REF] Piterbarg | Expansions and contractions of isotropic stochastic flows of homeomorphisms[END_REF] shows that the Coalescing Brownian flow is the weak limit of isotropic homeomorphic flows in some space of discontinuous functions, and deduces from the properties of the limit process a careful description of contraction and expansion regions of homeomorphic flows. Dorogovtsev's approximation [START_REF] Dorogovtsev | One Brownian stochastic flow[END_REF] is based on a representation of the Arratia flow with a Brownian sheet.

We propose an adaptation of Dorogovtsev's idea in the case of Wasserstein diffusions. First, we show that a process y satisfying (i)-(iv) admits a representation in terms of a Brownian sheet; we refer to the lectures of Walsh [START_REF] Walsh | An introduction to stochastic partial differential equations[END_REF] for a complete introduction to Brownian sheet and to Section 2 for the characterization of Brownian sheet which we use in this paper.

1 {y(u,s)=y(u ,s)} m(u, s) dw(u , s),

where m(u, s) = 1 0 1 {y(u,s)=y(v,s)} dv.

Remark 1.2. We refer to Appendix A to justify the use of the term "Wasserstein diffusion" for a process satisfying equation [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]. Indeed, we can write an Itô formula for this process for a smooth function u : P 2 (R) → R. As in the case of the standard Euclidean Brownian motion, the quadratic variation of the martingale term is proportional to the square of the gradient of u, in the sense of Lions' differential calculus on P 2 (R), which is the same as the differential calculus on the Wasserstein space (see [START_REF] Carmona | Probabilistic theory of mean field games with applications. I[END_REF]Section 5.4]).

The aim of this paper is to construct a sequence of smooth processes approaching y in the space L 2 ([0, 1], C[0, T ]). Therefore, we use the representation [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] in terms of a Brownian sheet of y and, given a positive parameter σ, we replace in the latter representation the indicator functions by a smooth function ϕ σ equal to 1 in the neighbourhood of 0 and whose support is included in the intervalσ 2 , σ 2 of small diameter σ. Fix σ > 0 and ε > 0. Given a Brownian sheet w on [0, 1] × [0, T ], we prove the existence of a process y σ,ε satisfying:

y σ,ε (u, t) = g(u) + t 0 1 0 ϕ σ (y σ,ε (u, s) -y σ,ε (u , s)) ε + m σ,ε (u, s) dw(u , s), (2)
where m σ,ε (u, s) := 1 0 ϕ2 σ (y σ,ε (u, s)-y σ,ε (v, s))dv can be seen as a kind of mass of particle y σ,ε (u) at time s. Remark that, due to the fact that the support of ϕ σ is small, only the particles located at a distance lower than σ 2 of particle u at time s are taken into account in the computation of the mass m σ,ε (u, s).

The smooth process (y σ,ε (u, t)) u∈[0,1],t∈[0,T ] offers several advantages. First, we are able to construct a strong solution (y σ,ε , w) to equation [START_REF] Andres | Particle approximation of the Wasserstein diffusion[END_REF], whereas in equation (1), we do not know if, given a Brownian sheet w, there exists an adapted solution y. Second, in Konarovskyi's process, the question of uniqueness of a solution to [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], even in the weak sense, or equivalently the question of uniqueness of a process in L 2 ([0, 1], C[0, T ]) satisfying conditions (i)-(iv), remains open. Here, pathwise uniqueness holds for equation [START_REF] Andres | Particle approximation of the Wasserstein diffusion[END_REF]. Moreover, the measure-valued process (µ σ,ε t ) t∈[0,T ] associated to the process of quantile functions (y σ,ε (•, t)) t∈[0,T ] does generally no longer consist of atomic measures. For example, if g(u) = u, (µ σ,ε t ) t∈[0,T ] is a process of absolutely continuous measures with respect to the Lebesgue measure.

Let L 2 [0, 1] be the usual space of square integrable functions from [0, 1] to R, and (•, •) L 2 the usual scalar product. We denote by L ↑ 2 [0, 1] the set of functions f ∈ L 2 [0, 1] such that there exists a non-decreasing and therefore càdlàg (i.e. right-continuous with left limits everywhere) element in the equivalence class of f . Let D((0, 1), C[0, T ]) be the space of right-continuous C[0, T ]-valued functions with left limits, equipped with the Skorohod metric.

We follow the definition given in [9, p.21]:

Definition 1.3. An (F t ) t∈[0,T ] -adapted process M is an L ↑ 2 [0, 1]-valued (F t ) t∈[0,T ] -martingale if M t belongs to L ↑ 2 [0, 1] for each t ∈ [0, T ], if E [ M t L 2 ] < ∞ and if for each h ∈ L 2 [0, 1], (M t , h) L 2 is a real-valued (F t ) t∈[0,T ] -martingale. The martingale is said to be square integrable if for each t ∈ [0, T ], E M t December 2018
Let us denote by L ↑ 2+ [0, 1] the set of all non-decreasing and càdlàg functions g : [0, 1] → R, where R := R ∪ {-∞, +∞}, such that there exists p > 2 for which

1 0 |g(u)| p du < +∞. Let Q + = Q ∩ [0, 1].
The following Theorem states the convergence of the mollified sequence (y σ,ε ) σ>0,ε>0 to a limit process satisfying properties (i) -(iv). It uses the framework introduced by Konarovskyi in [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF]:

Theorem 1.4. Let g ∈ L ↑ 2+ [0, 1].
For each positive σ and ε, there exists a solution y σ,ε to equation [START_REF] Andres | Particle approximation of the Wasserstein diffusion[END_REF]

such that (y σ,ε (u, t)) u∈[0,1],t∈[0,T ] belongs to L 2 ([0, 1], C[0, T ]) and almost surely, for each t ∈ [0, T ], y σ,ε (•, t) ∈ L ↑ 2 [0, 1]
. Furthermore, up to extracting a subsequence, the sequence (y σ,ε ) ε>0 converges in distribution in L 2 ([0, 1], C[0, T ]) for every σ ∈ Q + as ε tends to 0 to a limit y σ and the sequence

(y σ ) σ∈Q + converges in distribution in L 2 ([0, 1], C[0, T ]) as σ tends to 0 to a limit y. Let Y (t) := y(•, t). Then (Y (t)) t∈[0,T ] is a L ↑ 2 [0, 1]-valued process such that: (C1) Y (0) = g; (C2) (Y (t)) t∈[0,T ] is a square integrable continuous L ↑ 2 [0, 1]-valued (F t ) t∈[0,T ] -martingale, where F t := σ(Y (s), s t); (C3) almost surely, for every t > 0, Y (t) is a step function, i.e. there exist n 1, 0 = a 1 < a 2 < • • • < a n < a n+1 = 1 and z 1 < z 2 < • • • < z n such that for all u ∈ [0, 1] Y (t)(u) = y(u, t) = n k=1 z k 1 {u∈[a k ,a k+1 )} + z n 1 {u=1} ;
(C4) y belongs to D((0, 1), C[0, T ]) and for every u ∈ (0, 1), y(u, •) is a square integrable and continuous (F t ) t∈[0,T ] -martingale and P [∀u, v ∈ (0, 1), ∀s ∈ [0, T ], y(u, s) = y(v, s) implies ∀t s, y(u, t) = y(v, t)] = 1;

(C5) for each u and u in (0, 1),

y(u, •), y(u , •) t = t 0 1 {τ u,u s} m(u, s) ds,
where m(u, s) = 1 0 1 {y(u,s)=y(v,s)} dv and τ u,u = inf{t 0 : y(u, t) = y(u , t)} ∧ T .

Remark 1.5. More precisely, the filtration (F t ) t∈[0,T ] is given by:

F t = σ((Y (s), h) L 2 , s t, h ∈ L 2 [0, 1]).
Remark 1.6. By property (C4), the limit process y is said to be coalescent: if for a certain time t 0 , two particles y(u, t 0 ) and y(v, t 0 ) coincide, then they move together forever, i.e. y(u, t) = y(v, t) for every t t 0 .

It is interesting to wonder how the coalescence property of the process y translates to its smooth approximation y σ,ε : two paths (y σ,ε (u, t)) t∈[0,T ] and (y σ,ε (v, t)) t∈[0,T ] , starting from two distinct points g(u) and g(v), do not meet, which means that y σ,ε (•, t) is non-decreasing for each fixed t. If y σ,ε (u, •) and y σ,ε (v, •) get close enough, at distance smaller than σ, they begin to interact and to move together, whereas as long as they remain at distance greater than σ, they move "independently": more precisely, the covariation y σ,ε (u, •), y σ,ε (v, •) t is equal to zero for every time t τ σ u,v := inf{s 0 : |y σ,ε (u, s) -y σ,ε (v, s)| σ} (see figure 1).

Organisation of the article

We begin in Section 2 by proving Theorem 1.1, which states that a process y satisfying properties (i)-(iv) admits a representation in terms of a Brownian sheet. In Section 3, given a two-dimensional Brownian sheet, we prove the existence of a smooth process in the space L 2 ([0, 1], C[0, T ]) intended to approach Konarovskyi's process of coalescing particles. This smooth process can be seen as a cloud of point-particles interacting with all the particles at a distance smaller than σ, and in which two particles have independent trajectories conditionally to the fact that the distance between them is greater than σ. When the distance becomes smaller than σ, both trajectories are correlated, mimicking the coalescence property.

Section 4 is devoted to the proof of convergence when the parameter ε and the range of interaction σ tend to zero, using a tightness criterion in L 2 ([0, 1], C[0, T ]). In Section 5, we study the stochastic properties of the limit process, including the convergence of the mass process. The aim of this final part is to prove that the limit process y satisfies properties (C1)-(C5) of Theorem 1.4, in other words that our sequence of short-range interaction processes converges in distribution to the process of coalescing particles.

In Appendix A, we give an Itô formula in the Wasserstein space for the limit process y, after having recalled some basic definitions and properties of Lions' differential calculus on P 2 (R).

Singular representation of the process y

Let (Ω, F, P) be a probability space. Let us consider on (Ω, F, P) a random process y ∈ L 2 ((0, 1), C[0, T ]) satisfying properties (i)-(iv). We refer to [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF] for a comprehensive construction of y. We will give another one later in this paper.

The aim of this Paragraph is to prove Theorem 1.1. Before that, we recall the definition of a Brownian sheet given by Walsh in [23, p.269]. Let (E, E, ν) be a Euclidean space equipped with Lebesgue measure. A white noise based on ν is a random set function W on the sets A ∈ E of finite ν-measure such that

• W (A) is a N (0, ν(A)) random variable, • if A ∩ B = ∅, then W (A) and W (B) are independent and W (A ∩ B) = W (A) + W (B). Let T > 0. Consider E = [0, 1] × [0, T ] and ν the associated Lebesgue measure. The Brownian sheet w on [0, 1]×[0, T ] associated to the white noise W is the process (w(u, t)) u∈[0,1]×[0,T ] defined by w(u, t) := W ((0, u] × (0, t]).
Define the filtration

(G t ) t∈[0,T ] by G t := σ(w(u, s), u ∈ [0, 1], s t). Then in particular, (i) for each (G t ) t∈[0,T ] -progressively measurable function f defined on [0, 1] × [0, T ] such that T 0 1 0 f 2 (u, s)duds < +∞ almost surely, the process t 0 1 0 f (u, s)dw(u, s) t∈[0,T ]
is a local martingale (we often write dw(u, s) instead of w(du, ds));

(ii) for each f 1 and f 2 satisfying the same conditions as f ,

• 0 1 0 f 1 (u, s)dw(u, s), • 0 1 0 f 2 (u, s)dw(u, s) t = t 0 1 0 f 1 (u, s)f 2 (u, s)duds.
By Lévy's characterization of the Brownian motion, a process w satisfying (i) and (ii) is a Brownian sheet. Let us now prove Theorem 1.1.

Proof (Theorem 1.1). We take a Brownian sheet η on [0, 1] × [0, T ] independent of the process y, constructed by possibly extending the probability space (Ω, F, P). Then, we define (w(u, t)) u∈[0,1],t∈[0,T ] by w(0, •) ≡ 0, w(•, 0) ≡ 0 and:

w(du, dt) = η(du, dt) + y(u, dt)du - 1 m(u, t) 1 0 1 {y(u,t)=y(u ,t)} η(du , dt)du.
We denote by H t the filtration σ((y(u, s)) u∈[0,1],s t , (η(u, s)) u∈[0,1],s t ).

In order to prove that w is an (H t ) t∈[0,T ] -Brownian sheet on [0, 1] × [0, T ], let us consider two (H t ) t∈[0,T ] -progressively measurable functions f 1 and f 2 and compute, using independence of η and y:

• 0 1 0 f 1 (u, s)dw(u, s), • 0 1 0 f 2 (v, s)dw(v, s) t = V 1 + V 2 -V 3 -V 4 + V 5 , where V 1 := • 0 1 0 f 1 (u, s)dη(u, s), • 0 1 0 f 2 (v, s)dη(v, s) t = t 0 1 0 f 1 (u, s)f 2 (u, s)duds, since η is an (H t ) t∈[0,T ] -Brownian sheet; V 2 := • 0 1 0 f 1 (u, s)dy(u, s)du, • 0 1 0 f 2 (v, s)dy(v, s)dv t = t 0 1 0 1 0 f 1 (u, s)f 2 (v, s) 1 {y(u,s)=y(v,s)} m(u, s) dudvds,

Construction of a Wasserstein diffusion

December 2018 using property (iv) of process y;

V 3 := • 0 1 0 f 1 (u, s)dη(u, s), • 0 1 0 f 2 (v, s) m(v, s) 1 0 1 {y(v,s)=y(v ,s)} dη(v , s)dv t = t 0 1 0 1 0 f 1 (u, s)f 2 (v, s) m(v, s) 1 {y(v,s)=y(u,s)} dudvds = V 2 ,
since m(u, s) = m(v, s) whenever y(u, s) is equal to y(v, s). By similar computations,

V 4 := • 0 1 0 f 1 (u, s) m(u, s) 1 0 1 {y(u,s)=y(u ,s)} dη(u , s)du, • 0 1 0 f 2 (v, s)dη(v, s) t = V 2 ,
and

V 5 : = • 0 1 0 f 1 (u, s) m(u, s) 1 0 1 {y(u,s)=y(u ,s)} dη(u , s)du, • 0 1 0 f 2 (v, s) m(v, s) 1 0 1 {y(v,s)=y(v ,s)} dη(v , s)dv t = t 0 1 0 1 0 1 0 f 1 (u, s)f 2 (v, s) m(u, s)m(v, s) 1 {y(u,s)=y(u ,s)} 1 {y(v,s)=y(u ,s)} du dudvds = t 0 1 0 1 0 f 1 (u, s)f 2 (v, s) m(u, s) 2 1 0 1 {y(u,s)=y(u ,s)} du 1 {y(u,s)=y(v,s)} dudvds = t 0 1 0 1 0 f 1 (u, s)f 2 (v, s) m(u, s) 1 {y(u,s)=y(v,s)} dudvds = V 2 .
To sum up,

• 0 1 0 f 1 (u, s)dw(u, s), • 0 1 0 f 2 (v, s)dw(v, s) t = V 1 = t 0 1 0 f 1 (u, s)f 2 (u, s)duds,
whence w is an (H t ) t∈[0,T ] -Brownian sheet. Finally, we show that (y, w) satisfies equation (1): The result follows from the two below equalities:

W 2 = t 0 1 0 1 {y(u,s)=y(u ,s)} m(u, s) dy(u, s)du = t 0 dy(u, s) = y(u, t) -y(u, 0) = y(u, t) -g(u); W 3 = t 0 1 0 1 {y(u,s)=y(u ,s)} m(u, s) 1 0 1 {y(u ,s)=y(v,s)} m(v, s) dη(v, s)du = t 0 1 0 1 {y(u,s)=y(v,s)} m(u, s)m(v, s) 1 0 1 {y(u ,s)=y(v,s)} du dη(v, s) = t 0 1 0 1 {y(u,s)=y(v,s)} m(u, s) dη(v, s),
which implies that W 3 = W 1 and consequently equation [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF].

Therefore, every solution of the martingale problem (i)-(iv) has a representation in terms of a Brownian sheet. In the next Section, we will construct, given a Brownian sheet, an approximation of the process y.

Construction of a Wasserstein diffusion

3 Construction of a process with short-range interactions Let (Ω, F, P) be a probability space, on which we define a Brownian sheet w on [0, 1] × [0, T ]. We associate to that process the filtration G t := σ(w(u, s), u ∈ [0, 1], s t). Up to completing the filtration, we assume that G 0 contains all the P-null sets of F and that the filtration (G t ) t∈[0,T ] is right-continuous.

Fix σ > 0 and ε > 0. Let ϕ σ denote a smooth and even function, bounded by 1, equal to 1 on [0, σ 3 ] and equal to 0 on [ σ 2 , +∞). Recall that L ↑ 2+ [0, 1] represents the set of non-decreasing and càdlàg functions g : [0, 1] → R such that there exists p > 2 satisfying

1 0 |g(u)| p du < +∞.
The aim of this Section is to construct, for each initial quantile function g ∈ L ↑ 2+ [0, 1], a square integrable random variable y g σ,ε taking values in L 2 ([0, 1], C[0, T ]) such that almost surely, for every t ∈ [0, T ], the following equality holds in L 2 [0, 1]:

y g σ,ε (•, t) = g + t 0 1 0 ϕ σ (y g σ,ε (•, s) -y g σ,ε (u , s)) ε + 1 0 ϕ 2 σ (y g σ,ε (•, s) -y g σ,ε (v, s))dv dw(u , s). (3)
Remark 3.1. We add the parameter ε to the denominator in order to ensure that it is bounded by below. We also point out that relation (3) has to be compared with equation [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], where

x → 1 {x=0} is replaced by the function ϕ σ .

More precisely, we will prove the following Proposition. Recall that L ↑ 2 [0, 1] represents the set of functions f ∈ L 2 [0, 1] such that there is a non-decreasing and càdlàg element in the equivalence class of f .

Proposition 3.2. Let g ∈ L ↑ 2+ [0, 1]. There exists an L ↑ 2 [0, 1]-valued process (Y g σ,ε (t)) t∈[0,T ] = (y g σ,ε (•, t)) t∈[0,T ] such that: (A1) Y g σ,ε (0) = g; (A2) Y g σ,ε is a square integrable continuous L ↑ 2 [0, 1]-valued (F σ,ε t ) t∈[0,T ] -martingale, where F σ,ε t := σ(Y g σ,ε (s), s t); (A3) for every h, k ∈ L 2 [0, 1], (Y g σ,ε , h) L 2 , (Y g σ,ε , k) L 2 t = t 0 1 0 1 0 h(u)k(u ) m g σ,ε (u, u , s) (ε + m g σ,ε (u, s))(ε + m g σ,ε (u , s))
dudu ds,

where m g σ,ε (u, u , s) = 1 0 ϕ σ (y g σ,ε (u, s) -y g σ,ε (v, s))ϕ σ (y g σ,ε (u , s) -y g σ,ε (v, s))dv and m g σ,ε (u, s) = 1 0 ϕ 2 σ (y g σ,ε (u, s) -y g σ,ε (v, s))dv.

Existence of an approximate solution

Denote by M the set of random variables

z ∈ L 2 (Ω, C([0, T ], L 2 (0, 1))) such that (z(ω, •, t)) t∈[0,T ] is a (G t ) t∈[0,T ]
-progressively measurable process with values in L 2 (0, 1). We consider the following norm on M:

z M = E sup t T 1 0 |z(u, t)| 2 du 1/2 .
Throughout this Section, σ and ε are two fixed positive numbers. To begin, we want to prove that the map ψ : M → M, defined below, admits a unique fixed point. Fix g ∈ L ↑ 2+ [0, 1] an initial quantile function. For all z ∈ M, define:

ψ(z)(ω, u, t) := g(u) + t 0 1 0 ϕ σ (z(ω, u, s) -z(ω, u , s)) ε + m σ (ω, u, s) dw(ω, u , s), (4)
December 2018

where m σ (ω, u, s) = 1 0 ϕ 2 σ (z(ω, u, s) -z(ω, v, s))dv. We start by making sure that ψ is welldefined.

Proposition 3.3. For all z ∈ M, ψ(z) belongs to M. Furthermore, (ψ(z)(•, t)) t∈[0,T ] is an L 2 (0, 1)-valued continuous (G t ) t∈[0,T ] -martingale.

Remark 3.4. The definition of an L ↑ 2 [0, 1]-valued martingale was given in Definition 1.3. Up to replacing L ↑ 2 by L 2 , the definition of an L 2 (0, 1)-valued martingale is exactly the same.

Proof. We want to prove that (ψ(z)(•, t)) t∈[0,T ] is an L 2 (0, 1)-valued (G t ) t∈[0,T ] -martingale. Since z belongs to M, the process

(z(•, t)) t∈[0,T ] is (G t ) t∈[0,T ] -progressively measurable. Therefore (m σ (•, t)) t∈[0,T ] is also (G t ) t∈[0,T ]
-progressively measurable and we deduce that

(ψ(z)(•, t)) t∈[0,T ] is (G t ) t∈[0,T ] -progressively measurable. Then, we check that for each t ∈ [0, T ], ψ(z)(•, t) ∈ L 2 (0, 1) and E [ ψ(z)(•, t) L 2 ] < ∞.
We deduce this statement by recalling that g L 2 < +∞, because g ∈ L ↑ 2+ [0, 1], and by computing:

E t 0 1 0 ϕ σ (z(•, s) -z(u , s)) ε + m σ (•, s) dw(u , s) L 2 2 E t 0 1 0 ϕ σ (z(•, s) -z(u , s)) ε + m σ (•, s) dw(u , s) 2 L 2 = E 1 0 t 0 1 0 ϕ σ (z(u, s) -z(u , s)) ε + m σ (u, s) dw(u , s) 2 du = 1 0 E t 0 1 0 ϕ σ (z(u, s) -z(u , s)) ε + m σ (u, s) dw(u , s) 2 du = 1 0 E t 0 1 0 ϕ σ (z(u, s) -z(u , s)) ε + m σ (u, s) 2 du ds du ϕ σ 2 ∞ t ε 2 = t ε 2 < +∞. (5)
Furthermore, for each h ∈ L 2 [0, 1],

(ψ(z)(•, t), h) L 2 = (g, h) L 2 + t 0 1 0 1 0 h(u) ϕ σ (z(u, s) -z(u , s)) ε + m σ (u, s) dudw(u , s)
is a (G t ) t∈[0,T ] -local martingale. Then, we compute the quadratic variation:

E [ (ψ(z), h) L 2 , (ψ(z), h) L 2 t ] = t 0 1 0 1 0 1 0 h(u 1 )h(u 2 ) ϕ σ (z(u 1 , s) -z(u , s))ϕ σ (z(u 2 , s) -z(u , s)) (ε + m σ (u 1 , s))(ε + m σ (u 2 , s)) du 1 du 2 du ds t ε 2 h 2 L 2 .
Since it is finite, the local martingale is actually a martingale. Moreover, by Doob's inequality (see Theorem 2.2 in [9, p.22])

ψ(z) M = E sup t T 1 0 |ψ(z)(u, t)| 2 du 1/2 g L 2 + E sup t T 1 0 t 0 1 0 ϕ σ (z(u, s) -z(u , s)) ε + m σ (u, s) dw(u , s) 2 du 1/2 g L 2 + 2E 1 0 T 0 1 0 ϕ σ (z(u, s) -z(u , s)) ε + m σ (u, s) dw(u , s) 2 du 1/2
.

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The last term is finite by [START_REF] Cardaliaguet | Notes on Mean Field Games (from P.-L. Lions' lectures at collège de France)[END_REF]. Thus ψ(z) M is finite and ψ(z) belongs to M, which concludes the proof.

Let us now prove that ψ has a unique fixed point: Proposition 3.5. Let σ > 0 and ε > 0. Then the map ψ : M → M defined by (4) has a unique fixed point in M, denoted by y g σ,ε . Proof. For all n ∈ N, denote by ψ n the n-fold composition of ψ, where ψ 0 denotes the identity function of M. We want to prove that ψ n is a contraction for n large enough.

Let z 1 and z 2 be two elements of M. We define

h n (t) := E sup s t 1 0 |ψ n (z 1 )(u, s) -ψ n (z 2 )(u, s)| 2 du .
Let us remark that h n (T ) = ψ n (z 1 )-ψ n (z 2 ) 2 M and recall that, by Proposition 3.3,

(ψ(z 1 )(•, t)- ψ(z 2 )(•, t)) t∈[0,T ] is a (G t ) t∈[0,T ] -martingale.
We denote by m σ,1 and m σ,2 the masses associated respectively to z 1 and z 2 . By Doob's inequality, we have:

h 1 (t) = E sup s t 1 0 |ψ(z 1 )(u, s) -ψ(z 2 )(u, s)| 2 du = E sup s t 1 0 s 0 1 0 ϕ σ (z 1 (u, r) -z 1 (u , r)) ε + m σ,1 (u, r) - ϕ σ (z 2 (u, r) -z 2 (u , r)) ε + m σ,2 (u, r) dw(u , r) 2 du 4E 1 0 t 0 1 0 ϕ σ (z 1 (u, s) -z 1 (u , s)) ε + m σ,1 (u, s) - ϕ σ (z 2 (u, s) -z 2 (u , s)) ε + m σ,2 (u, s) 2 du dsdu .
Furthermore, we compute:

ϕ σ (z 1 (u, s) -z 1 (u , s)) ε + m σ,1 (u, s) - ϕ σ (z 2 (u, s) -z 2 (u , s)) ε + m σ,2 (u, s) 2 2 ϕ σ (z 1 (u, s) -z 1 (u , s)) -ϕ σ (z 2 (u, s) -z 2 (u , s)) ε + m σ,1 (u, s) 2 + ϕ σ (z 2 (u, s) -z 2 (u , s)) (ε + m σ,1 (u, s))(ε + m σ,2 (u, s)) (m σ,1 (u, s) -m σ,2 (u, s)) 2 .
Moreover, we have:

|m σ,1 (u, s) -m σ,2 (u, s)| 1 0 |ϕ 2 σ (z 1 (u, s) -z 1 (v, s)) -ϕ 2 σ (z 2 (u, s) -z 2 (v, s))|dv Lip(ϕ 2 σ ) 1 0 |(z 1 (u, s) -z 1 (v, s)) -(z 2 (u, s) -z 2 (v, s))|dv Lip(ϕ 2 σ ) |z 1 (u, s) -z 2 (u, s)| + 1 0 |z 1 (v, s) -z 2 (v, s)|dv .
We obtain the following upper bound:

ϕ σ (z 1 (u, s) -z 1 (u , s)) ε + m σ,1 (u, s) - ϕ σ (z 2 (u, s) -z 2 (u , s)) ε + m σ,2 (u, s) 2 4 Lip ϕ σ ε 2 + 4 Lip(ϕ 2 σ ) ε 2 2 |z 1 (u, s) -z 2 (u, s)| 2 + |z 1 (u , s) -z 2 (u , s)| 2 + 1 0 |z 1 (v, s) -z 2 (v, s)| 2 dv .

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Finally, we deduce that there is a constant C σ,ε depending only on σ and ε such that

h 1 (t) C σ,ε E t 0 1 0 |z 1 (u, s) -z 2 (u, s)| 2 duds C σ,ε t 0 E sup r s 1 0 |z 1 (u, r) -z 2 (u, r)| 2 du ds = C σ,ε t 0 h 0 (s)ds.
Applied to ψ n (z 1 ) and ψ n (z 2 ) instead of z 1 and z 2 , those computations show that for every t ∈ [0, T ], h n+1 (t) C σ,ε t 0 h n (s)ds. Using the fact that h 0 is non-decreasing with respect to t, it follows that h n (T )

(Cσ,εT ) n n!
h 0 (T ), whence we have:

ψ n (z 1 ) -ψ n (z 2 ) 2 M (C σ,ε T ) n n! z 1 -z 2 2 M .
Thus, for n large enough, the map ψ n is a contraction. By completeness of M under the norm • M (remark that M is a closed subset of L 2 (Ω, C([0, T ], L 2 (0, 1))), it follows that ψ has a unique fixed point in M.

We denote by y g σ,ε the unique fixed point of ψ. Remark that by construction it satisfies equation ( 3) almost surely and for every t ∈ [0, T ].

Non-decreasing property

Define, for each t ∈ [0, T ], Y g σ,ε (t) := y g σ,ε (•, t). So far, by Proposition 3.5, we have established that (Y g σ,ε (t)) t∈[0,T ] is an L 2 [0, 1]-valued process, satisfying property (A1) of Proposition 3.2. Since Y g σ,ε belongs to M, and by Proposition 3.3, (Y g σ,ε (t)) t∈[0,T ] is a square integrable continuous L 2 [0, 1]-valued martingale, with respect to the filtration (G t ) t∈[0,T ] . Therefore, it is also an (F σ,ε t ) t∈[0,T ] -martingale, where F σ,ε t := σ(Y g σ,ε (
s), s t). In order to obtain property (A2), it remains to prove the following statement:

Proposition 3.6. (Y g σ,ε (t)) t∈[0,T ] is an L ↑ 2 [0, 1]-valued process.
We will start by proving three Lemmas and then we will conclude the proof of Proposition 3.6. For every x ∈ R, we consider the following stochastic differential equation:

z(x, t) = x + t 0 1 0 ϕ σ (z(x, s) -y g σ,ε (u , s)) ε + 1 0 ϕ 2 σ (z(x, s) -y g σ,ε (v, s))dv dw(u , s), (6)
where y g σ,ε is the unique solution of equation ( 3).

Lemma 3.7. Let x ∈ R. For almost every ω ∈ Ω, equation (6) has a unique solution in C[0, T ], denoted by (z(ω, x, t)) t∈[0,T ] . Moreover, (z(x, t)) t∈[0,T ] is a real-valued (G t ) t∈[0,T ] -martingale.
Proof. We get existence and uniqueness of the solution by applying a fixed-point argument. The proof is the same as the proof of Proposition 3.5. We obtain the martingale property by the same argument as in Proposition 3.3.

Then, take x 1 , x 2 ∈ R. After some computations similar to those of the proof of Proposition 3.5, we have for every t ∈ [0, T ]:

E sup s t |z(x 1 , s) -z(x 2 , s)| 2 2|x 1 -x 2 | 2 + C σ,ε t 0 E sup r s |z(x 1 , r) -z(x 2 , r)| 2 ds.
December 2018 By Gronwall's Lemma, we deduce that:

E sup t T |z(x 1 , t) -z(x 2 , t)| 2 C σ,ε |x 1 -x 2 | 2 .
By Kolmogorov's Lemma, there is a modification z of z in C(R × [0, T ]). We define y g σ,ε (u, t) := z(g(u), t). In particular, u → y g σ,ε (u, •) is measurable and, since g is a càdlàg function, y g σ,ε

belongs to D((0, 1), C[0, T ]). Remark 3.8. In the case where g is continuous, it is straightforward to see that y g σ,ε belongs to

C([0, 1] × [0, T ]).
Furthermore, y g σ,ε belongs to M. Indeed,

E sup t T 1 0 y g σ,ε (u, t) 2 du E 1 0 sup t T y g σ,ε (u, t) 2 du = 1 0 E sup t T y g σ,ε (u, t) 2 du.
By Lemma 3.7, for every u ∈ [0, 1], ( y g σ,ε (u, t)) t∈[0,T ] is a martingale, we have by Doob's inequality:

E sup t T y g σ,ε (u, t) 2 CE y g σ,ε (u, T ) 2 2Cg(u) 2 + 2CE   T 0 1 0 ϕ σ ( y g σ,ε (u, s) -y g σ,ε (u , s)) ε + 1 0 ϕ 2 σ ( y g σ,ε (u, s) -y g σ,ε (v, s))dv 2 du ds   2Cg(u) 2 + 2C T ε 2 . Therefore, y g σ,ε M 2C g 2 L 2 + 2C T ε 2 < +∞. Moreover, ( y g σ,ε (•, t)) t∈[0,T ] is an L 2 [0, 1]-valued (G t ) t∈[0,T ] -martingale. Indeed, for every h ∈ L 2 [0, 1], for every t ∈ [0, T ], E [( y g σ,ε (•, t), h) L 2 ]
is finite. Fix 0 s t T , and A s ∈ G s . We have:

E 1 0 y g σ,ε (u, t)h(u)du - 1 0 y g σ,ε (u, s)h(u)du 1 As = 1 0 E ( y g σ,ε (u, t) -y g σ,ε (u, s))1 As h(u)du = 0.
Lemma 3.9. We have

E sup t T 1 0 | y g σ,ε (u, t) -y g σ,ε (u, t)| 2 du = 0. Therefore, y g σ,ε = y g σ,ε in M. Proof. Since ( y g σ,ε (•, t) -y g σ,ε (•, t)) t∈[0,T ] is an L 2 [0, 1]-valued martingale, then by [9, p.21-22] 1 0 | y g σ,ε (u, t) -y g σ,ε (u, t)| 2 du is a real-valued submartingale. By Doob's inequality, E sup s t 1 0 y g σ,ε (u, s) -y g σ,ε (u, s) 2 du CE 1 0 y g σ,ε (u, t) -y g σ,ε (u, t) 2 du CE 1 0 t 0 1 0 (θ σ,ε ( y g σ,ε (u, s), u , s) -θ σ,ε (y g σ,ε (u, s), u , s))dw(u , s) 2 du CE 1 0 t 0 1 0 θ σ,ε ( y g σ,ε (u, s), u , s) -θ σ,ε (y g σ,ε (u, s), u , s) 2 du dsdu , where θ σ,ε (x, u , s) = ϕσ(x-y g σ,ε (u ,s)) ε+ 1 0 ϕ 2 σ (x-y g σ,ε (v,s))dv
. Using the same constant C σ,ε as in the proof of Proposition 3.5, we have:

E sup s t 1 0 y g σ,ε (u, s) -y g σ,ε (u, s) 2 du C σ,ε E 1 0 t 0 y g σ,ε (u, s) -y g σ,ε (u, s) 2 dsdu C σ,ε t 0 E sup r s 1 0 y g σ,ε (u, r) -y g σ,ε (u, r) 2 du ds.

Construction of a Wasserstein diffusion

December 2018 By Gronwall's Lemma, we deduce that E sup s t

1 0 | y g σ,ε (u, s) -y g σ,ε (u, s)| 2 du = 0 for every t ∈ [0, T ].
This implies the statement of the Lemma.

Lemma 3.10. Almost surely, for every u 1 , u 2 ∈ Q such that u 1 < u 2 , we have for every t 0,

y g σ,ε (u 1 , t) y g σ,ε (u 2 , t). Furthermore, if g(u 1 ) < g(u 2 ) (resp. g(u 1 ) = g(u 2 )), then for every t 0, y g σ,ε (u 1 , t) < y g σ,ε (u 2 , t) (resp. y g σ,ε (u 1 , t) = y g σ,ε (u 2 , t)). Proof. Let (u 1 , u 2 ) ∈ Q 2 such that 0 u 1 < u 2 1. For u = u 1 , u 2 ,
we have:

y g σ,ε (u, t) = g(u) + t 0 1 0 θ σ,ε ( y g σ,ε (u, s), u , s)dw(u , s), where θ σ,ε (x, u , s) = ϕσ(x-y g σ,ε (u ,s)) ε+ 1 0 ϕ 2 σ (x-y g σ,ε (v,s))dv
. Therefore, we have (writing y instead of y g σ,ε and θ instead of θ σ,ε ):

y(u 2 , t) -y(u 1 , t) = g(u 2 ) -g(u 1 ) + t 0 1 0 (θ( y(u 2 , s), u , s) -θ( y(u 1 , s), u , s))dw(u , s) = g(u 2 ) -g(u 1 ) + t 0 ( y(u 2 , s) -y(u 1 , s))dM s (7)
where

M t = t 0 1 0 1 { y(u 2 ,s) = y(u 1 ,s)} θ( y(u 2 ,s),u ,s)-θ( y(u 1 ,s),u ,s) y(u 2 ,s)-y(u 1 ,s)
dw(u , s). Observe that:

θ( y(u 2 , s), u , s) -θ( y(u 1 , s), u , s) = y(u 2 ,s) y(u 1 ,s) ∂ x θ(x, u , s)dx,
and that ∂ x θ(x, u , s) = ϕ σ (x-y(u ,s)) ε+ 1 0 ϕ 2 σ (x-y(v,s))dv - ϕσ(x-y(u ,s)) 1 0 (ϕ 2 σ ) (x-y(v,s))dv (ε+ 1 0 ϕ 2 σ (x-y(v,s))dv) 2 . Therefore, ∂ x θ is bounded uniformly in (x, u , s) ∈ R × [0, 1] × [0, T ] by C σ,ε := ϕ σ L∞ ε + ϕσ L∞ (ϕ 2 σ ) L∞ ε 2 . We deduce that E [ M, M T ] = E T 0 1 0 1 { y(u 2 ,s) = y(u 1 ,s)} θ( y(u 2 , s), u , s) -θ( y(u 1 , s), u , s) y(u 2 , s) -y(u 1 , s) 2 du ds E T 0 1 0 (C σ,ε ) 2 du ds T (C σ,ε ) 2 ,
and thus M is a (G t ) t∈[0,T ] -martingale on [0, T ]. We resolve the stochastic differential equation ( 7):

y g σ,ε (u 2 , t) -y g σ,ε (u 1 , t) = (g(u 2 ) -g(u 1 )) exp M t -1 2 M, M t . If g(u 1 ) < g(u 2 ) (resp. g(u 1 ) = g(u 2 )), then almost surely for every t ∈ [0, T ], y g σ,ε (u 1 , t) < y g σ,ε (u 2 , t) (resp. =). Thus it is true almost surely for every (u 1 , u 2 ) ∈ Q 2 such that u 1 < u 2 .
Therefore the proof of Proposition 3.6 is complete:

Proof (Proposition 3.6). For each t ∈ [0, T ], Y g σ,ε (t) = y g σ,ε (•, t) has a modification y g σ,ε (•, t) be- longing to L ↑ 2 [0, 1].
We precise the properties of y g σ,ε in the following Corollary, which derives directly from Proposition 3.6. From now on, we will always use this version of the process.

Corollary 3.11. The following two statements hold:

• for almost every u ∈ (0, 1), ( y g σ,ε (ω, u, t)) t∈[0,T ] is a (F σ,ε t ) t∈[0,T ]
-martingale, and it is continuous for almost every (u, ω) ∈ (0, 1) × Ω.

December 2018

• almost surely, for every t ∈ [0, T ], u → y g σ,ε (u, t) is càdlàg and non-decreasing.

We complete the proof of Proposition 3.2.

Proof (Proposition 3.2). Thanks to Proposition 3.6, the proof of properties (A1) and (A2) has been completed. It remains to compute the quadratic variation. Recall that for every u ∈ [0, 1], (

y g σ,ε (u, t)) t∈[0,T ] is a (G t ) t∈[0,T ] -martingale and that y g σ,ε (u, t) = g(u) + t 0 1 0 θ σ,ε ( y g σ,ε (u, s), u , s)dw(u , s).
Therefore, for every u, u ∈ [0, 1],

y g σ,ε (u, •), y g σ,ε (u , •) t = • 0 1 0 θ σ,ε ( y g σ,ε (u, s), v, s)dw(v, s), • 0 1 0 θ σ,ε ( y g σ,ε (u , s), v, s)dw(v, s) t = t 0 1 0 θ σ,ε ( y g σ,ε (u, s), v, s)θ σ,ε ( y g σ,ε (u , s), v, s)dvds. Therefore, for every h, k ∈ L 2 [0, 1], (Y g σ,ε , h) L 2 , (Y g σ,ε , k) L 2 t = t 0 1 0 1 0 h(u)k(u ) 1 0 θ σ,ε ( y g σ,ε (u, s), v, s)θ σ,ε ( y g σ,ε (u , s), v, s)dvdudu ds = t 0 1 0 1 0 h(u)k(u ) 1 0 θ σ,ε (y g σ,ε (u, s), v, s)θ σ,ε (y g σ,ε (u , s), v, s)dvdudu ds = t 0 1 0 1 0 h(u)k(u ) m g σ,ε (u, u , s) (ε + m g σ,ε (u, s))(ε + m g σ,ε (u , s))
dudu ds, which completes the proof.

We conclude this Section with a property on the quadratic variation of two fixed particles, which will be useful to obtain lower bounds on the mass in the next Section. Corollary 3.12. For almost every u, u ∈ [0, 1],

y g σ,ε (u, •), y g σ,ε (u , •) t = t 0 1 0 m g σ,ε (u, u , s) (ε + m g σ,ε (u, s))(ε + m g σ,ε (u , s)) dvds. (8)
Proof. This statement follows clearly from the proof of Proposition 3.2, from the fact that for almost every u ∈ (0, 1), (

y g σ,ε (u, t)) t∈[0,T ] is a continuous martingale.
4 Convergence of the process (y g σ,ε ) σ,ε∈Q + From now on, for the sake of simplicity, we fix a function g in L ↑ 2+ [0, 1] and y σ,ε will denote the version y g σ,ε starting from g. We denote by p a number such that p > 2 and g ∈ L p (0, 1). We begin by proving the tightness of the sequence (y σ,ε ) σ,ε∈Q + in the space L 2 ([0, 1], C[0, T ]) in Paragraph 4.1. We will then pass to the limit in distribution, first when ε → 0 and then when σ → 0 and prove, in Paragraph 4.3, that the limit process is also a martingale.

Tightness of the collection (y

σ,ε ) σ>0,ε>0 in L 2 ([0, 1], C[0, T ])
Recall that for all σ > 0, the map ϕ σ is smooth, even, bounded by 1, equal to 1 on 0, σ-η 2 and equal to 0 on σ 2 , +∞ , where η is chosen so that η < σ 3 . Recall that y σ,ε is solution of the following equation:

y σ,ε (u, t) = g(u) + t 0 1 0 ϕ σ (y σ,ε (u, s) -y σ,ε (u , s)) ε + 1 0 ϕ 2 σ (y σ,ε (u, s) -y σ,ε (v, s))dv dw(u , s).
We begin by proving that the collection (y σ,ε ) σ>0,ε>0 satisfies a compactness criterion in the space L 2 ([0, 1], C[0, T ]). We recall the following criterion (see [18, Theorem 1, p.71]):

Proposition 4.1. Let K be a subset of L 2 ([0, 1], C[0, T ]). K is relatively compact in L 2 ([0, 1], C[0, T ]
) if and only if:

(H1) for every 0 u 1 < u 2 1, u 2 u 1 f (u, •)du, f ∈ K is relatively compact in C[0, T ], (H2) lim h→0 + sup f ∈K 1-h 0 f (u + h, •) -f (u, •) 2 C[0,T ] du = 0.
By Ascoli's Theorem, (H1) is satisfied if and only if for every 0

u 1 < u 2 1, -for every t ∈ [0, T ], u 2 u 1 f (u, t)du is uniformly bounded, -lim η→0 + sup f ∈K sup |t 2 -t 1 |<η u 2 u 1 (f (u, t 2 ) -f (u, t 1 ))du = 0.
In order to prove tightness for the collection (y σ,ε ) σ>0,ε>0 , we will prove the following Proposition: Proposition 4.2. Let δ > 0. The following statements hold:

(K1) there exists M > 0 such that for all σ > 0, ε > 0, P

1 0 y σ,ε (u, •) 2 C[0,T ] du M 1 -δ, (
K2) for all k 1, there exists η k > 0 such that for all σ > 0, ε > 0,

P 1 0 sup |t 2 -t 1 |<η k |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )|du 1 k 1 - δ 2 k , ( K3
) for all k 1, there exists h k > 0 such that for all σ > 0, ε > 0,

P ∀h ∈ (0, h k ), 1-h 0 y σ,ε (u + h, •) -y σ,ε (u, •) 2 C[0,T ] du 1 k 1 - δ 2 k . Proposition 4.2 will be proved in Paragraph 4.1.2. It implies tightness of (y g σ,ε ) σ>0,ε>0 in L 2 ([0, 1], C[0, T ]): Corollary 4.3. For all g ∈ L ↑ 2+ [0, 1], the collection (y g σ,ε ) σ>0,ε>0 is tight in L 2 ([0, 1], C[0, T ]). Proof (Corollary 4.3). Let δ > 0. Let M , (h k ) k 1 , (η k ) k 1 be such that the statements of Proposition 4.2 hold for δ. Denote K δ the closed set of all functions f ∈ L 2 ([0, 1], C[0, T ]) satisfying: (L1) 1 0 f (u, •) 2 C[0,T ] du M .
December 2018

(L2) for all k 1, 1 0 sup |t 2 -t 1 |<η k |f (u, t 2 ) -f (u, t 1 )|du 1 k . (L3) for all k 1, ∀h ∈ (0, h k ), 1-h 0 f (u + h, •) -f (u, •) 2 C[0,T ] du 1 k . Let 0 u 1 < u 2 1.
We deduce from (L1) that for every t ∈ [0, T ], and every f ∈ K δ ,

u 2 u 1 f (u, t)du u 2 u 1 f (u, t) 2 du 1/2 1 0 f (u, •) 2 C[0,T ] du 1/2 √ M . We deduce from (L2) that for every k 1, sup f ∈K δ sup |t 2 -t 1 |<η k u 2 u 1 (f (u, t 2 ) -f (u, t 1 ))du sup f ∈K δ 1 0 sup |t 2 -t 1 |<η k |f (u, t 2 ) -f (u, t 1 )|du 1 k .
Therefore, by Ascoli's Theorem, condition (H1) of Proposition 4.1 is satisfied. Furthermore, by (L3), condition (H2) is also satisfied uniformly for

f ∈ K δ . Therefore, K δ is compact in L 2 ([0, 1], C[0, T ]). By Proposition 4.2, for all σ > 0, ε > 0, P [y σ,ε ∈ K δ ] 1 -3δ.
This concludes the proof.

To prove Proposition 4.2, we will first give in the next Paragraph an estimation of the inverse of the mass function (see Lemma 4.6). This Lemma is an equivalent in our case of short-range interacting particles of Lemma 2.16 in [START_REF]A system of coalescing heavy diffusion particles on the real line[END_REF], stated in the case of a system of coalescing particles.

Estimation of the inverse of mass

Recall that m σ,ε (u, t) = 1 0 ϕ 2 σ (y σ,ε (u, t) -y σ,ε (v, t))dv.
We define a modified mass

M σ,ε (u, t) :=      (ε + m σ,ε ) 2 m σ,ε (u, t) if m σ,ε (u, t) > 0, + ∞ otherwise.
Clearly, M σ,ε (u, t) m σ,ε (u, t) for every u ∈ [0, 1] and t ∈ [0, T ]. By Corollary 3.11, there exists a (non-random) Borel set A in [0, 1], Leb(A) = 1, such that for all u ∈ A, (y σ,ε (u, t)) t∈[0,T ] is almost surely a continuous (F σ,ε t ) t∈[0,T ] -martingale. Recall also that almost surely, for every t ∈ [0, T ], u → y σ,ε (u, t) is càdlàg and non-decreasing. Moreover, we assume that for every u, u ∈ A, equality (8) holds.

Lemma 4.4. There exist C > 0 and γ ∈ (0, 1) such that for each σ, ε > 0, t ∈ (0, T ] and for every u ∈ A and every h > 0 satisfying u -h ∈ (0, 1),

P T 0 1 {Mσ,ε(u,s)<γh} ds t C [g(u) -g(u -h)] h t . (9)
Proof. Fix σ > 0 and ε > 0. Let h > 0 be such that u -h belongs to A. If g(u -h) = g(u), then for every t ∈ [0, T ], y σ,ε (u-h, t) = y σ,ε (u, t). By the non-decreasing and càdlàg property, for every v ∈ (u -h, u), we have y σ,ε (v, t) = y σ,ε (u, t). We deduce that m σ,ε (u, t)

u u-h ϕ 2 σ (y σ,ε (u, t) - y σ,ε (v, t))dv = u u-h ϕ 2 σ (0)dv = h. Therefore, M σ,ε (u, t) h γh for every t ∈ [0, T ],
and ( 9) is satisfied.

Consider now the case where g(u -h) < g(u). Choose k in ( h 3 , 2h 3 ) such that u -k ∈ A. Denote by N and N the two following (F σ,ε t ) t∈[0,T ] -martingales:

N t = y σ,ε (u, t) -y σ,ε (u -h, t), N t = y σ,ε (u, t) -y σ,ε (u -k, t).
December 2018 Denote by G s and H s respectively the events M σ,ε (u, s) < h

2 6
and { N s > σ+η 2 }. We want to prove the existence of a constant C 1 independent of h and u such that for all σ > 0, ε > 0 and t > 0,

P T 0 1 {Gs} ds t C 1 [g(u) -g(u -h)] h t . (10)
Decompose this probability in two terms:

P T 0 1 {Gs} ds t P T 0 1 {Gs∩Hs} ds t 2 + P T 0 1 {Gs∩H s } ds t 2 , (11)
where H s denotes the complement of the event H s .

• First step:

Study of G s ∩ H s . Fix s ∈ [0, T ]. Under G s ∩ H s , we have M σ,ε (u, s) < h 2 6 and N s > σ+η 2 .
We want to show that it implies the following inequality:

2m σ,ε (u, u -h, s) (ε + m σ,ε (u, s))(ε + m σ,ε (u -h, s)) 1 M σ,ε (u, s) 3/4 M σ,ε (u -h, s) 1/4 . (12)
Suppose, by contradiction, that ( 12) is false. Using Cauchy-Schwarz inequality,

m σ,ε (u, u - h, s) m σ,ε (u, s) 1/2 m σ,ε (u -h, s) 1/2
, and we would deduce that:

1 M σ,ε (u, s) 3/4 M σ,ε (u -h, s) 1/4 2 M σ,ε (u, s) 1/2 M σ,ε (u -h, s) 1/2 , and thus M σ,ε (u -h, s) 2 4 M σ,ε (u, s). Using the fact that M σ,ε m σ,ε , we can deduce that m σ,ε (u, s) + m σ,ε (u -h, s) M σ,ε (u, s) + M σ,ε (u -h, s) (1 + 2 4 ) h 2 6 < h 3 . (13)
We distinguish three cases depending on the value of N s = y σ,ε (u, s) -y σ,ε (u -h, s).

• N s σ -η: For each v ∈ [u -h, u], one of the two terms y σ,ε (u, s) -y σ,ε (v, s) and y σ,ε (v, s) -y σ,ε (u -h, s) is lower than σ-η 2 , which means that one of those terms belongs to the preimage of 1 by the function ϕ σ . Hence m σ,ε (u, s)

+ m σ,ε (u -h, s) = 1 0 ϕ 2 σ (y σ,ε (u, s) -y σ,ε (v, s)) + ϕ 2 σ (y σ,ε (u -h, s) -y σ,ε (v, s)) dv u u-h dv = h.
This is in contradiction with [START_REF] Konarovskyi | Modified Massive Arratia Flow and Wasserstein Diffusion[END_REF]. Therefore inequality [START_REF]A system of coalescing heavy diffusion particles on the real line[END_REF] is satisfied in this case.

• N s ∈ (σ -η, σ): Introduce Med := {v : y σ,ε (u, s) -y σ,ε (v, s) ∈ [ σ-η 2 , σ+η 2
]}, which is a set of particles more or less at half distance between particle u and particle u -h. Since η < σ 3 , we have

N s > σ -η σ+η 2 and thus Med ⊂ [u -h, u]. Let v ∈ [u -h, u]. We distinguish three new cases: -if y σ,ε (u, s) -y σ,ε (v, s) < σ-η 2 , then ϕ σ (y σ,ε (u, s) -y σ,ε (v, s)) = 1. -if y σ,ε (u, s) -y σ,ε (v, s) > σ+η 2 , and since N s σ, y σ,ε (v, s) -y σ,ε (u -h, s) is lower than σ-η 2 and thus ϕ σ (y σ,ε (u -h, s) -y σ,ε (v, s)) = 1. -otherwise, v belongs to Med. It follows that: h = u u-h (1 {yσ,ε(u,s)-yσ,ε(v,s)< σ-η 2 } + 1 {yσ,ε(u,s)-yσ,ε(v,s)> σ+η 2 } + 1 {v∈Med} )dv u u-h (ϕ 2 σ (y σ,ε (u, s) -y σ,ε (v, s)) + ϕ 2 σ (y σ,ε (u -h, s) -y σ,ε (v, s)) + 1 {v∈Med} )dv m σ,ε (u, s) + m σ,ε (u -h, s) + Leb(Med).

December 2018

By inequality [START_REF] Konarovskyi | Modified Massive Arratia Flow and Wasserstein Diffusion[END_REF], we deduce that Leb(Med)

> 2h 3 . As Med is an interval included in [u -h, u] and since k ∈ ( h 3 , 2h 3 ) we deduce that u -k ∈ Med, i.e. N s ∈ [ σ-η 2 , σ+η 2
], which is in contradiction with the hypothesis N s > σ+η 2 . Thus inequality ( 12) is also true in this case.

• N s σ: In this case, the two particles u and u -h do not have any interaction. In other words, since the support of

ϕ σ is included in [-σ 2 , σ 2 ], ϕ σ (y σ,ε (u, s) -y σ,ε (v, s)) and ϕ σ (y σ,ε (u -h, s) -y σ,ε (v, s))
can not be simultaneously non-zero, whence we deduce that m σ,ε (u, u -h, s) = 0. Inequality (12) follows clearly.

Therefore, inequality ( 12) is proved. By Corollary 3.12, it follows that, on G s ∩ H s :

d ds N, N s = 1 M σ,ε (u, s) + 1 M σ,ε (u -h, s) - 2m σ,ε (u, u -h, s) (ε + m σ,ε (u, s))(ε + m σ,ε (u -h, s)) 1 M σ,ε (u, s) + 1 M σ,ε (u -h, s) - 1 M σ,ε (u, s) 3/4 M σ,ε (u -h, s) 1/4 1 4M σ,ε (u, s) + 3 4M σ,ε (u -h, s) 1 4M σ,ε (u, s) 2 4 h ,
where we have applied a convexity inequality:

∀a, b > 0, a 3/4 b 1/4 3a 4 + b 4 . To sum up, we showed that G s ∩ H s implies d ds N, N s 2 4 h . If T 0 1 {Gs∩Hs} ds t 2 , we get N, N T = T 0 d ds N, N s ds T 0 d ds N, N s 1 {Gs∩Hs} ds 2 4 h T 0 1 {Gs∩Hs} ds 2 3 t h .
Hence, since N is a continuous square integrable (F σ,ε t ) t∈[0,T ] -martingale, there exists a standard (F σ,ε t ) t∈[0,T ] -Brownian motion β such that N t = g(u) -g(u -h) -β( N, N t ). Since N remains positive on [0, T ] by Lemma 3.10 (because g(u -h) < g(u)), we deduce that sup [0, N,N T ] β g(u) -g(u -h). Therefore,

P T 0 1 {Gs∩Hs} ds t 2 P   sup [0, 2 3 t h ] β g(u) -g(u -h)   = P 2 3 h sup [0,t] β g(u) -g(u -h) C 2 [g(u) -g(u -h)] h t , ( 14
)
where β is a rescaled Brownian motion and C 2 does not depend on u, h, σ, ε and t.

• Second step:

Study of G s ∩ H s .
Under this event, we have M σ,ε (u, s) < h 2 6 and N s σ+η 2 . In particular, by the assumption η < σ 3 , we have N s σ -η. We claim that the following inequality holds true:

2m σ,ε (u, u -k, s) (ε + m σ,ε (u, s))(ε + m σ,ε (u -k, s)) 1 M σ,ε (u, s) 3/4 M σ,ε (u -k, s) 1/4 . ( 15
)
To prove it, it is sufficient to imitate the proof of the case N s σ -η of the previous step.

We should notice that we did not use the hypothesis N s > σ+η 2 in that case.

December 2018

Using inequality (15) as in the first step, we show that d ds N , N s 2 4

h . Therefore,

P T 0 1 {Gs∩H s } ds t 2 P N , N T 2 3 t h . There exists a (F σ,ε t ) t∈[0,T ] -Brownian mo- tion β such that N t = g(u) -g(u -k) -β( N , N t ).
Finally, we obtain the existence of a constant C 3 independent of u, h, k, σ, ε and t such that:

P T 0 1 {Gs∩H s } ds t 2 P   sup [0, 2 3 t h ] β g(u) -g(u -k)   C 3 [g(u) -g(u -k)] h t C 3 [g(u) -g(u -h)] h t . (16)
Putting together inequality [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF] and inequalities ( 14) and ( 16), we conclude the proof of inequality [START_REF] Konarovskii | On an infinite system of diffusing particles with coalescing[END_REF]. Thus inequality ( 9) is proved for every h such that u -h ∈ A. Let h > 0 be such that u -h ∈ (0, 1). Let h 1 ∈ ( h 2 , h) be such that u -h 1 ∈ A.

P T 0 1 {Mσ,ε(u,s)< γh 2 } ds t P T 0 1 {Mσ,ε(u,s)<γh 1 } ds t C [g(u) -g(u -h 1 )] h 1 t C [g(u) -g(u -h)] h t .
Up to replacing γ by γ 2 , inequality (9) follows for every h > 0 such that u -h ∈ (0, 1).

Remark 4.5. Similarly, there exist C > 0 and γ ∈ (0, 1) such that for each σ > 0, ε > 0, t ∈ (0, T ] and for every u ∈ A and every h > 0 satisfying u + h ∈ (0, 1),

P T 0 1 {Mσ,ε(u,s)<γh} ds t C [g(u + h) -g(u)] h t .
Thanks to Lemma 4.4 and to the above remark, we obtain the following result, which has to be compared with Proposition 4.3 in [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF]: Lemma 4.6. Let g ∈ L p (0, 1). For all β ∈ (0, 3 2 -1 p ), there is a constant C > 0 depending only on β and g Lp such that for all σ, ε > 0 and 0 s < t T , we have the following inequality:

E t s 1 0 1 M β σ,ε (u, r) dudr C √ t -s. ( 17
)
Remark 4.7. Observe that by the assumption p > 2, made at the beginning of Section 4, there exists some β > 1 such that (17) holds.

Proof. By Fubini-Tonelli Theorem, we have:

E t s 1 0 1 M β σ,ε (u, r) dudr = 1 0 E t s +∞ 0 1 {M -β σ,ε (u,r)>x} dxdr du 2 β (t -s) + 1 0 +∞ 2 β E t s 1 {Mσ,ε(u,r)<x -1/β } dr dxdu 2 β √ T √ t -s + 1 0 +∞ 2 β γ β E t s 1 {Mσ,ε(u,r)<γx -1/β } dr γ -β dxdu.
December 2018

Furthermore, we compute:

E t s 1 {Mσ,ε(u,r)<γx -1/β } dr = t-s 0 P t s 1 {Mσ,ε(u,r)<γx -1/β } dr > α dα t-s 0 P T 0 1 {Mσ,ε(u,r)<γx -1/β } dr > α dα.
Using Lemma 4.4, we obtain a constant C 1 independent of σ and ε such that for all x > 2 β :

1 1 2 E t s 1 {Mσ,ε(u,r)<γx -1/β } dr du 1 1 2 t-s 0 C 1 g(u) -g(u -x -1/β ) x -1/β α dαdu 2C 1 1 1/2 (g(u) -g(u -x -1/β ))du x 1/(2β) √ t -s.
Moreover, we have for each x > 2 β , using Hölder's inequality:

1 1 2 g(u) -g(u -x -1/β ) du = 1 0 1 [ 1 2 ,1] (u) -1 [ 1 2 -x -1/β ,1-x -1/β ] (u) g(u)du g Lp (2x -1/β ) 1-1 p . (18)
Therefore,

1 1 2 +∞ 2 β γ β E t s 1 {Mσ,ε(u,r)<x -1/β } dr dxdu C 2 +∞ 2 β γ β g Lp √ t -s x 1 2β x 1 β (1-1 p ) dx C 3 g Lp √ t -s,
where C 2 and C 3 are independent of σ, ε, and t. The last inequality holds because 1

β 3 2 -1 p > 1.
We conclude the proof of the Lemma by using a similar argument for u belonging to [0, 1 2 ] and using g(u + x -1/β ) -g(u) instead of g(u) -g(u -x -1/β ).

Corollary 4.8. There is a constant C such that for every t ∈ [0, T ] and for every σ, ε > 0,

E 1 0 y 2 σ,ε (u, t)du C.
Proof. We have:

E 1 0 y 2 σ,ε (u, t)du 1/2 E 1 0 g(u) 2 du 1/2 + E 1 0 (y σ,ε (u, t) -g(u)) 2 du 1/2 .
Since g belongs to L 2 (0, 1), the first term of the right hand side is bounded. Furthermore, by Corollary 3.12 and Fubini-Tonelli Theorem: Proposition 4.9. Let g ∈ L ↑ 2+ [0, 1] and δ be positive. Then there exists M > 0 such that for all σ > 0 and ε > 0, P

E 1 0 (y σ,ε (u, t) -g(u)) 2 du = 1 0 E [ y σ,ε (u, •), y σ,ε (u, •) t ] du = 1 0 E t 0 1 M σ,ε (u, s) ds du C √ t,
1 0 y σ,ε (u, •) 2 C[0,T ] du M δ.
Proof. Using again Fubini-Tonelli Theorem,

E 1 0 sup t T |y σ,ε (u, t)| 2 du = 1 0 E sup t T |y σ,ε (u, t)| 2 du.
Moreover, for almost every u

∈ [0, 1], y σ,ε (u, •) is a (F σ,ε t ) t∈[0,T ] -martingale.
Hence by Doob's inequality, there is a constant C 1 independent of u, σ and ε such that:

E sup t T |y σ,ε (u, t)| 2 C 1 E |y σ,ε (u, T )| 2 .
Therefore, by Corollary 4.8,

E 1 0 sup t T |y σ,ε (u, t)| 2 du C 1 1 0 E |y σ,ε (u, T )| 2 du C 2 , ( 19
)
where C 2 is independent of σ and ε. We conclude by Markov's inequality: there is a constant C > 0 such that for all σ, ε > 0,

P 1 0 y σ,ε (u, •) 2 C[0,T ] du M E 1 0 sup t T |y σ,ε (u, t)| 2 du M C M .
For M large enough, that last quantity is smaller than δ.

Then, we show criterion (K2):

Proposition 4.10. Let g ∈ L p [0, 1] and δ > 0. Then for all k 1, there exists η k > 0 such that for every σ, ε > 0,

P 1 0 sup |t 2 -t 1 |<η k |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )|du 1 k δ 2 k .
Proof. By Markov's inequality, it is sufficient to prove that:

lim η→0 + sup σ>0,ε>0 E 1 0 sup |t 2 -t 1 |<η |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )|du = 0. ( 20
)
Fix δ > 0 and β ∈ (1, 3 2 -1 p ). For every u ∈ (0, 1), define

K 1 (u) := E y σ,ε (u, •) C[0,T ] , K 2 (u) := E T 0 1 M β σ,ε (u, s)
ds .

Since y σ,ε is uniformly bounded for σ > 0 and ε > 0 in L 2 ([0, 1], C[0, T ]) (see inequality [START_REF] Sturm | A monotone approximation to the Wasserstein diffusion, Singular phenomena and scaling in mathematical models[END_REF]) and by Lemma 4.6, December 2018

Therefore, there exists C > 0 such that

1 0 1 {K 1 (u) C} du δ and 1 0 1 {K 2 (u) C} du δ.
We define:

K 1 := {u ∈ (0, 1) : K 1 (u) C}, K 2 := {u ∈ (0, 1) : K 2 (u) C}.
The collection (y σ,ε (u, •)) σ>0,ε>0,u∈K 1 ∩K 2 is tight in C[0, T ]. We use Aldous' tightness criterion to prove this claim (see [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 16.10]). We prove the two following statements:

lim a→∞ sup σ>0,ε>0,u∈K

1 ∩K 2 P y σ,ε (u, •) C[0,T ] a = 0.
-for all α > 0 and r > 0, there is η 0 such that for all η ∈ (0, η 0 ), for all σ > 0,

ε > 0 and u ∈ K 1 ∩ K 2 , if τ is a stopping time for y σ,ε (u, •) such that τ T , then P [|y σ,ε (u, τ + η) -y σ,ε (u, τ )| r] α. By Markov's inequality, for all a > 0, σ > 0, ε > 0 and u ∈ K 1 ∩ K 2 , P y σ,ε (u, •) C[0,T ] a 1 a E y σ,ε (u, •) C[0,T ] = K 1 (u) a C a ,
whence we obtain the first statement. Moreover, for all u ∈ K 1 ∩ K 2 , by Hölder's inequality,

E |y σ,ε (u, τ + η) -y σ,ε (u, τ )| 2 = E τ +η τ 1 M σ,ε (u, s) ds K 2 (u) 1 β η 1-1 β C 1 β η 1-1 β ,
whence we obtain the second statement. By Aldous' tightness criterion, there exists a compact L of the set D[0, T ] of càdlàg functions on [0, T ] such that for all σ > 0, ε > 0 and u Back to [START_REF] Villani | Topics in optimal transportation[END_REF], we have:

∈ K 1 ∩ K 2 , P [y σ,ε (u, •) ∈ L] 1 -δ. Since C[0, T ] is closed in D[
E 1 0 sup |t 2 -t 1 |<η |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )|du = 1 0 E sup |t 2 -t 1 |<η |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )| du = 1 0 E 1 {u∈K 1 ∩K 2 ,yσ,ε(u,•)∈L} sup |t 2 -t 1 |<η |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )| du + 1 0 E 1 {u∈K 1 ∩K 2 ,yσ,ε(u,•)∈L} sup |t 2 -t 1 |<η |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )| du. (21)
The first term on the right hand side of ( 21) is bounded by:

1 0 E 1 {u∈K 1 ∩K 2 ,yσ,ε(u,•)∈L} du 1/2 1 0 E sup |t 2 -t 1 |<η |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )| 2 du 1/2 .
We have:

1 0 E 1 {u∈K 1 ∩K 2 ,yσ,ε(u,•)∈L} du 1 0 1 {u∈K 1 ∩K 2 } P [y σ,ε (u, •) / ∈ L] du + 1 0 1 {K 1 (u) C} du + 1 0 1 {K 2 (u) C} du 3δ.

Construction of a Wasserstein diffusion December 2018

Moreover,

1 0 E sup |t 2 -t 1 |<η |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )| 2 du 4 1 0 E sup t T |y σ,ε (u, t)| 2 du 4M,
where M is a constant independent of σ > 0 and ε > 0 by inequality [START_REF] Sturm | A monotone approximation to the Wasserstein diffusion, Singular phenomena and scaling in mathematical models[END_REF]. It remains to handle the second term on the right hand side of [START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF]. Since L is a compact set of C[0, T ], there exists η > 0 such that for every f ∈ L, ω f (η) := sup |t-s|<η |f (t) -f (s)| < δ. Therefore, there exists η > 0 such that:

1 0 E 1 {u∈K 1 ∩K 2 ,yσ,ε(u,•)∈L} sup |t 2 -t 1 |<η |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )| du δ.
Back to equality [START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF], we have proved that there is η > 0 such that for every σ > 0 and ε > 0:

E 1 0 sup |t 2 -t 1 |<η |y σ,ε (u, t 2 ) -y σ,ε (u, t 1 )|du δ + √ 12δM .
This proves convergence [START_REF] Villani | Topics in optimal transportation[END_REF] and thus concludes the proof of the Proposition.

Then, to obtain criterion (K3), we state the following Proposition:

Proposition 4.11. Let g ∈ L ↑ 2+ [0, 1] and δ > 0. Then for all k 1, there is h k > 0 such that for all σ, ε > 0,

P 1-h k 0 y σ,ε (u + h k , •) -y σ,ε (u, •) 2 C[0,T ] du 1 k δ 2 k . If 1-h k 0 y σ,ε (u + h k , •) -y σ,ε (u, •) 2 C[0,T ] du 1 k
, we deduce by monotonicity of u → y σ,ε (u, t) for every t ∈ [0, T ] that for every h ∈ (0, h k ),

1-h 0 y σ,ε (u + h, •) -y σ,ε (u, •) 2 C[0,T ] du 1-h k 0 y σ,ε (u + h, •) -y σ,ε (u, •) 2 C[0,T ] du + 1-h k 1-2h k +h y σ,ε (u + h k , •) -y σ,ε (u + h k -h, •) 2 C[0,T ] du 2 1-h k 0 y σ,ε (u + h k , •) -y σ,ε (u, •) 2 C[0,T ] du 2 k .
Therefore, the latter Proposition implies the following Corollary, which is equivalent to criterion (K3):

Corollary 4.12. Let g ∈ L ↑ 2+ [0, 1] and δ > 0. Then for all k 1, there is h k > 0 such that for all σ, ε > 0,

P ∀h ∈ (0, h k ), 1-h 0 y σ,ε (u + h, •) -y σ,ε (u, •) 2 C[0,T ] du 2 k 1 - δ 2 k .

December 2018

Proof (Proposition 4.11). Let h ∈ (0, 1). By Corollary 3.11, for almost every u ∈ (0, 1 -h), N u,t := y σ,ε (u+h, t)-y σ,ε (u, t) is a martingale. By Fubini-Tonelli Theorem and Doob's inequality, we have:

E 1-h 0 N u,• 2
C[0,T ] du = 1-h 0 E N u,• 2
C[0,T ] du C 1-h 0 E N 2 u,T du. ( 22
) Let us split E N 2 u,T in two terms E N 2 u,T 1 {N u,T 1} + E N 2 u,T 1 {N u,T >1} .

Study of

1-h 0 E N 2 u,T 1 {N u,T 1} du. Let u ∈ (0, 1 -h) be such that N u,
• is a martingale. By Lemma 3.10, if g(u+h)-g(u) = 0, then N u,T = 0 almost surely, thus E N 2 u,T 1 N u,T 1 = 0. From now on, we suppose that g(u + h) -g(u) > 0. N u,• is a square integrable continuous martingale, starting from g(u+h)-g(u) > 0 and positive by Lemma 3.10. Therefore, there exists a standard Brownian motion

β u such that N u,t = N u,0 + β u ( N u,• , N u,• t ). Recall that N u,0 = g(u + h) -g(u) is a deterministic quantity. If N u,0
1, then the inequality

E N 2 u,T 1 {N u,T 1}
N u,0 is obvious. Otherwise, we have

E N 2 u,T 1 {N u,T 1} = +∞ 0 P N 2 u,T 1 {N u,T 1} λ dλ 1 0 P N 2 u,T λ dλ N 2 u,0 + 1 N 2 u,0 P N u,T λ 1/2 dλ. (23)
Let us estimate P [N u,T κ] for a real number κ > N u,0 . We define the following stopping times:

τ -N u,0 := inf{t 0 : N u,0 + β u (t) 0};

τ κ-N u,0 := inf{t 0 : N u,0 + β u (t) κ}; τ := inf{t 0 : N u,t κ} ∧ T.

On the first hand, we know that almost surely, for all t ∈ [0, T ], N u,t > 0, hence τ -N u,0 N u,• , N u,• T . On the other hand, if N u,T κ, N u,τ is equal to κ by continuity of N u,• , hence N u,• , N u,• τ τ κ-N u,0 . It follows from both inequalities that τ κ-N u,0 τ -N u,0 . Therefore,

P [N u,T κ] P τ κ-N u,0 τ -N u,0 = N u,0 κ , (24)
by a usual martingale equality. Using inequality [START_REF] Walsh | An introduction to stochastic partial differential equations[END_REF] and N u,0 1, we obtain:

E N 2 u,T 1 {N u,T 1} N 2 u,0 + 1 N 2 u,0 N u,0 λ 1/2 dλ N 2 u,0 + 2N u,0 3N u,0 .
Therefore, we have:

1-h 0 E N 2 u,T 1 {N u,T 1} du 3 1-h 0 N u,0 du.

Study of

1-h 0 E N 2 u,T 1 {N u,T >1} du.
Recall that g belongs to L p (0, 1) for some p > 2. Fix December 2018

β ∈ (1, 3 2 -1 p ). We compute:

1-h 0 E N 2 u,T 1 {N u,T >1} du 2 1-h 0 E (N u,T -N u,0 ) 2 1 {N u,T >1} du + 2 1-h 0 E N 2 u,0 1 {N u,T >1} du 2 1-h 0 E (N u,T -N u,0 ) 2β du 1 β 1-h 0 P [N u,T > 1] du 1-1 β + 2 1-h 0 N 2 u,0 du.
Furthermore, we have P [N u,T > 1] N u,0 : that inequality is obvious if N u,0 1 and otherwise, it is a consequence of inequality (24).

Then, we want to give an upper bound for E (N u,T -N u,0 ) 2β . Using Burkholder-Davis-Gundy inequality, there exists

C β such that E (N u,T -N u,0 ) 2β C β E N u,• , N u,• β T .
We compute the quadratic variation of the martingale N u,t = y σ,ε (u + h, t) -y σ,ε (u, t):

E N u,• , N u,• β T = E T 0 1 M σ,ε (u, s) + 1 M σ,ε (u + h, s) - 2m σ,ε (u, u + h, s) (ε + m σ,ε (u, s))(ε + m σ,ε (u + h, s)) ds β . By Cauchy-Schwarz inequality m σ,ε (u, u + h, s) m 1/2 σ,ε (u, s)m 1/2
σ,ε (u + h, s), we deduce that the sum of the three terms in the integral is non-negative and thus that it is bounded by 1 Mσ,ε(u,s) + 1 Mσ,ε(u+h,s) , whence we obtain:

E N u,• , N u,• β T T β-1 E T 0 1 M σ,ε (u, s) + 1 M σ,ε (u + h, s) β ds C β,T E T 0 ds M β σ,ε (u, s) + E T 0 ds M β σ,ε (u + h, s)
.

By Lemma 4.6, we deduce that

1-h 0 E N u,• , N u,• β T du is bounded, because β < 3 2 -1
p . Therefore, we can conclude that there is a constant C T,β such that:

1-h 0 E N 2 u,T 1 {N u,T >1} du 2C T,β 1-h 0 N u,0 du 1-1/β + 2 1-h 0 N 2 u,0 du.

Conclusion:

Putting together the studies of both cases, we have proved that there is a positive constant C satisfying, for all σ, ε and h ∈ (0, 1):

1-h 0 E N 2 u,T du C 1-h 0 N u,0 du + C 1-h 0 N u,0 du 1-1/β + C 1-h 0 N 2 u,0 du. ( 25
)
Recall that there is p > 2 such that g ∈ L p (0, 1). As for inequality [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF], we get:

1-h 0 N u,0 du = 1-h 0 (g(u + h) -g(u))du g Lp (2h) 1-1 p .
December 2018 Furthermore, define α := p-2 p-1 ∈ (0, 1). We have

1-h 0 N 2 u,0 du = 1-h 0 (g(u + h) -g(u)) α (g(u + h) -g(u)) 2-α du 1-h 0 (g(u + h) -g(u))du α 1-h 0 (g(u + h) -g(u)) 2-α 1-α du 1-α g Lp (2h) 1-1 p α C p g Lp 1-α , because 2-α 1-α = p. Therefore 1-h 0 N 2 u,0 du = 1-h 0 (g(u + h) -g(u)) 2 du C 1-α p g Lp h p-2 p . ( 26
)
It follows from (25) that there is C β such that for each σ, ε > 0,

1-h 0 E (y σ,ε (u + h, T ) -y σ,ε (u, T )) 2 du C β g Lp h p-1 p + h p-1 p (1-1 β ) + h p-2 p
, for every β < 3 2 -1 p , i.e. such that 0 < 1 -1 β < p-2 3p-2 . Thus, there is q > 0 depending on p (e.g.

q = (p-1)(p-2) 2p(3p-2) by choosing 1 -1 β = p-2 2(3p-2)
) and a constant C such that for each σ, ε > 0,

1-h 0 E (y σ,ε (u + h, T ) -y σ,ε (u, T )) 2 du C g Lp h q . (27)
Therefore, by ( 22) and Markov's inequality, there is C such that for each σ, ε > 0,

P 1-h 0 y σ,ε (u + h, •) -y σ,ε (u, •) 2 C[0,T ] du 1 k kC g Lp h q ,
whence it is sufficient to choose h k so that kC g Lp h q k < δ 2 k .

Convergence when ε → 0

Fix σ ∈ Q + . By Prokhorov's Theorem, it follows from Corollary 4.3 that the collection of laws of the sequence (y σ,ε ) ε∈Q + is relatively compact in P(L 2 ([0, 1], C[0, T ])). In particular, up to extracting a subsequence, we may suppose that (y σ,ε ) ε∈Q + converges in distribution in L 2 ([0, 1], C[0, T ]) to a limit, denoted by y σ . For every t ∈ [0, T ], let us denote by e t (f ) := f (•, t) the continuous evaluation function:

L 2 ([0, 1], C[0, T ]) → L 2 [0, 1]. We define Y σ (t) := e t (y σ ) = y σ (•, t)
. Under the same model as Proposition 3.2, we obtain:

Proposition 4.13. Fix σ ∈ Q + . Suppose that g ∈ L ↑ 2+ [0, 1]. (Y σ (t)) t∈[0,T ] is a L ↑ 2 [0, 1]-valued process such that: (B1) Y σ (0) = g; (B2) (Y σ (t)) t∈[0,T ] is a square integrable continuous L ↑ 2 [0, 1]-valued (F σ t ) t∈[0,T ] -martingale, where F σ t = σ(Y σ (s), s t); (B3) for every h, k ∈ L 2 [0, 1], (Y σ , h) L 2 , (Y σ , k) L 2 t = t 0 1 0 1 0 h(u)k(u ) m σ (u, u , s) m σ (u, s)m σ (u , s) dudu ds,
where m σ (u, u , s)

= 1 0 ϕ σ (y σ (u, s) -y σ (v, s))ϕ σ (y σ (u , s) -y σ (v, s))dv and m σ (u, s) = 1 0 ϕ 2 σ (y σ (u, s) -y σ (v, s))dv. Proof. Fix t ∈ [0, T ]. We want to prove that Y σ (t) belongs to L ↑ 2 [0, 1]. For each ε ∈ Q + , Y σ,ε (t) belongs with probability 1 to the set K := f ∈ L 2 (0, 1) : ∀u, u , ∀r, r , if 0 < u < u + r < u < u + r < 1, then 1 r u+r u f 1 r u +r u f which is closed in L 2 (0, 1). Recall that the sequence (y σ,ε ) ε∈Q + converges in distribution to y σ in L 2 ([0, 1], C[0, T ]). Therefore, (Y σ,ε (t)) ε∈Q + converges in distribution to Y σ (t) in L 2 [0, 1].
Because K is closed, the limit Y σ (t) also belongs to K with probability 1. Therefore, almost surely, for every t

∈ [0, T ] ∩ Q, Y σ (t) ∈ K. Let ω ∈ Ω ,
where Ω is such that P [Ω ] = 1 and for every ω ∈ Ω , f is non-increasing, we have f (u) =

1 0 sup s T |y σ (v, s)| 2 (ω)dv < +∞ and for every t ∈ [0, T ] ∩ Q, Y σ (t)(ω) ∈ K. Let t ∈ [0, T ] and (t n ) be a sequence in [0, T ] ∩ Q tending to t. For every n ∈ N and each u, u , r, r such that 0 < u < u + r < u < u + r < 1, 1 r u+r u y σ (v, t n )(ω)dv 1 r u +r u y σ (v, t n )(ω)dv. Since y σ (ω) belongs to L 2 ([0, 1], C[0, T ]), and since u+r u y σ (v, t n ) 2 (ω)dv 1 0 sup s T |y σ (v, s)| 2 (ω)dv < +∞, 1 r u+r u y σ (v, t n )(ω)dv tends to 1 r u+r u y σ (v, t)(ω)dv
lim h→0 + 1 h (u+h)∧1 u f (v)dv. Choose a sequence (u n ) u. By monotonicity, f (u) f (u n ). Fix δ > 0. There exists h > 0 such that u + h < 1 and | f (u) -1 h u+h u f | < δ. Since f ∈ L 2 , there exists N such that for all n N , | 1 h un+h un f -1 h u+h u f | < δ. Therefore, f (u n ) 1 h un+h un f f (u) + 2δ for all n N . Thus f (u n ) → f (u).
In addition, f has left limits because of its monotonicity. Hence f is a càdlàg function. Furthermore, f = f almost everywhere. Indeed, for every δ > 0, there exists

F ∈ C[0, 1] such that f -F L 1 (0,1) < δ. Define F (u) = lim h→0 + 1 h (u+h)∧1 u F (v)dv. By continuity of F , F (u) = F (u)
for every u ∈ (0, 1). Thus we have:

1 0 |f (u) -f (u)|du 1 0 |f (u) -F (u)|du + 1 0 | f (u) -F (u)|du δ + 1 0 lim h→0 + 1 h (u+h)∧1 u |f (v) -F (v)|dvdu δ + lim inf h→0 + 1 0 |f (v) -F (v)|dv 2δ,
where we used Fatou's Lemma to obtain the last line. Thus

1 0 |f (u) -f (u)|du = 0, whence f = f almost everywhere. Thus f belongs to L ↑ 2 [0, 1]: Y σ is a L ↑ 2 [0, 1]-valued process. Property (B1). (Y σ,ε (0)) ε∈Q + converges in law to Y σ (0) in L 2 [0, 1]. Therefore, Y σ (0) = g.

December 2018

Property (B2). By inequality [START_REF] Sturm | A monotone approximation to the Wasserstein diffusion, Singular phenomena and scaling in mathematical models[END_REF], Moreover, we know from property (A2) that for each h ∈ L 2 (0, 1), each l 1, 0 s 1 s 2 . . . s l s t and each bounded and continuous function f l : (L 2 (0, 1)) l → R:

E Y σ,ε 2 L 2 ([0,1],C[0,T ]) is bounded uniformly in ε ∈ Q + . We deduce that for every t ∈ [0, T ], E Y σ (t) 2 L 2 ([0,1]) < +∞, thus the process Y σ is square integrable. Furthermore, Y σ is a continuous L ↑ 2 [0, 1]-valued process. Indeed, for each sequence (t n ) n 0 converging to a time t, Y σ (t n ) -Y σ (t) 2 L 2 = 1 0 (y σ (u, t n ) -y σ (u, t))
E 1 0 h(u)(y σ,ε (u, t) -y σ,ε (u, s))du f l (y σ,ε (•, s 1 ), . . . , y σ,ε (•, s l )) = 0. ( 28
)
Since

1 0 h(u)b(u, t)du h L 2 1 0 sup [0,T ] |b(u, •)| 2 du 1/2 for every b ∈ L 2 ([0, 1], C[0, T ]), the function ϕ : b ∈ L 2 ([0, 1], C[0, T ]) → 1 0 h(u)(b(u, t) -b(u, s))du f l (b(•, s 1 ), . . . , b(•, s l )) is contin- uous. Furthermore, we prove that (ϕ(y σ,ε )) ε∈Q + is bounded in L 2 : E ϕ(y σ,ε ) 2 f l 2 ∞ h 2 L 2 E 1 0 (y σ,ε (u, t) -y σ,ε (u, s)) 2 du C f l 2 ∞ h 2 L 2 ,
where C is independent of ε by Corollary 4.8. We deduce that (ϕ(y σ,ε )) ε∈Q + is uniformly integrable. By continuity of ϕ and since (y σ,ε

) ε∈Q + converges in law to y σ in L 2 ([0, 1], C[0, T ]), we get: E [ϕ(y σ,ε )] -→ ε→0 E [ϕ(y σ )].
Since by equality ( 28), E [ϕ(y σ,ε )] = 0 for each ε ∈ Q + , we have:

E 1 0 h(u)(y σ (u, t) -y σ (u, s))duf l (Y σ (s 1 ), . . . , Y σ (s l )) = 0. ( 29
)
Therefore, Y σ (•) is a square integrable continuous (F σ t ) t∈[0,T ] -martingale.

Property (B3). We know, by property (A3), that for every l 1, for every 0 s 1 s 2 . . . s l s t, for every bounded and continuous f l : (L 2 (0, 1)) l → R and for every h and k in L 2 (0, 1):

E 1 0 1 0 h(u)k(u )[(y σ,ε (u, t) -g(u))(y σ,ε (u , t) -g(u )) -(y σ,ε (u, s) -g(u))(y σ,ε (u , s) -g(u ))]dudu f l (Y σ,ε (s 1 ), . . . , Y σ,ε (s l )) = E 1 0 1 0 h(u)k(u ) t s m σ,ε (u, u , r) (ε + m σ,ε (u, r))(ε + m σ,ε (u , r)) drdudu f l (Y σ,ε (s 1 ), . . . , Y σ,ε (s l )) . (30)
First, we want to obtain the convergence of the left hand side of (30). We proceed in the same way as for the proof of equality (29); to get a uniform integrability property, we have now to prove the existence of β > 1 such that

sup ε∈Q + E 1 0 h(u)(y σ,ε (u, t) -g(u))du 1 0 k(u )(y σ,ε (u , t) -g(u ))du β (31)
December 2018 is finite. Therefore, it is sufficient to prove the existence of β > 1 such that

sup ε∈Q + E 1 0 h(u)(y σ,ε (u, t) -g(u))du 2β is finite for every h ∈ L 2 [0, 1]. By Cauchy-Schwarz inequality, E 1 0 h(u)(y σ,ε (u, t) -g(u))du 2β E h 2β L 2 1 0 (y σ,ε (u, t) -g(u)) 2 du β h 2β L 2 E 1 0 (y σ,ε (u, t) -g(u)) 2β du . ( 32
)
We deduce by Burkholder-Davis-Gundy inequality and Fubini's Theorem that there are some constants independent of ε such that

E 1 0 (y σ,ε (u, t) -g(u)) 2β du C 1 1 0 E y σ,ε (u, •), y σ,ε (u, •) β t du C 2 E 1 0 t 0 1 M β σ,ε (u, r) drdu .
By Lemma 4.6, there exists β > 1 such that E

1 0 t 0 1 M β σ,ε (u,r)
drdu is bounded uniformly for ε ∈ Q + . Thus (31) is finite. It is also finite if we replace t by s.

To obtain the convergence of the right hand side of (30), we start by using Skorohod's representation Theorem1 : there exists a sequence ( y σ,ε ) ε∈Q + defined on a common probability space ( Ω, P) that converges to y σ in L 2 ([0, 1], C[0, T ]) almost surely, where y σ,ε (resp. y σ ) has same distribution as y σ,ε (resp. y σ ). We denote by m σ,ε (resp. m σ ) the mass associated to y σ,ε (resp. y σ ).

Furthermore, on the probability space ( Ω × [0, 1], P ⊗ Leb | [0,1] ), y σ,ε converges in probability in the space C[0, T ] to y σ . Indeed, for every δ > 0, we have:

P ⊗ Leb | [0,1] {(ω, u) : ( y σ,ε -y σ )(ω, u) C[0,T ] δ} = E Leb{u : ( y σ,ε -y σ )(ω, u) C[0,T ] δ} E 1 ∧ 1 δ 2 1 0 ( y σ,ε -y σ )(ω, u) 2 C[0,T ] du .
We know that, for every fixed δ > 0, 1 ∧ 1

δ 2 1 0 ( y σ,ε -y σ )(ω, u) 2 C[0,T ]
du converges to 0 almost surely, and it is bounded by 1, so we deduce that the latter term tends to 0. We deduce from the convergence in probability that there exists a subsequence

(ε n ) n , ε n → 0, such that for almost every (ω, u) ∈ Ω × [0, 1], ( y σ,εn -y σ )(ω, u) C[0,T ] → 0.
We want to prove that,

E 1 0 1 0 h(u)k(u ) t s m σ,εn (u, u , r) (ε n + m σ,εn (u, r))(ε n + m σ,εn (u , r)) drdudu f l ( Y σ,εn (s 1 ), . . . , Y σ,εn (s l )) -→ n→∞ E 1 0 1 0 h(u)k(u ) t s m σ (u, u , r) m σ (u, r) m σ (u , r) drdudu f l ( Y σ (s 1 ), . . . , Y σ (s l )) . ( 33
)
On the one hand, almost surely and for almost every u ∈ (0, 1), y σ,εn (u,

•) → y σ (u, •) in C[0, T ].
Then for almost every u, u ∈ (0, 1),

m σ,εn (u, u , r) = 1 0 ϕ σ ( y σ,εn (u, r) -y σ,εn (v, r))ϕ σ ( y σ,εn (u , r) -y σ,εn (v, r))dv -→ n→∞ m σ (u, u , r), (34)
ε n + m σ,εn (u, r) = ε n + 1 0 ϕ 2 σ ( y σ,εn (u, r) -y σ,εn (v, r))dv -→ n→∞ m σ (u, r). (35)
Therefore, in order to obtain (33), it remains to justify that there exists β > 1 such that:

sup n∈N E 1 0 1 0 h(u)k(u ) t s m σ,εn (u, u , r) (ε n + m σ,εn (u, r))(ε n + m σ,εn (u , r)) drdudu β is finite. By Cauchy-Schwarz inequality, m σ,εn (u, u , r) m 1/2 σ,εn (u, r) m 1/2
σ,εn (u , r), so that it is sufficient to prove that there is β > 1 such that

sup n∈N E   1 0 1 0 h(u)k(u ) t s 1 M 1/2 σ,εn (u, r) M 1/2 σ,εn (u , r) drdudu β   is finite, and thus that sup n∈N E 1 0 t s 1 M β σ,εn (u,r)
drdu is finite, using Cauchy-Schwarz inequality as in the proof of (32). By Lemma 4.6, this statement holds. We conclude that we have the following equality:

E 1 0 1 0 h(u)k(u )[(y σ (u, t) -g(u))(y σ (u , t) -g(u )) -(y σ (u, s) -g(u))(y σ (u , s) -g(u ))]dudu f l (Y σ (s 1 ), . . . , Y σ (s l )) = E 1 0 1 0 h(u)k(u ) t s m σ (u, u , r) m σ (u, r)m σ (u , r) drdudu f l (Y σ (s 1 ), . . . , Y σ (s l )) , (36)
whence we obtain property (B3), since

1 0 1 0 h(u)k(u ) t 0 mσ(u,u ,r) mσ(u,r)mσ(u ,r) drdudu is (F σ t ) t∈[0,T ] - measurable.
Property (B3) implies the following Corollary: Corollary 4.14. Let ψ be a non-negative and bounded map: [0, 1] → R. Then for every l ∈ N\{0}, 0 s 1 s 2 . . . s l s t and for every bounded and continuous function

f l : L 2 [0, 1] l → R, we have: E 1 0 ψ(u) (y σ (u, t) -g(u)) 2 -(y σ (u, s) -g(u)) 2 - t s 1 m σ (u, r) dr du f l (Y σ (s 1 ), . . . , Y σ (s l )) = 0.
Proof. We use the following notations: z(u, •) := y σ (u, •) -g(u) and

F l = f l (Y σ (s 1 ), . . . , Y σ (s l )).
Let us consider an orthonormal basis (e i ) i 1 in the Hilbert space L 2 (ψ(x)dx). We denote by

Construction of a Wasserstein diffusion

December 2018

[•, •] L 2 (ψ) the scalar product of L 2 (ψ(x)dx): [h, k] L 2 (ψ) = 1 0 hkψ
. By Parseval's formula, we have:

E 1 0 ψ(u)(z(u, t) 2 -z(u, s) 2 )duF l = E i 1 ([z(•, t), e i ] 2 L 2 (ψ) -[z(•, s), e i ] 2 L 2 (ψ) )F l = i 1 E ((z(•, t), e i ψ) 2 L 2 -(z(•, s), e i ψ) 2 L 2 )F l = i 1 E 1 0 1 0 e i (u)ψ(u)e i (u )ψ(u ) t s m σ (u, u , r) m σ (u, r)m σ (u , r) drdudu F l ,
by applying equality (36) with h = k = e i . By definition of m σ (u, u , r), we have:

E 1 0 ψ(u)(z(u, t) 2 -z(u, s) 2 )duF l = E t s 1 0 i 1 ϕ σ (y σ (•, r) -y σ (v, r)) m σ (•, r) , e i 2 L 2 (ψ) dvdrF l = E t s 1 0 1 0 ϕ 2 σ (y σ (u, r) -y σ (v, r)) m 2 σ (u, r) ψ(u)dudvdrF l = E 1 0 t s 1 m σ (u, r) drψ(u)duF l , since m σ (u, r) = 1 0 ϕ 2 σ (y σ (u, r) -y σ (v, r))dv.
We deduce the following estimation, by analogy with Lemma 4.6:

Lemma 4.15. For all β ∈ (0, 3 2 -1 p ), there is a constant C > 0 such that for all σ > 0 and 0 s < t T , we have the following inequality:

E t s 1 0 1 m β σ (u, r) dudr C √ t -s.
Proof. We use again the sequence ( y σ,εn ) n∈N obtained by Skorohod's representation Theorem, as in the proof of convergence (33). Therefore, by Fatou's Lemma,

E t s 1 0 1 m β σ (u, r) dudr lim inf n→∞ E t s 1 0 1 M β σ,εn (u, r) dudr C √ t -s,
where C is obtained thanks to Lemma 4.6.

By Burkholder-Davis-Gundy inequality, we obtain immediately the following Corollary:

Corollary 4.16. For each β ∈ (0, 3 2 -1 p ), sup σ∈Q + sup t T E 1 0 (y σ (u, t) -g(u)) 2β du < +∞.

Convergence when σ → 0

Recall that by Corollary 4.3 and Prokhorov's Theorem, the collection of laws of the sequence

(y σ,ε ) σ,ε∈Q + is relatively compact in P(L 2 ([0, 1], C[0, T ]))
. By construction, the collection of laws of the sequence (y σ ) σ∈Q + inherits the same property. Thus, up to extracting a subsequence, we may suppose that (y σ ) σ∈Q + converges in distribution to a limit, denoted by y, in L 2 ([0, 1], C[0, T ]). As before, we define Y (t) := y(•, t). We state the first part of Theorem 1.4 in the following Proposition: December 2018 For every ε > 0, we get by Markov's inequality:

P ⊗ Leb{(ω, u) : ( y σ -y)(ω, u) C[0,T ] ε} = E Leb{u : ( y σ -y)(ω, u) C[0,T ] ε} E 1 ∧ 1 ε 2 1 0 ( y σ -y)(ω, u) 2 C[0,T ] du . (37)
Since ( y σ ) σ∈Q + converges almost surely to y in L 2 ([0, 1], C[0, T ]), the right hand side tends to 0. Therefore, ( y σ ) σ∈Q + converges in probability to y in C[0, T ] on the probability space ( Ω × [0, 1], P ⊗ Leb). Thus there exists a subsequence (σ n ) n tending to 0 along which y σn converges on an almost sure event of Ω × [0, 1] to y in C[0, T ]. Therefore, there is Ω , P[Ω ] = 1, such that for every ω ∈ Ω , there exists a Borel set

A = A(ω) in [0, 1], Leb(A) = 1, such that for all u ∈ A, y σn (u, •) -y(u, •) C[0,T ] tends to zero.
Remark that the extraction (σ n ) n does not depend on ω. From now on, we forget the tildes and the extraction in our notation.

Let ω ∈ Ω. Fix u ∈ A(ω) and t ∈ [0, T ]. We set v ∈ A such that y(v, t) = y(u, t). Then there exist σ 0 > 0 and δ > 0 such that for all σ ∈ (0,

σ 0 ) ∩ Q + , |y σ (v, t) -y σ (u, t)| δ. For all σ min(σ 0 , δ), we have |y σ (v, t) -y σ (u, t)| σ and thus ϕ σ (y σ (v, t) -y σ (u, t)) = 0. Hence, lim σ→0 1 -ϕ 2 σ (y σ (v, t) -y σ (u, t)) = 1.
Thus we have shown that for all v ∈ A,

1 {y(v,t) =y(u,t)} lim inf σ→0 1 -ϕ 2 σ (y σ (v, t) -y σ (u, t)) , since 1 -ϕ 2
σ is non-negative. By Fatou's Lemma and since Leb(A) = 1, we deduce that:

1 -m(u, t) = 1 0 1 {y(v,t) =y(u,t)} dv lim inf σ→0 1 0 1 -ϕ 2 σ (y σ (v, t) -y σ (u, t)) dv,
whence for all u ∈ A and t ∈ [0, T ], lim sup n→∞ m σn (u, t) m(u, t).

We deduce from Lemma 5.2 the following Corollary. Set N (t) := 1 0 du m(u,t) . By a classical combinatorial argument, N (t) is the number of equivalence classes at time t relatively to the equivalence relation u ∼ t v ⇐⇒ y(u, t) = y(v, t). In other words, if

N (t) < ∞, Y (t) is a càdlàg step function taking N (t) distinct values: there exist 0 = a 1 < a 2 < • • • < a N (t) < a N (t)+1 = 1 and y 1 < y 2 < • • • < y N (t) such that for all u ∈ [0, 1] Y (t)(u) = N (t) k=1 y k 1 {u∈[a k ,a k+1 )} + y N (t) 1 {u=1} . Corollary 5.3. For every time t ∈ [0, T ], E t 0 N (s)ds is finite.
Proof. By Lemma 5.2, there is a subsequence (σ n ) such that almost surely, for every t ∈ [0, T ] and for almost every u ∈ [0, 1], lim sup n→∞ m σn (u, t) m(u, t). Therefore, (u,t) . By Fatou's Lemma, we deduce that:

1 m(u,t) lim inf n→∞ 1 mσ n
E t 0 N (s)ds E t 0 1 0 lim inf n→∞ 1 m σn (u, t) duds lim inf n→∞ E t 0 1 0 1 m σn (u, s) duds C √ t,
by Lemma 4.15.

Corollary 5.4. Almost surely, for every t > 0, N (t) is finite and t → N (t) is non-increasing on (0, T ].

Proof. We begin by proving the coalescence property. Let

u 1 , u 2 , h ∈ Q be such that 0 < u 1 < u 1 + h < u 2 < u 2 + h < 1. Define y h (u 1 , t) = 1 h u 1 +h u 1 y(v, t)dv = (Y (t), 1 h 1 (u 1 ,u 1 +h) ) L 2 and y h (u 2 , t) = (Y (t), 1 h 1 (u 2 ,u 2 +h) ) L 2 . By Proposition 4.17, Z(t) = y h (u 2 , t)-y h (u 1 , t) is a continuous R-valued (F t ) t∈[0,T ] -martingale,
almost surely non-negative. As a consequence, Z(t) = 0 for every t τ 0 = inf{s 0, Z(s) = 0}. In other terms, the following coalescence property holds: for every

u 1 , u 2 , h ∈ Q such that 0 < u 1 < u 1 + h < u 2 < u 2 + h < 1, y h (u 1 , t 0 ) = y h (u 2 , t 0 ) implies y h (u 1 , t) = y h (u 2 ,
t) for every t t 0 almost surely.

On a full event Ω of (Ω, P), the latter statement is true and T 0 N (s)ds is finite (by Corollary 5.3). Fix ω ∈ Ω . In particular, for almost every t ∈ (0, T ), N (t) is finite. Let t 0 ∈ (0, T ) be such that N (t 0 ) < +∞.

There exist 0 = a 1 < a 2 < • • • < a N (t 0 ) < a N (t 0 )+1 = 1 and z 1 < z 2 < • • • < z N (t 0 ) , depending on ω, such that for all u ∈ [0, 1], Y (t 0 )(u) = N (t 0 ) k=1 z k 1 {u∈[a k ,a k+1 )} + z N (t 0 ) 1 {u=1} . Fix k ∈ {1, . . . , N (t 0 )}. By the coalescence property, almost surely, for all u 1 , u 2 , h ∈ Q such that a k < u 1 < u 1 + h < u 2 < u 2 + h < a k+1 , since y h (u 1 , t 0 ) = z k = y h (u 2 , t 0 ), we have y h (u 1 , t) = y h (u 2 , t) for every t t 0 . Fix t t 0 . By monotonicity of Y (t), we deduce that Y (t) is constant on (u 1 , u 2 + h). Thus Y (t) is constant on (a k , a k+1 ). Therefore, since Y (t) is càdlàg, there exist z 1 z 2 . . . z N (t 0 ) , depending on ω, such that for all u ∈ [0, 1], Y (t)(u) = N (t 0 ) k=1 z k 1 {u∈[a k ,a k+1 )} + z N (t 0 ) 1 {u=1} .
We deduce that N (t) N (t 0 ) < +∞, for every t t 0 . Therefore, for every ω ∈ Ω , t → N (t) is finite and non-increasing on (0, T ]. This concludes the proof of the Lemma. Therefore, Corollary 5.4 concludes the proof of Proposition 5.1. Then, Proposition 4.17 and Proposition 5.1 imply the following property, by applying Proposition 2.3 of [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF]: Proposition 5.5. There exists a modification y of y in L 2 ([0, 1], C[0, T ]) such that y belongs to D((0, 1), C[0, T ]). In particular, for every t ∈ [0, T ], y(•, t) and y(•, t) are equal in L 2 [0, 1] almost surely. Moreover, for every u ∈ (0, 1), y(u, •) is a square integrable and continuous (F t ) t∈[0,T ]martingale and

P [∀u, v ∈ (0, 1), ∀s ∈ [0, T ], y(u, s) = y(v, s) implies ∀t s, y(u, t) = y(v, t)] = 1.
From now on, we denote by y (instead of y) the version of the limit process in D((0, 1), C[0, T ]).

Remark 5.6. The proof can be found in Appendix B of [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF]. It should be noticed that the difficult part of the proof relies on the construction of a version y such that for every u ∈ (0, 1), y(u, •) is continuous at time t = 0. This concludes the proof of properties (C3) and (C4) of Theorem 1.4. The aim of the next two Paragraphs is to prove property (C5), in two steps.

Quadratic variation of y(u, •)

The following Proposition shows that the quadratic variation of a particle is proportional to the inverse of its mass: December 2018 Proposition 5.7. Let y be the version in D((0, 1), C[0, T ]) of the limit process given by Proposition 5.5. For every u ∈ (0, 1),

y(u, •), y(u, •) t = t 0 1 m(u, s) ds,
where m(u, s) = 1 0 1 {y(u,s)=y(v,s)} dv. Proof. By Corollary 4.14, for every positive ψ ∈ L ∞ (0, 1), we have:

E 1 0 ψ(u)[(y σ (u, t) -g(u)) 2 -(y σ (u, s) -g(u)) 2 ]f l (Y σ (s 1 ), . . . , Y σ (s l ))du = E 1 0 ψ(u) t s 1 m σ (u, r) dr f l (Y σ (s 1 ), . . . , Y σ (s l ))du . ( 38
)
To obtain the convergence of the left hand side of (38), we proceed in the same way as for the proof of equality (36). The uniform integrability property follows from Corollary 4.16. Therefore, the left hand side of (38) converges when σ → 0 to

E 1 0 ψ(u)[(y(u, t) -g(u)) 2 -(y(u, s) -g(u)) 2 ]f l (Y (s 1 ), . . . , Y (s l ))du .
We also get a uniform integrability property for the right hand side of (38) by the same argument as in the proof of property (B3) (see Proposition 4.13). Assume that there exists a sequence (σ n ) of rational numbers tending to 0, a probability space ( Ω, P), a modification ( m σn , y σn ) n∈N of (m σn , y σn

) n∈N on L 1 ([0, 1], C[0, T ])×L 2 ([0, 1], C[0, T ]
) and a modification ( m, y) of (m, y) on the same space such that for almost each ω ∈ Ω and almost every (u, t) ∈ [0, 1]×[0, T ], the sequence ( m σn (ω, u, t), y σn (ω)) n∈N converges to ( m(ω, u, t), y(ω)) in R × L 2 ([0, 1], C[0, T ]). This will be proved in Lemma 5.8.

It follows that for every ψ ∈ L ∞ (0, 1):

E 1 0 ψ(u) ( y(u, t) -g(u)) 2 -( y(u, s) -g(u)) 2 - t s dr m(u, r) f l ( Y (s 1 ), . . . , Y (s l ))du = 0.
By Fubini's Theorem, we deduce that for almost every u ∈ (0, 1),

E ( y(u, t) -g(u)) 2 -( y(u, s) -g(u)) 2 - t s dr m(u, r) f l ( Y (s 1 ), . . . , Y (s l )) = 0. (39)
We want to prove that (39) holds for every u ∈ (0, 1). Let u ∈ (0, 1). Choose δ > 0 such that u ∈ (δ, 1 -δ). Let (u p ) p∈N be a decreasing sequence in (δ, 1 -δ) converging to u such that for every p ∈ N, equality (39) holds at point u p , (y σn,ε (u p , t)) t∈[0,T ] is a square integrable continuous (F σn,ε t ) t∈[0,T ] -martingale for every n ∈ N and ε ∈ Q + and lim sup n→∞ m σn (u p , t) m(u p , t) almost surely for all t ∈ [0, T ]. Such a sequence exists by Corollary 3.11 and Lemma 5.2. We will use these different properties later in this proof.

Almost surely, for every r ∈ (0, T ], y(•, r) is right-continuous at point u and is a step function. Therefore, m(•, r) = 1 0 1 { y(•,r)= y(v,r)} dv is also right continuous at point u for every positive time r. In order to prove (39) at point u, it is thus sufficient to show the following uniform integrability property: there exists β > 1 such that

sup p∈N E ( y(u p , t) -g(u p )) 2 -( y(u p , s) -g(u p )) 2 - t s dr m(u p , r) β < +∞. ( 40
)
December 2018

First, by monotonicity, for all p ∈ N, E g(u p ) 2β g(δ) 2β + g(1 -δ) 2β . Then, the following statement holds: there exists β > 1 such that for every t ∈ [0, T ], sup p∈N E y(u p , t) 2β < +∞. Indeed, for every p ∈ N, by monotonicity,

1 δ δ 0 y(v, t)dv y(u p , t) 1 δ 1 1-δ y(v, t)dv.
Therefore, we have: Because ( y σn,ε (u p , t)) t∈[0,T ] is a square integrable (F σn,ε t ) t∈[0,T ] -martingale and y σn,ε (u p , •),

E y(u p , t) 2β E 1 δ δ 0 y(v, t)dv 2β + E 1 δ 1 1-δ y(v, t)dv 2β 2 δ E 1 0 y(v, t) 2β dv , (41)
y σn,ε (u p , •) t = t 0 dr Mσ n,ε (up,r)
, we obtain by Burkholder-Davis-Gundy inequality:

E   t 0 dr M σn,ε (u p , r) β   CE ( y σn,ε (u p , t) -g(u p )) 2β .
We have already seen that E g(u p ) 2β is uniformly bounded for p ∈ N. By the same argument as for inequality (41), E y σn,ε (u p , t) 2β 2 δ E 1 0 y σn,ε (v, t) 2β dv , which is uniformly bounded for n ∈ N and ε ∈ Q + . This concludes the proof of (40).

Therefore, equality (39) holds for every u ∈ (0, 1), for every bounded and continuous f l and for every 0 s 1 . . . s l s t. Thus for every u ∈ (0, 1), the process ( y(u, t) -g(u)) 2t 0 ds m(u,s) t∈[0,T ] is an (F t ) t∈[0,T ] -martingale. This concludes the proof of the Proposition.

In the proof of Proposition 5.7, we used the following Lemma: Lemma 5.8. There exists a sequence (σ n ) of rational numbers tending to 0, a sequence of processes ( m σn , y σn ) n∈N and a process ( m, y) defined on the same probability space such that • for all n ∈ N, ( m σn , y σn ) and (m σn , y σn ) (resp. ( m, y) and (m, y)) have same law on

L 1 ([0, 1], C[0, T ]) × L 2 ([0, 1], C[0, T ]).
• for almost each ω ∈ Ω and for almost every (u, t) in [0, 1]×[0, T ], the sequence ( m σn (ω, u, t), y σn (ω)) n∈N converges to ( m(ω, u, t), y(ω)) in R × L 2 ([0, 1], C[0, T ]).

Remark 5.9. The Borel subset of [0, 1] × [0, T ] on which we have the convergence can depend on ω.

Before giving the proof of Lemma 5.8, we give the following definition and state the following Lemma, which will be useful in the proof. Let us define in L 1 ([0, 1] × [0, 1], C[0, T ]):

C σ (u 1 , u 2 , t) := t 0 1 m σ (u 1 , s) + 1 m σ (u 2 , s) - 2m σ (u 1 , u 2 , s) m σ (u 1 , s)m σ (u 2 , s)
ds.

Lemma 5.10. There exists a sequence

(σ n ) in Q + tending to 0 such that (y σn , C σn ) n∈N converges in distribution to (y, C) in L 2 ([0, 1], C[0, T ]) × L 1 ([0, 1] × [0, 1], C[0, T ]).
For almost every u 1 , u 2 ∈ [0, 1], the limit process C(u 1 , u 2 , •) is the quadratic variation of y(u 1 , •) -y(u 2 , •) relatively to the filtration generated by Y and C.

We start by giving the proof of Lemma 5.8 and then we give the proof of Lemma 5.10.

Proof (Lemma 5.8). By Skorohod's representation Theorem, we deduce from Lemma 5.10 that there exists a sequence ( y σn , C σn ) n and a random variable ( y, C) defined on the same probability space such that

• for all n ∈ N, ( y σn , C σn ) and (y σn , C σn ) (resp. ( y, C) and (y, C)) have same law,

• the sequence ( y σn , C σn ) n converges almost surely to ( y, C) in the space

L 2 ([0, 1], C[0, T ]) × L 1 ([0, 1] × [0, 1], C[0, T ]).
We apply to ( y σn ) n the argument in the proof of Lemma 5.2 and we prove that, up to extracting another subsequence (independent of ω), for almost every u ∈ [0, 1] and almost surely, lim sup n→∞ m σn (u, t) m(u, t) for every t ∈ [0, T ].

For each t ∈ [0, T ], we may suppose that for each n ∈ N, y σn (•, t) is a càdlàg function, so that for every u ∈ (0, 1),

m σn (u, t) = 1 0 ϕ 2 σn ( y σn (u, t) -y σn (v, t))dv = lim p→∞ p (u+ 1 p )∧1 u 1 0 ϕ 2 σn ( y σn (u , t) -y σn (v, t))dvdu
is a measurable function with respect to y σn (•, t). We deduce that ( m σn (u, t), y σn ) has the same law as (m σn (u, t), y σn ) for every u ∈ (0, 1).

From now on, we forget the hats in our notation. We may suppose that y is the version in D((0, 1), C[0, T ]) given by Proposition 5.5. Let Ω be such that P [Ω ] = 1 and for all ω ∈ Ω , we have the following convergences in R:

1 0 sup t T |y σn (u, t) -y(u, t)| 2 (ω)du -→ n→∞ 0, ( 42
) 1 0 1 0 sup t T |C σn (u 1 , u 2 , t) -C(u 1 , u 2 , t)|(ω)du 1 du 2 -→ n→∞ 0. (43)
Fix ω ∈ Ω . Thanks to (42), we already have the convergence of (y σn (ω)) n to y(ω) in L 2 ([0, 1], C[0, T ]). It remains to show that for almost every (u, t) ∈ [0, 1] × [0, T ], (m σn (ω, u, t)) n converges to m(ω, u, t) = 1 0 1 {y(u,t)=y(v,t)} (ω)dv. We already know that for every ω ∈ Ω , every t ∈ [0, T ] and almost every u ∈ (0, 1), lim sup n→∞ m σn (ω, u, t) m(ω, u, t). u min and u med belongs to (u inf , u sup ), we have t 0 τ u 1 ,u 2 for (u 1 , u 2 ) = (u, u max ), (u, u min ), (u max , u min ) and (u, u med ).

We deduce from (44) and the fact that u, u max , u min , u med belongs to A that there exists N such that for each n N , f σn (t 0 , u 1 , u 2 ) δ for (u 1 , u 2 ) = (u, u max ), (u, u min ), (u max , u min ) and (u, u med ). It implies that for each n N ,

1 0 |ϕ σn (y σn (u 1 , t 0 ) -y σn (v, t 0 )) -ϕ σn (y σn (u 2 , t 0 ) -y σn (v, t 0 ))| 2 dv m σn (u 1 , t 0 )m σn (u 2 , t 0 ) = f σn (t 0 , u 1 , u 2 ) δ. ( 45
)
Since the mass m σn is bounded by 1, we deduce in particular that for all n N ,

1 0 |ϕ σn (y σn (u 1 , t 0 ) -y σn (v, t 0 )) -ϕ σn (y σn (u 2 , t 0 ) -y σn (v, t 0 ))| 2 dv δ. (46)
Inequalities ( 45) and ( 46) are satisfied for (u 1 , u 2 ) = (u, u max ), (u, u min ), (u max , u min ) and (u, u med ).

Let n N and d := y σn (u max , t 0 ) -y σn (u min , t 0 ) 0. We distinguish three cases:

• d σ n : Recall that ϕ σn is equal to 0 on [ σn 2 , +∞). Thus for all v ∈ [0, 1], ϕ σn (y σn (u max , t 0 )y σn (v, t 0 )) and ϕ σn (y σn (u min , t 0 ) -y σn (v, t 0 )) can not be simultaneously different from 0 because d σ n . Therefore, selecting (u 1 , u 2 ) = (u max , u min ), inequality (45) implies:

1 0 ϕ 2 σn (y σn (u max , t 0 ) -y σn (v, t 0 ))dv + 1 0 ϕ 2 σn (y σn (u min , t 0 ) -y σn (v, t 0 ))dv m σn (u max , t 0 )m σn (u min , t 0 ) δ,
that is:

1 m σn (u min , t 0 ) + 1 m σn (u max , t 0 ) δ.
Thus, we obtain δ 2, which is excluded by definition of δ.

• d σ n -η: Recall that η is chosen so that η < σn 3 . Define the two following sets

V max = {v ∈ [u min , u max ] : y σn (u max , t 0 ) -y σn (v, t 0 ) σn-η 2 }, V min = {v ∈ [u min , u max ] : y σn (u max , t 0 ) -y σn (v, t 0 ) > σn-η 2 }.
Clearly, we have: Leb(V max ) + Leb(V min ) = u max -u min m(u, t 0 ) -2δ. Recall that ϕ σn is equal to 1 on [0, σn-η 2 ]. Thus, for each v ∈ V max , ϕ σn (y σn (u max , t 0 ) -y σn (v, t 0 )) = 1, and for each v ∈ V min , using d σ n -η, ϕ σn (y σn (u min , t 0 ) -y σn (v, t 0 )) = 1. We have

m σn (u, t 0 ) Vmax ϕ 2 σn (y σn (u, t 0 )-y σn (v, t 0 ))dv+ V min ϕ 2 σn (y σn (u, t 0 )-y σn (v, t 0 ))dv. ( 47
)
We can deduce from inequality (46) applied to (u 1 , u 2 ) = (u, u max ) that:

Vmax |ϕ σn (y σn (u, t 0 ) -y σn (v, t 0 )) -ϕ σn (y σn (u max , t 0 ) -y σn (v, t 0 ))| 2 dv δ.

By Minkowski's inequality | f 1 L 2 -f 2 L 2 | f 1 -f 2 L 2 , we obtain: Vmax ϕ 2 σn (y σn (u, t 0 ) -y σn (v, t 0 ))dv 1/2 -Leb(V max ) 1/2 √ δ, whence Vmax ϕ 2 σn (y σn (u, t 0 ) -y σn (v, t 0 ))dv -Leb(V max ) (m 1/2 σn (u, t 0 )+Leb(V max ) 1/2 ) √ δ 2 √ δ.
December 2018

Similarly, applying inequality (46) to (u, u min ), we obtain:

V min ϕ 2 σn (y σn (u, t 0 ) -y σn (v, t 0 ))dv -Leb(V min ) 2 √ δ.
Thus, by inequality (47), we conclude:

m σn (u, t 0 ) Leb(V max ) + Leb(V min ) -4 √ δ m(u, t 0 ) -2δ -4 √ δ.
• d ∈ (σ n -η, σ n ): We now define three distinct sets

V max = {v ∈ [u min , u max ] : y σn (u max , t 0 ) -y σn (v, t 0 ) < σn-η 2 }, V med = {v ∈ [u min , u max ] : y σn (u max , t 0 ) -y σn (v, t 0 ) ∈ [ σn-η 2 , σn+η 2 ]}, V min = {v ∈ [u min , u max ] : y σn (u max , t 0 ) -y σn (v, t 0 ) > σn+η 2 }
. By definition of those sets, and since d ∈ (σ n -η, σ n ), we have ∀v ∈ V max , ϕ σn (y σn (u max , t 0 ) -y σn (v, t 0 )) = 1, ∀v ∈ V min , ϕ σn (y σn (u min , t 0 ) -y σn (v, t 0 )) = 1.

Moreover, we have y

σn (u max , t 0 ) -y σn (u med , t 0 ) ∈ [ σn-η 2 , σn+η 2
]. Indeed, if y σn (u max , t 0 ) -y σn (u med , t 0 ) was greater than σn+η 2 , we would have, for all v ∈ [u min , u med ], ϕ σn (y σn (u max , t 0 ) -y σn (v, t 0 )) = 0 and ϕ σn (y σn (u min , t 0 ) -y σn (v, t 0 )) = 1. By inequality (46) applied to (u 1 , u 2 ) = (u max , u min ), we would deduce that:

δ 1 0 |ϕ σn (y σn (u max , t 0 ) -y σn (v, t 0 )) -ϕ σn (y σn (u min , t 0 ) -y σn (v, t 0 ))| 2 dv u med u min dv = u med -u min u max -u min 2 -δ.
However, since δ < usup-u inf 6

and u max -u min u sup -u inf -2δ, we have u maxu min > 4δ, which is in contradiction with the above inequality. Similarly, y σn (u max , t 0 )y σn (u med , t 0 ) can not be smaller than σn-η 2 , otherwise y σn (u med , t 0 )-y σn (u min , t 0 ) would be greater than σn+η 2 and we would obtain the same contradiction. Therefore, y σn (u max , t 0 )y σn (u med , t 0 ) ∈ [ σn-η 2 , σn+η 2 ], which implies that u med ∈ V med and in particular that ∀v ∈ V med , ϕ σn (y σn (u med , t 0 ) -y σn (v, t 0 )) = 1.

As in the previous case, we deduce that

m σn (u, t 0 ) Leb(V max ) + Leb(V med ) + Leb(V min ) -6 √ δ = u max -u min -6 √ δ m(u, t 0 ) -2δ -6 √ δ.
Actually, putting all the cases together, we have proved that for each n N , m σn (u, t 0 ) m(u, t 0 ) -2δ -6 √ δ. Hence, for all δ < usup-u inf 6

, we have:

lim inf n→∞ m σn (u, t 0 ) m(u, t 0 ) -2δ -6 √ δ.
By letting δ converge to 0, we have for every t 0 ∈ B, lim inf n→∞ m σn (u, t 0 ) m(u, t 0 ) for every u ∈ A. Therefore, there exists a subsequence (σ n ) such that for almost every ω, for almost every t ∈ [0, T ] and almost every u ∈ [0, 1], m σn (ω, u, t) → n→∞ m(ω, u, t).

It remains to give the proof of Lemma 5.10.

Proof (Lemma 5.10). The first step will be to prove that the sequence (y

σ , C σ ) σ∈Q + is tight in L 2 ([0, 1], C[0, T ]) × L 1 ([0, 1] × [0, 1], C[0, T ]).
We have already proved that (y σ ) σ∈Q + is tight in L 2 ([0, 1], C[0, T ]). We will use a tightness criterion to prove that the sequence

(C σ ) σ∈Q + is tight in L 1 ([0, 1] × [0, 1], C[0, T ]).
The space changed in comparison with L 2 ([0, 1], C[0, T ]), but the criterion remains very semilar to the one of Proposition 4.2.

We have, similarly to Proposition 4.2, three criteria to prove. We want to show the following criterion:

First criterion: Let δ > 0. There is M > 0 such that for all σ in Q +

, P [ C σ M ] δ, where C σ := 1 0 1 0 sup t T |C σ (u 1 , u 2 , t)|du 1 du 2 .
That statement follows from Markov's inequality and the existence of a constant C independent of σ such that:

E 1 0 1 0 sup t T |C σ (u 1 , u 2 , t)|du 1 du 2 2E 1 0 T 0 dtdu 1 m σ (u 1 , t) + 2E 1 0 T 0 dtdu 2 m σ (u 2 , t) C.
The existence of C is a consequence of Lemma 4.15.

Then, we prove the following criterion: Second criterion: Let δ > 0. For each k 1, there exists η k > 0 such that for all σ in Q + ,

P 1 0 1 0 sup |t 2 -t 1 |<η k |C σ (u 1 , u 2 , t 2 ) -C σ (u 1 , u 2 , t 1 )|du 1 du 2 1 k δ 2 k .
The proof is very close to Proposition 4.10. We start by defining for every u 1 , u 2 ∈ (0, 1):

K 1 (u 1 , u 2 ) := E C σ (u 1 , u 2 , •) C[0,T ] and K 2 (u 1 ) := E T 0 1 m β σ (u 1 ,s)
ds . Fix δ > 0. There exists

C > 0 such that 1 0 1 0 1 {K 1 (u 1 ,u 2 ) C} du 1 du 2 δ and 1 0 1 {K 2 (u) C} du δ. Define the following set K := {(u 1 , u 2 ) : K 1 (u 1 , u 2 ) C, K 2 (u 1 ) C, K 2 (u 2 ) C}.
By Aldous' tightness criterion, the collection (C σ (u 1 , u 2 , •)) σ∈Q + ,(u 1 ,u 2 )∈K is tight in C[0, T ]. This fact relies on the following inequality, where η > 0 and τ is a stopping time for C σ (u 1 , u 2 , •):

E [|C σ (u 1 , u 2 , τ + η) -C σ (u 1 , u 2 , τ )|] = E τ +η τ 1 m σ (u 1 , s) + 1 m σ (u 2 , s) - 2m σ (u 1 , u 2 , s) m σ (u 1 , s)m σ (u 2 , s) ds 2E τ +η τ 1 m σ (u 1 , s) + 1 m σ (u 2 , s) ds ,
and the rest of the proof is an adaptation of the proof of Proposition 4.10.

Finally we show the third criterion: Third criterion: Let δ > 0. For each k 1, there is H > 0 such that for all σ in Q + ,

P ∀h = (h 1 , h 2 ), 0 < h 1 < H, 0 < h 2 < H, 1-h 1 0 1-h 2 0 sup t T |C σ (u 1 + h 1 , u 2 + h 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 1 k 1 - δ 2 k . ( 48
)
December 2018

Let h 1 > 0 and begin by estimating

E σ := E 1-h 1 0 1 0 sup t T |C σ (u 1 + h 1 , u 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 .
We compute (for the sake of simplicity, we will write from now on y σ (u) instead of y σ (u, •) if there is no possibility of confusion):

C σ (u 1 + h 1 , u 2 , t) -C σ (u 1 , u 2 , t) = y σ (u 1 + h 1 ) -y σ (u 2 ), y σ (u 1 + h 1 ) -y σ (u 2 ) t -y σ (u 1 ) -y σ (u 2 ), y σ (u 1 ) -y σ (u 2 ) t = y σ (u 1 + h 1 ) -y σ (u 1 ), y σ (u 1 + h 1 ) -y σ (u 2 ) t + y σ (u 1 ) -y σ (u 2 ), y σ (u 1 + h 1 ) -y σ (u 1 ) t .
Therefore,

sup t T |C σ (u 1 + h 1 , u 2 , t) -C σ (u 1 , u 2 , t)| sup t T | y σ (u 1 + h 1 ) -y σ (u 1 ), y σ (u 1 + h 1 ) -y σ (u 2 ) t | + sup t T | y σ (u 1 ) -y σ (u 2 ), y σ (u 1 + h 1 ) -y σ (u 1 ) t |. (49)
Then, we use Kunita-Watanabe's inequality on the first term of the right hand side:

| y σ (u 1 + h 1 ) -y σ (u 1 ), y σ (u 1 + h 1 ) -y σ (u 2 ) t | | y σ (u 1 + h 1 ) -y σ (u 1 ), y σ (u 1 + h 1 ) -y σ (u 1 ) t | 1 2 | y σ (u 1 + h 1 ) -y σ (u 2 ), y σ (u 1 + h 1 ) -y σ (u 2 ) t | 1 2 | y σ (u 1 + h 1 ) -y σ (u 1 ), y σ (u 1 + h 1 ) -y σ (u 1 ) T | 1 2 | y σ (u 1 + h 1 ) -y σ (u 2 ), y σ (u 1 + h 1 ) -y σ (u 2 ) T | 1 2 .
By doing the same computation on the second term of the right hand side of (49), by Cauchy-Schwarz inequality and by the substitution of u 1 + h 1 by u 1 , we obtain:

E σ 2E 1-h 1 0 1 0 y σ (u 1 + h 1 ) -y σ (u 1 ), y σ (u 1 + h 1 ) -y σ (u 1 ) T du 1 du 2 1/2 × E 1 0 1 0 y σ (u 1 ) -y σ (u 2 ), y σ (u 1 ) -y σ (u 2 ) T du 1 du 2 1/2 2E 1-h 1 0 y σ (u 1 + h 1 ) -y σ (u 1 ), y σ (u 1 + h 1 ) -y σ (u 1 ) T du 1 1/2 C 1/2 ,
where C is the same constant as the one in the first criterion. By Fubini's Theorem:

E σ 2C 1/2 E 1-h 1 0 (y σ (u 1 + h 1 , T ) -y σ (u 1 , T ) + g(u 1 ) -g(u 1 + h 1 )) 2 du 1 1/2 2C 1/2 E 1-h 1 0 (y σ (u 1 + h 1 , T ) -y σ (u 1 , T )) 2 du 1 1/2 + 2C 1/2 E 1-h 1 0 (g(u 1 + h 1 ) -g(u 1 )) 2 du 1 1/2
.

December 2018

We recall inequalities (26) and (27). Therefore, there are α > 0 and C > 0 such that for each σ ∈ Q + and each h 1 > 0,

E σ Ch α 1 .
We deduce that for each n ∈ N, by Markov's inequality,

p n := P 1-1 2 n 0 1 0 sup t T |C σ (u 1 + 1 2 n , u 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 1 2 nα 2 2 nα 2 C 1 2 n α = C 2 nα 2
.

Since α > 0, n 0 p n converges. By Borel-Cantelli's Lemma, for each k 1, there is n 0 0 such that, with probability greater than 1 -δ 2 k , for all n n 0 ,

1-1 2 n 0 1 0 sup t T |C σ (u 1 + 1 2 n , u 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 1 2 nα 2 .
Furthermore, up to choosing a greater n 0 , we can suppose that for all n n 0 , we also have:

1 0 1-1 2 n 0 sup t T |C σ (u 1 , u 2 + 1 2 n , t) -C σ (u 1 , u 2 , t)|du 1 du 2 1 2 nα 2 .
We will now extend these estimations to more general perturbations. Let h = (h 1 , h 2 ) be such that 0 < h 1 < 1 2 n 0 , 0 < h 2 < 1 2 n 0 . We decompose:

1-h 1 0 1-h 2 0 sup t T |C σ (u 1 + h 1 , u 2 + h 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 1-h 1 0 1 0 sup t T |C σ (u 1 + h 1 , u 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 + 1 0 1-h 2 0 sup t T |C σ (u 1 , u 2 + h 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 . ( 50
)
Suppose h 1 0. Since h 1 < 1 2 n 0 , there exists a sequence (ε n ) n>n 0 with values in {0, 1} such that h 1 = n n 0 +1 εn 2 n . Moreover, we have for every q 1:

1-h 1 0 1 0 sup t T |C σ (u 1 + h 1 , u 2 , t) -C σ (u 1 + n n 0 +q εn 2 n , u 2 , t)|du 1 du 2 q-1 k=1 1-h 1 0 1 0 sup t T |C σ (u 1 + n n 0 +k εn 2 n , u 2 , t) -C σ (u 1 + n n 0 +k+1 εn 2 n , u 2 , t)|du 1 du 2 q-1 k=1 1-1 2 n 0 +k 0 1 0 sup t T |C σ (u 1 + 1 2 n 0 +k , u 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 q-1 k=1 1 2 (n 0 +k) α 2 . (51)
We want to let q tend to +∞ in (51). To do that, we prove that:

1-h 1 0 1 0 sup t T |C σ (u 1 + n n 0 +q εn 2 n , u 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 -→ q→+∞ 0. (52)
December 2018

By definition of C σ ,

1-h 1 0 1 0 sup t T |C σ (u 1 + n n 0 +q εn 2 n , u 2 , t) -C σ (u 1 , u 2 , t)|du 1 du 2 1-h 1 0 T 0 1 m σ (u 1 + n n 0 +q εn 2 n , s) - 1 m σ (u 1 , s) dsdu 1 + 1-h 1 0 1 0 T 0 2 m σ (u 2 , s) m σ (u 1 + n n 0 +q εn 2 n , u 2 , s) m σ (u 1 + n n 0 +q εn 2 n , s) - m σ (u 1 , u 2 , s) m σ (u 1 , s) dsdu 1 du 2 . (53)
For each s ∈ [0, T ], m σ (•, s) is right-continuous. Therefore, m σ (u 1 + n n 0 +q εn 2 n , s) converges to m σ (u 1 , s) when q → +∞. Furthermore, there is β > 1 such that almost surely, which is almost surely finite. Thus the first term of the right hand side of (53) tends almost surely to 0 for every h 1 < 1 2 n 0 . A similar argument shows that the second term of the right hand side of (53) also converges to 0. Hence we have justified convergence (52).

sup u∈ 0, 1 2 n 0 -1 1-u 0 T 0 1 m σ (u 1 + u, s) - 1 m σ (u 1 , s) β dsdu 1 < +∞.
When q → ∞ in inequality (51), we obtain: .

Then, we proceed similarly for the second term of the right hand side of (50) and we finally obtain, for each h = (h 1 , h 2 ) such that 0 < h 1 < 1 2 n 0 and 0 < h 2 < 1 2 n 0 , .

Choosing H = 1 2 n 0 such that CH α/2 1 k , we get (48) for each σ in Q + .

Conclusion of the proof. By Simon's tightness criterion on L 1 ([0, 1] × [0, 1], C[0, T ]), the collection of laws of (C σ ) σ∈Q + is relatively compact in P(L 1 ([0, 1] × [0, 1], C[0, T ])). Therefore the collection of laws of (y σ , C σ ) σ∈Q + is also relatively compact in P(L 2 ([0, 1], C[0, T ]) × L 1 ([0, 1] × [0, 1], C[0, T ])). Thus there is a subsequence, (y σn , C σn ) n 1 converges in distribution in L 2 ([0, 1], C[0, T ]) × L 1 ([0, 1] × [0, 1], C[0, T ]). We denote by (y, C) the limit. We want to prove that for almost every u 1 , u 2 ∈ [0, 1], C(u 1 , u 2 , •) is the quadratic variation of y(u 1 , •) -y(u 2 , •) relatively to the filtration generated by Y and C. Let l 1, 0 s 1 s 2 . . . s l s t and f l : (L 2 (0, 1)) l × L 1 ([0, 1] × [0, 1]) l → R be a bounded and continuous function. For every non-negative ψ 1 , ψ 2 ∈ L ∞ (0, 1), we have for every n 1: -(y(u 1 , s) -y(u 2 , s) -g(u 1 ) + g(u 2 )) 2 -C(u 1 , u 2 , t) + C(u 1 , u 2 , s) du 1 du 2 f l (Y (s 1 ), . . . , Y (s l ), C(s 1 ), . . . , C(s l )) = 0. By Fubini's Theorem, we obtain that for almost every u 1 , u 2 ∈ (0, 1), for all rational numbers (s 1 , . . . , s l , s, t) such that 0 s 1 s 2 . . . s l s t: E (y(u 1 , t) -y(u 2 , t) -g(u 1 ) + g(u 2 )) 2 -(y(u 1 , s) -y(u 2 , s) -g(u 1 ) + g(u 2 )) 2 -C(u 1 , u 2 , t) + C(u 1 , u 2 , s) f l (Y (s 1 ), . . . , Y (s l ), C(s 1 ), . . . , C(s l )) = 0.

By continuity in time, the latter equality remains true for every 0 s 1 s 2 . . . s l s t. Furthermore, for almost every u 1 , u 2 , (C σn (u 1 , u 2 , t)) t∈[0,T ] is a non-decreasing bounded variation process. This remains true for the limit (C(u 1 , u 2 , t)) t∈[0,T ] . Therefore, we deduce that C(u 1 , u 2 , t) = y(u 1 ) -y(u 2 ), y(u 1 ) -y(u 2 ) t , for almost every u 1 , u 2 ∈ (0, 1), with respect to the filtration generated by (Y, C).

We conclude this Paragraph by using Fatou's Lemma to extend the statement of Lemma 4.15 to the limit process: Proposition 5.11. Let g ∈ L p (0, 1). For all β ∈ (0, 3 2 -1 p ), there is a constant C > 0 depending only on β and g Lp such that for all 0 s < t T , we have the following inequality:

E t s 1 0 1 m(u, r) β dudr C √ t -s.
December 2018 is a bounded variation process for almost every u, u ∈ (0, 1

Therefore, there exists a (non-random) subset A of [0, 1], such that for every u, u ∈ A, (55) and (56) holds almost surely. Let u, u ∈ A. If g(u) = g(u ) then τ u,u = 0 almost surely, thus (54) is clear. Up to exchanging u and u , assume that g(u) < g(u ). Let δ < 2(g(u ) -g(u)). Almost surely, by (55), there exists σ 0 such that for all σ ∈ (0, σ 0 ) ∩ Q + , sup Define τ δ u,u := inf{t 0 : |y(u, t)-y(u , t)| δ}∧T . Therefore, for all t < τ δ u,u and for all σ < σ 0 , |y σ (u, t) -y σ (u , t)| δ 2 . Let σ < min(σ 0 , δ 2 ). For all t < τ δ u,u , we have |y σ (u, t) -y σ (u , t)| σ and thus m σ (u, u , t) = 0, hence K σ (u, u , t) = t 0 mσ(u,u ,s) mσ(u,s)mσ(u ,s) ds = 0 for t τ δ u,u . By (56), we obtain sup Thus for every δ > 0, for every u, u ∈ A and t τ δ u,u , y(u), y(u ) t = 0. Since τ δ u,u → τ u,u when δ → 0, we have for each u, u ∈ A: y(u), y(u ) t∧τ u,u = 0.

(57)

It remains to show that (57) holds for every (u, u ) ∈ (0, 1) 2 . Let (u, u ) ∈ (0, 1) 2 . As previously, we may assume that g(u) < g(u ). By continuity of the processes (y(u, t)) t∈[0,T ] and (y(u , t)) t∈[0,T ] , the first time of coalescence τ u,u is almost surely positive. Fix l 1, 0 s 1 s 2 . . . s l s t and a bounded and continuous function f l : (L 2 (0, 1)) l → R. Suppose that s > 0. We want to prove that: E (y(u, t ∧ τ u,u )y(u , t ∧ τ u,u ) -y(u, s ∧ τ u,u )y(u , s ∧ τ u,u ))f l (Y (s 1 ), . . . , Y (s l )) = 0. (58) Let ε > 0. For each v ∈ (u, u + ε) ∩ A and v ∈ (u , u + ε) ∩ A (since A is of plain measure in (0, 1), both sets are non-empty), since we have equality (57), 0 = E (y(v, t ∧ τ v,v )y(v , t ∧ τ v,v ) -y(v, s ∧ τ v,v )y(v , s ∧ τ v,v ))f l (Y (s 1 ), . . . , Y (s l )) . (59) Let t 0 ∈ (0, s). We define η := sup{h 0 : y(u + h, t 0 ) = y(u, t 0 ) and y(u + h, t 0 ) = y(u , t 0 )}.

By the coalescence property given by Proposition 5.5, under the event {τ u,u > t 0 }, we know that for every r t 0 , for each v ∈ (u, u+η) and v ∈ (u , u +η), y(v, r) = y(u, r) and y(v , r) = y(u , r), whence τ v,v = τ u,u . Thus, by equality (59), we deduce that for each v ∈ (u, u + ε) ∩ A and v ∈ (u , u + ε) ∩ A, 0 = E 1 {η>ε} 1 {τ u,u >t 0 } (y(u, t ∧ τ u,u )y(u , t ∧ τ u,u ) -y(u, s ∧ τ u,u )y(u , s ∧ τ u,u ))f l (Y (s 1 ), . . . , Y (s l )) + E 1 {η ε}∪{τ u,u t 0 } (y(v, t ∧ τ v,v )y(v , t ∧ τ v,v ) -y(v, s ∧ τ v,v )y(v , s ∧ τ v,v )) f l (Y (s 1 ), . . . , Y (s l )) . (60) Let h > 0 be such that (u, u + ε) and (u , u + ε) are contained in (h, 1 -h). Thus for every v ∈ (u, u + ε) ∩ A, for every r ∈ [0, T ], by inequality (41) and by Doob's inequality, we deduce that:

E sup r T y(v, r) 2β 2 h E 1 0 sup r T y(x, r) 2β dx C β h E 1 0 y(x, T ) 2β dx C β h ,
for a β arbitrarily chosen in (1, 3 2 -1 p ) (by Corollary 5.12). Thus, there exists β > 1 such that E (y(v, t ∧ τ v,v )y(v , t ∧ τ v,v ) β is uniformly bounded for v ∈ (u, u + ε) and v ∈ (u , u + ε). Let α = 1 -1 β . Therefore, we deduce from (60) that there is a constant C depending only on u, u and α such that: E 1 {η>ε} 1 {τ u,u >t 0 } (y(u, t ∧ τ u,u )y(u , t ∧ τ u,u ) -y(u, s ∧ τ u,u )y(u , s ∧ τ u,u )) f l (Y (s 1 ), . . . , Y (s l ))

C P [η ε] α + P τ u,u t 0 α . (61)

We divide the left hand side of inequality (61) into two parts by writing 1 {η>ε} 1 {τ u,u >t 0 } = 1 -1 {η ε}∪{τ u,u t 0 } and we estimate the second term in the same way as above. We deduce that there is a constant C such that:

E (y(u, t ∧ τ u,u )y(u , t ∧ τ u,u ) -y(u, s ∧ τ u,u )y(u , s ∧ τ u,u ))f l (Y (s 1 ), . . . , Y (s l )) C P [η ε] α + P τ u,u t 0 α .

Figure 1 :

 1 Figure 1: Two simulations, based on the same underlying Brownian sheet, for the limit process (µt) t∈[0,T ] (on top) and for the process (µ σ,ε t ) t∈[0,T ] with positive σ and ε (on bottom). The horizontal axis represents time. On the vertical axis, we put the position of the particles (initially, we took five particles on [0, 1]).

1 0 K 1 (u)du and 1 0

 111 K 2 (u)du are uniformly bounded for σ > 0 and ε > 0.

  0, T ] with respect to Skorohod's topology, and y σ,ε (u, •) ∈ C[0, T ] almost surely, we may suppose that L is a compact set of C[0, T ].

  (and the same is true for u and r ). Thus almost surely Y σ (t) belongs to K for every t ∈ [0, T ]. It remains to prove that it implies that Y σ (t) belongs to L ↑ 2 [0, 1]. Let f ∈ K. Define, for each u ∈ (0, 1), f (u) := lim inf h→0 + 1 h (u+h)∧1 u f (v)dv. First, remark that f is non-decreasing. Then, since h → 1 h u+h u

n h 1 < 1 2 n 0 -1 for every q 1 ,

 11 u 1 , s) β dsdu 1 < +∞, by Lemma 4.6. Therefore, since n n 0 +qεn 2 u 1 + u, s) -1 m σ (u 1 , s) β dsdu 1 ,

2 +∞ k=1 1 2 (n 0 +k) α 2 C α 2 n 0 α 2

 21222 |C σ (u 1 + h 1 , u 2 , t) -C σ (u 1 , u 2 , t)|du 1 du

ψ 1 2 -ψ 1

 121 (u 1 )ψ 2 (u 2 ) (y σn (u 1 , t) -y σn (u 2 , t) -g(u 1 ) + g(u 2 )) (y σn (u 1 , s) -y σn (u 2 , s) -g(u 1 ) + g(u 2 )) 2 -C σn (u 1 , u 2 , t) + C σn (u 1 , u 2 , s) du 1 du 2 f l (Y σn (s 1 ), . . . , Y σn (s l ), C σn (s 1 ), . . . , C σn (s l )) = 0, since the process (C σn (t)) t∈[0,T ] := (C σn (•, •, t)) t∈[0,T ] is (F σn t ) t∈[0,T ] -adapted.By the convergence in distribution, we obtain when n goes to ∞: (u 1 )ψ 2 (u 2 ) (y(u 1 , t) -y(u 2 , t) -g(u 1 ) + g(u 2 ))2

  ,s) is finite. Thus for almost every u and u in (0, 1), K(u, u , •) is a bounded variation process. This concludes the proof of the Lemma.We use the latter Lemma to prove Proposition 5.13.Proof (Proposition 5.13). By Lemma 5.14 and Skorohod's representation Theorem, we may suppose that(y σ , K σ ) σ∈Q + converges almost surely in L 2 ([0, 1], C[0, T ]) × L 1 ([0, 1] × [0, 1], C[0, T ]) to (y, K).As previously, up to extracting a subsequence, we deduce that for almost every(ω, u, u ) ∈ Ω × [0, 1] × [0, 1], sup t T |y σ (u, t) -y(u, t)|(ω)|K σ (u, u , t) -K(u, u , t)|(ω) -→ σ→0 0.

  σ (u , t) -y(u , t)| δ 4.

  |K(u, u , t)| = 0.December 2018

  We will now use Lemma 4.6 and its Corollary 4.8 to prove Proposition 4.2. We start by (K1):

	December 2018
	4.1.2 Proof of Proposition 4.2
	by Lemma 4.6.

  , since for almost every u ∈ (0, 1), y σ (u, •) is continuous at time t, and (y σ (u, t n ) -y σ (u, t)) 2 4 sup t T |y σ (u, t)| 2 which is almost surely integrable.

2 du -→ n→∞ 0 by dominated convergence Theorem

  Since lim sup n→∞ m σn (u p , t) m(u p , t) and by Fatou's

	by Hölder's inequality. By Fatou's Lemma
							1				1
						E	0	y(v, t) 2β dv	lim inf n→∞	E	0	y σn (v, t) 2β dv ,
	which is finite by Corollary 4.16, for a β chosen in (1, 3 2 -1 p ).
		Let us keep the same exponent β ∈ (1, 3 2 -1 p ). It remains to show that for every t ∈
	[0, T ], sup p∈N E	t 0	dr m(up,r)
	Lemma,
	E	0	t	dr m(u p , r)	β	E		0	t	lim inf n→∞	dr m σn (u p , r)

β < +∞.

  ). It follows from Kunita-Watanabe's inequality that: (u, u , t k+1 ) -K σ,ε (u, u , t k )| = (u), y σ,ε (u ) t k+1 -y σ,ε (u), y σ,ε (u ) t k | . By Proposition 5.11, we know that almost surely and for almost every u ∈ (0, 1),

	p-1 k=0	|K σ,ε p-1 k=0 | y σ,ε p-1 t k+1	1 2	t k+1	1 2
							d y σ,ε (u), y σ,ε (u) s	d y σ,ε (u ), y σ,ε (u ) s
				k=0		t k				t k
				1 2	tp t 0	d y σ,ε (u), y σ,ε (u) s +	1 2	tp t 0	d y σ,ε (u ), y σ,ε (u ) s
			=	1 2	tp t 0	ds M σ,ε (u, s)	+	1 2	tp t 0	ds M σ,ε (u , s)	,
	Therefore, for every p 1 2 tp t 0 ds tp ds m(u,s) + 1 2 t 0 m(u ,s)	1 and 0		t 0	t 1	. . .	t p , p-1 k=0 |K(u, u , t k+1 ) -K(u, u , t k )|

L 2 < +∞, and continuous if the process t → M t is a continuous function from [0, T ] to L 2 [0, 1].

L2([0, 1], C[0, T ]) is a Polish space. Its separability can be proved using the separability of C([0, 1] × [0, T ]) and the density of C([0, 1] × [0, T ]) in L2([0, 1], C[0, T ]).

Construction of a Wasserstein diffusion

Proposition 4.17. Suppose that g ∈ L ↑ 2+ [0, 1]. (Y (t)) t∈[0,T ] is a L ↑ 2 [0, 1]-valued process such that:

(C1) Y (0) = g;

(C2) (Y (t)) t∈[0,T ] is a square integrable continuous L ↑ 2 [0, 1]-valued (F t ) t∈[0,T ] -martingale, where F t = σ(Y (s), s t).

Proof. We refer to the proof of Proposition 4.13.

Remark 4.18. It should be noticed at this point that a new difficulty arises when we want to obtain a property analogous to (B3). Indeed, whereas it was straightforward to prove (34) and ( 35), the convergence of m σ (u, t) = 1 0 ϕ 2 σ (y σ (u, t) -y σ (v, t))dv to m(u, t) = 1 0 1 {y(u,t)=y(v,t)} dv is not obvious, due to the singularity of the indicator function. It will be the main goal of the next Section to prove this convergence.

In Section 5, we will study the martingale properties of the limit process Y and compute its quadratic variation (property (C5) of Theorem 1.4). To obtain this, we will first prove that for every positive t, Y (t) is a step function (see property (C3)). It implies that y has a version in D((0, 1), C[0, T ]) (see property (C4)) by an argument given in ( [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF]Proposition 2.3]).

Properties of the limit process Y

The aim of this Section is to complete the proof of Theorem 1.4. Properties (C3) and (C4) will be proved in Paragraph 5.1 and property (C5) will be proved in two steps in Paragraph 5.2 and Paragraph 5.3.

Coalescence properties and step functions

In this Paragraph, we will prove the following Proposition: Proposition 5.1. Almost surely, for every t > 0, Y (t) is a step function.

Recall that Y (0) = g is not necessarily a step function, since g can be chosen arbitrarily in

If we denote for each t ∈ [0, T ] by µ t the measure associated to the quantile function

1 means that for every positive time t, µ t is a finite weighted sum of Dirac measures. We begin by the following Lemma. Recall the definition of the mass: m(u, t) = 1 0 1 {y(u,t)=y(v,t)} dv. Lemma 5.2. There exists a probability space ( Ω, P) on which the sequence ( y σ ) σ∈Q + converges almost surely to y in L 2 ([0, 1], C[0, T ]) and where, for each σ ∈ Q + , y σ (resp. y) has same law as y σ (resp. y). Furthermore, there is a subsequence (σ n ) n , σ n → 0, such that for almost every (ω, u) ∈ Ω × (0, 1) and for every time t ∈ [0, T ], lim sup n→∞ m σn (u, t) m(u, t).

Proof. Recall that (y σ ) σ∈Q + converges in distribution in L 2 ([0, 1], C[0, T ]) to y. By Skorohod's representation Theorem, we deduce that there exists a sequence ( y σ ) σ∈Q + and a random variable y defined on a common probability space ( Ω, P) such that for every σ ∈ Q + , the laws of y σ and y σ are the same, the laws of y and y are also equal and the sequence ( y σ ) σ∈Q + converges almost surely to y in L 2 ([0, 1], C[0, T ]).

December 2018

Proof of inequality:

lim inf n→∞ m σn (ω, u, t) m(ω, u, t). By the coalescence property given by Proposition 5.5, for every u 1 , u 2 and for all t > τ u 1 ,u 2 , y(u 1 , t) = y(u 2 , t). Therefore, since C(u 1 , u 2 , •) is the quadratic variation of y(u 1 , •) -y(u 2 , •), t → C(u 1 , u 2 , t) remains constant on (τ u 1 ,u 2 , T ). Thus we obtain:

By (43), the latter term tends to 0. We also recall that

Therefore, for every ε > 0, using Markov's inequality as in (37), and since f σn 0:

which tends to 0 when n → ∞, whence we obtain a convergence in probability with respect to the probability space Ω × [0, T ] × [0, 1] × [0, 1]. Up to extracting another subsequence (independent of the choice of ω), we deduce the existence of an almost sure event on which (f σn ) converges to 0.

Let Ω , P [Ω ] = 1, be such that for every ω ∈ Ω , we have

Let us consider a Borel set A (depending on ω and t 0 ) of measure 1 such that for all u 1 , u 2 ∈ A, By Burkholder-Davis-Gundy inequality, we deduce the following estimation: Corollary 5.12. For each β ∈ (0, 3 2

Covariation of y(u, •) and y(u , •)

In this Paragraph, we want to complete the proof of property (C5) of Theorem 1.4. It remains to prove the following Proposition:

Proposition 5.13. Let y be the version in D((0, 1), C[0, T ]) of the limit process given by Proposition 5.5. For every u, u ∈ (0, 1),

where τ u,u = inf{t 0 : y(u, t) = y(u , t)} ∧ T .

As in the previous Paragraph, we will need to prove the convergence of the joint law of y σ and a quadratic covariation. More precisely, define:

ds.

We state the following result:

Lemma 5.14. For every sequence (σ n ) n of rational numbers tending to 0, we can extract a subsequence ( σ n ) n such that the sequence (y σn , K σn ) n→∞ converges in distribution to

Proof (Lemma 5.14). We follow the same structure as in the proof of Lemma 5.10. First, we define

(ε+mσ,ε(u,s))(ε+mσ,ε(u ,s)) ds. We show that K σ,ε satisfies the three criteria of tightness in L 1 ([0, 1] × [0, 1], C[0, T ]). For the first criterion, we want to bound

uniformly for σ, ε ∈ Q + . This follows from Kunita-Watanabe's inequality:

and from Cauchy-Schwarz inequality:

which is bounded uniformly for σ, ε ∈ Q + by Corollary 4.8. We refer to the proof of Lemma 5.10 for the second and the third criteria of tightness, and for the rest of the proof, which follows in the same way. It remains to explain why (K(u, u , t)) t∈[0,T ] December 2018 Let δ > 0. Since τ u,u > 0 almost surely, we choose t 0 ∈ (0, s) such that P τ u,u t 0 α δ. Since t 0 > 0, we know by Proposition 5.1 that y(•, t 0 ) is almost surely a step function, so η > 0 almost surely. Therefore, we can choose ε > 0 so that P [η ε] α δ. This concludes the proof of equality (58).

Recall that we suppose that t s > 0. By continuity of time of y(u, •) and y(u , •), equality (58) also holds for s = 0. Therefore, y(u, t ∧ τ u,u )y(u , t ∧ τ u,u ) is a (F t ) t∈[0,T ] -martingale and y(u), y(u ) t∧τ u,u = 0. This concludes the proof of Proposition 5.13.

A Appendix: Itô's formula for the Wasserstein diffusion

We assume, to simplify the notations, that g(1) is finite, but the proof can be easily adapted to functions g with g(u) -→ u→1 +∞. Let y be a process in

Recall that the process y(•, t) t∈[0,T ] can be considered as the quantile function of (µ t ) t∈[0,T ] , by setting

The latter process has every feature of a Wasserstein diffusion. We describe in this Paragraph the dynamics of the process (µ t ) t∈[0,T ] , after having introduced a differential calculus on P 2 (R) due to Lions ([15], [START_REF] Cardaliaguet | Notes on Mean Field Games (from P.-L. Lions' lectures at collège de France)[END_REF]). We prove that, for a smooth function U : P 2 (R) → R, the process (U (µ t )) t∈[0,T ] is a semi-martingale with quadratic variation proportional to the square of the gradient of U (see Theorem A.3). This result is a generalization of the formula given by Konarovskyi and von Renesse in [START_REF] Konarovskyi | Modified Massive Arratia Flow and Wasserstein Diffusion[END_REF]. We compare it to a similar result obtained by von Renesse and Sturm [START_REF] Von Renesse | Entropic measure and Wasserstein diffusion[END_REF] for the Wasserstein diffusion on [0, 1] (see Remark A.4).

In order to describe the dynamics of (µ t ) t∈[0,T ] , we begin by a discretization in space and by writing the classical Itô formula for that discretized process. Let introduce µ n

, where [n] denotes the set {1, . . . , n}. Fix U : P 2 (R) → R a continuous function, with respect to the Wasserstein distance W 2 on P 2 (R). Let define U n (x 1 , . . . , x n ) := U ( 1 n j∈[n] δ x j ). Remark that U ( µ n t ) = U n y( 1 n , t), y( 2 n , t), . . . , y(1, t) . Assuming that U n belongs to C 2 (R n ), and using that y( k n , •) is a square integrable continuous martingale on [0, T ], we have (recall that g(1) is finite):

In order to write the derivatives of U n in terms of derivatives of U , we should introduce a differential calculus on P 2 (R), well-adapted to the differentiation of empirical measures. P.L. Lions introduces in his lectures at Collège de France (see Section 6.1 of Cardaliaguet's notes [START_REF] Cardaliaguet | Notes on Mean Field Games (from P.-L. Lions' lectures at collège de France)[END_REF]) a differential calculus on P 2 (R) by using the Hilbertian structure of L 2 (Ω). We set U (X) := U (Law(X)) for all X ∈ L 2 (Ω).

A function U : P 2 (R) → R is said to be L-differentiable (or differentiable in the sense of Lions) at a point µ 0 ∈ P 2 (R) if there is a random variable X 0 with law µ 0 such that U is Fréchet-differentiable at X 0 . The definition does not depend on the choice of the representative X 0 of the law µ 0 , and if X 0 and X 1 have the same law, then the laws of D U (X 0 ) and D U (X 1 ) are equal (see e.g. [START_REF] Cardaliaguet | Notes on Mean Field Games (from P.-L. Lions' lectures at collège de France)[END_REF]). Furthermore, if D U : L 2 (Ω) → L 2 (Ω) is a continuous function, then for all µ 0 ∈ P 2 (R), there exists a measurable function R → R, denoted by ∂ µ U (µ 0 ), such that for each X ∈ L 2 (Ω) with law µ 0 , we have D U (X) = ∂ µ U (µ 0 )(X) almost surely (see [START_REF] Cardaliaguet | Notes on Mean Field Games (from P.-L. Lions' lectures at collège de France)[END_REF]).

December 2018

In [START_REF] Carmona | Probabilistic theory of mean field games with applications. I[END_REF], Carmona and Delarue prove that the L-differentiability of U : P 2 (R) → R implies the differentiability of U n on R n , and that we have for each k ∈ [n]:

Furthermore, assume that U is L-differentiable and that (µ, v) ∈ P 2 (R)×R → ∂ µ U (µ)(v) ∈ R is continuous. Moreover, we assume that for every µ ∈ P 2 (R), the map v ∈ R → ∂ µ U (µ)(v) ∈ R is differentiable on R in the classical sense and that its derivative is given by a jointly continuous function (µ, v) → ∂ v ∂ µ U (µ)(v). We also assume that for every v ∈ R, the map µ → ∂ µ U (µ)(v) is L-differentiable and its derivative is denoted by

Therefore, we obtain from equation (62):

By property of coalescence, if τ k n , l n s, we have y( k n , s) = y( l n , s), so that the last term in the latter equation is equal to:

Observe that the difference between

s} is an interval. We want to let n tend to +∞ in order to obtain an Itô formula for the limit process. We start by proving the convergence of a subsequence of (( µ n t ) t∈[0,T ] ) n 1 to (µ t ) t∈[0,T ] with respect to the L 2 -Wasserstein distance.

Proposition A.1. There exists a subsequence (( µ

) n 1 converges almost surely to µ t with respect to the Wasserstein distance W 2 .

Remark A.2. We point out that the extraction function ϕ does not depend on t ∈ [0, T ].

Proof. To obtain the statement of the Proposition, it is sufficient to prove that:

Let V be a uniform random variable on [0, 1], defined on a probability space ( Ω, F, P). Therefore, µ t is the law of y(V, t) and µ n t the law of k∈

). Hence we have:

December 2018

Therefore, it is sufficient to show that:

Fixing u ∈ (0, 1), t ∈ (0, T ),

converges almost surely to 0 by the right-continuity of y(•, t) at point u. To prove (64), we have to show a uniform integrability property, i.e. that sup n 1 E T 0 1 0

We compute:

, where M t = y( k n , t) -y(u, t). Recall that by property (i) of the process y, M 0 = g( k n ) -g(u). We deduce that:

Since g belongs to L ↑ 2+ [0, 1], there exists p > 2 such that g ∈ L p (0, 1). Therefore, we can choose β > 1 such that 2β p. By Burkholder-Davis-Gundy inequality and the martingale property of M , we have:

By property (iv),

so that there is a constant C β satisfying:

December 2018

To conclude, we use the following statement: provided β < 3 2 -1 p , there is a constant C β such that for each t and u:

This statement is Proposition 5.11 for the limit process that we constructed in this paper, or in [START_REF] Konarovskyi | On asymptotic behavior of the modified Arratia flow[END_REF]Prop. 4.3] for the process constructed by Konarovskyi. This completes the proof.

By similar arguments of convergence, equation (63) leads to the following Itô formula for (µ t ) t∈[0,T ] , by letting n tend to ∞. The estimation (65) is the key of the proof of those convergences.

Theorem A.3. Let U : P 2 (R) → R be smooth enough so that U and its derivatives ∂ µ U , ∂ v ∂ µ U and ∂ 2 µ U exist, are uniformly continuous and bounded. Almost surely, for each t ∈ [0, T ], we have: m(u,s) du and L 2 U (µ s ) := m i,j=1 ∂ 2 i,j V -→ α dµ s 1 0 α i (y(u, s))α j (y(u, s))du. Remark that we have some restrictions on the domain of the generator L 1 . We know that for measures with finite support, 1 0 du m(u,s) is finite and is equal to the cardinality of the support (see the Paragraph preceding Corollary 5.3). The fact that the generator of the martingale problem is not defined on the whole Wasserstein space is related to the fact that the process (µ t ) t∈[0,T ] takes values, for every positive time t, on the space of measures with finite support.

We compare this result to Theorem 7.17 in [START_REF] Von Renesse | Entropic measure and Wasserstein diffusion[END_REF]. The generator of the martingale in the case of von Renesse and Sturm's Wasserstein diffusion is L = L 1 + L 2 + βL 3 , with L 1 = L 2 and L 3 similar to L 1 up to the lack of the mass function, whereas L 2 , which is the part of the generator considering the gaps of the measure µ, does not appear in our model.