A new approach for the construction of a Wasserstein diffusion

We propose in this paper a construction of a diffusion process on the Wasserstein space P\_2(R) of probability measures with a second-order moment. This process was introduced in several papers by Konarovskyi (see e.g."A system of coalescing heavy diffusion particles on the real line", 2017) and consists of the limit when N tends to infinity of a system of N coalescing and mass-carrying particles. It has properties analogous to those of a standard Euclidean Brownian motion, in a sense that we will precise in this paper. We also compare it to the Wasserstein diffusion on P\_2(R) constructed by von Renesse and Sturm (see Entropic measure and Wasserstein diffusion). We obtain that process by the construction of a system of particles having short-range 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 [Kon17b,Kon11,KvR15,Kon17a]. It is a close relative to the Wasserstein diffusion, introduced by von Renesse and Sturm [vRS09]. 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 [vRS09], 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: 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

Konarovskyi's model
In [Kon17b], 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 [Kon11,KvR15,Kon17a]). 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 [Kon17a]), 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 [Arr79] and subject of many interest, among others in [Pit98,Dor04,NT08,LJR04]. 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 [Kon17b]), 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.
By transporting the Lebesgue measure on [0, 1] by the map y(·, t), we obtain a measurevalued process (µ t ) t∈[0,T ] defined by: µ t := Leb | [0,1] • y(·, t) −1 . In other words, u → y(u, t) is the quantile functile associated to µ t . An important feature of this process is that for each positive t, µ t is an atomic measure with a finite number of atoms, or in other words that y(·, t) is a step function.
More generally, Konarovskyi proves in [Kon17a] that this construction also holds for a greater family of initial measures µ 0 . He constructs a process y g in D([0, 1], C[0, T ]) satisfying (ii) − (iv) and: (i) for all u ∈ [0, 1], y g (u, 0) = g(u), for every non-decreasing càdlàg function g from [0, 1] into R such that there exists p > 2 satisfying 1 0 |g(u)| p du < ∞. In other words, he generalizes the construction of a diffusion starting from any probability measure µ 0 satisfying R |x| p dµ 0 (x) < ∞ for a certain p > 2, where µ 0 = Leb | [0,1] • g −1 , which means that g is the quantile function of the initial measure. The property that y g (·, t) is a step function for each t > 0 remains true for this larger class of functions g.
The process y g is said to be coalescent: almost surely, for every u, v ∈ [0, 1] and for every t ∈ (τ u,v , T ], we have y g (u, t) = y g (v, t) (recall that τ u,v = inf{t 0 : y g (u, t) = y g (v, t)} ∧ T ). This property is a consequence of (ii), (iii) and of the fact that for each t > 0, y g (·, t) is a step function (see [KvR15,p.12]). Therefore, we can rewrite the formula for the mass as follows: 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) ( [Lio,Car13]), 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 [Pit98] 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 [Dor04] 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 [Wal86] for a complete introduction to Brownian sheet and to Section 2 for the characterization of Brownian sheet which we use in this paper.
Remark 1.2. We refer to Appendix A to justify the use of the term "Wasserstein diffusion" for a process satisfying equation (1). 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 [CD17,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 (1) 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 interval − σ 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: where m σ,ε (u, s) := 1 0 ϕ 2 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 (2), 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 (1), 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 (2). 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.
Remark 1.5. More precisely, the filtration (F t ) t∈[0,T ] is given by: 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 .
July 2018 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]).

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).
2 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 [Kon17a] 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 [Wal86,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 Define the filtration (G t ) t∈[0,T ] by G t := σ(w(u, s), u ∈ [0, 1], s t). Then in particular, 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 , 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.
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: using property (iv) of process y; since m(u, s) = m(v, s) whenever y(u, s) is equal to y(v, s). By similar computations, and To sum up, The result follows from the two below equalities: which implies that W 3 = W 1 and consequently equation (1).
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 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.
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.

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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 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: Furthermore, we compute: Moreover, we have: We obtain the following upper bound:

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Finally, we deduce that there is a constant C σ,ε depending only on σ and ε such that 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: 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 it satisfies, by construction, almost surely and for every t ∈ [0, T ], equation (3). In the particular case where g(u) = u, the process y id σ,ε has a continuous modification with respect to the couple of variables (u, t).
Proof. Take u 1 , u 2 ∈ [0, 1]. After some computations similar to those of the proof of Proposition 3.5, we have for every t ∈ [0, T ]: By Gronwall's Lemma, we have: By Kolmogorov's Lemma (see e.g. [LG13,p.19]) in the space C[0, T ] equipped with the · ∞norm, we deduce that there is a modification of y id σ,ε whose paths are α-Hölder continuous in space and continuous in time, for all α ∈ 0, 1 2 . In particular, this modification belongs to Remark 3.7. We need some further assumptions about the regularity of g to extend the result of Proposition 3.6 to y g σ,ε . In order to apply Kolmogorov's Lemma, g should be ( 1 2 + η)-Hölder continuous, for some positive η.
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, by the same arguments as for Proposition 3.6, we prove that there is C σ,ε such that for every By Lemma 3.9, for every u ∈ [0, 1], ( y g σ,ε (u, t)) t∈[0,T ] is a martingale, we have by Doob's inequality: is finite. Fix 0 s t T , and A s ∈ G s . We have: . Using the same constant C σ,ε as in the proof of Proposition 3.5, we have: This implies the statement of the Lemma.
We complete the proof of Proposition 3.8 by constructing a càdlàg modification of y g σ,ε (·, t).

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Therefore, sup n E 1 0 t 0 | y g σ,ε (u n , s)| p dsdu < +∞. Thus, passing to the limit inferior in (9), inequality (8) becomes: Thus, by Gronwall's inequality, In the following Corollary, we precise the properties of y g σ,ε . From now on, we will always use this version of the process.
Proof. The second point has already been proved in Proposition 3.8. Furthermore, The third inequality has been deduced from inequality (9). Therefore, by Gronwall's Lemma, we obtain This concludes the proof.
We complete the proof of Proposition 3.2.
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 3.13. For almost every u, u ∈ [0, 1], 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, and from the fact that y g σ,ε = y g σ,ε in M.
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.
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 [Kon17b], stated in the case of a system of coalescing particles.

Estimation of the inverse of mass
We define a modified mass 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 (10) 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), By the non-decreasing and càdlàg property, for every 3 ) such that u − k ∈ A. Denote by N and N the two following (F σ,ε t ) t∈[0,T ] -martingales: 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, Decompose this probability in two terms: where H s denotes the complement of the event H s .
, which means that one of those terms belongs to the preimage of 1 by the function ϕ σ . Hence m σ,ε (u, s) This is in contradiction with (15). Therefore inequality (14) is satisfied in this case.
which is a set of particles more or less at half distance between particle u and particle u − h.
It follows that: By inequality (15), we deduce that , which is in contradiction with the hypothesis N s > σ+η 2 . Thus inequality (14) 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 can not be simultaneously non-zero, whence we deduce that m σ,ε (u, u − h, s) = 0. Inequality (14) follows clearly. Therefore, inequality (14) is proved. By Corollary 3.13, it follows that, on G s ∩ H s :

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where we have applied a convexity inequality: Since N remains positive on [0, T ] by Lemma 3.11 (because g(u − h) < g(u)), we deduce that sup [0, N,N T ] β g(u) − g(u − h). Therefore, 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: 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. Using inequality (17) as in the first step, we show that d ds N , N s ( N , N t ). Finally, we obtain the existence of a constant C 3 independent of u, h, k, σ, ε and t such that: Putting together inequality (13) and inequalities (16) and (18), we conclude the proof of inequality (12). Thus inequality (11) 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.
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 [Kon17a]: 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: Remark 4.7. Observe that by the assumption p > 2, made at the beginning of Section 4, there exists some β > 1 such that (19) holds.
Proof. By Fubini-Tonelli Theorem, we have: Furthermore, we compute: Using Lemma 4.4, we obtain a constant C 1 independent of σ and ε such that for all x > 2 β : Moreover, we have for each x > 2 β , using Hölder's inequality: Therefore, 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( Corollary 4.8. There is a constant C such that for every t ∈ [0, T ] and for every σ, ε > 0, Proof. We have: Since g belongs to L 2 (0, 1), the first term of the right hand side is bounded. Furthermore, by Corollary 3.13 and Fubini-Tonelli Theorem: by Lemma 4.6.
Proof. Using again Fubini-Tonelli Theorem, 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: Therefore, by Corollary 4.8, 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, For M large enough, that last quantity is smaller than δ.

Construction of a Wasserstein diffusion
July 2018 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 ∈ K 1 ∩ K 2 , P [y σ,ε (u, ·) ∈ L] 1 − δ. Since C[0, T ] is closed in D[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 ].
Back to (22), we have: The first term on the right hand side of (23) is bounded by: We have: Moreover, where M is a constant independent of σ > 0 and ε > 0 by inequality (21). It remains to handle the second term on the right hand side of (23). 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: Back to equality (23), we have proved that there is η > 0 such that for every σ > 0 and ε > 0: This proves convergence (22) and thus concludes the proof of the Proposition.
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, by a usual martingale equality. Using inequality (25) and N u,0 1, we obtain: Therefore, we have:

Study of
Recall that g belongs to L p (0, 1) for some p > 2. Fix β ∈ (1, 3 2 − 1 p ). We compute: 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 (26).
By Lemma 4.6, we deduce that T du is bounded, because β < 3 2 − 1 p . Therefore, we can conclude that there is a constant C T,β such that:
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): First, we want to obtain the convergence of the left hand side of (32). We proceed in the same way as for the proof of equality (31); to get a uniform integrability property, we have now to prove the existence of β > 1 such that is finite. Therefore, it is sufficient to prove the existence of β > 1 such that is finite for every h ∈ L 2 [0, 1]. By Cauchy-Schwarz inequality, We deduce by Burkholder-Davis-Gundy inequality and Fubini's Theorem that there are some constants independent of ε such that By Lemma 4.6, there exists β > 1 such that E drdu is bounded uniformly for ε ∈ Q + . Thus (33) is finite. It is also finite if we replace t by s.
Let us consider an orthonormal basis (e i ) i 1 in the Hilbert space L 2 (ψ(x)dx). We denote by [·, ·] 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: by applying equality (38) with h = k = e i . By definition of m σ (u, u , r), we have: 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: Proof. We use again the sequence ( y σ,εn ) n∈N obtained by Skorohod's representation Theorem, as in the proof of convergence (35). Therefore, by Fatou's Lemma, where C is obtained thanks to Lemma 4.6.

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: 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 (36) and (37), 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 ([Kon17a, 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 L ↑ 2+ [0, 1]. If we denote for each t ∈ [0, T ] by µ t the measure associated to the quantile function Y (t), that is µ t = Leb | [0,1] • Y (t) −1 , Proposition 5.1 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.
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 ]).
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 [Kon17a]. 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: 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), where m(u, s) = 1 0 1 {y(u,s)=y(v,s)} dv.
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 ]): 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, . 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 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 ].
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 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 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: The existence of C is a consequence of Lemma 4.15.
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 , ·): 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 + , Let h 1 > 0 and begin by estimating 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): Therefore, Then, we use Kunita-Watanabe's inequality on the first term of the right hand side: By doing the same computation on the second term of the right hand side of (51), by Cauchy-Schwarz inequality and by the substitution of u 1 + h 1 by u 1 , we obtain: where C is the same constant as the one in the first criterion. By Fubini's Theorem: We recall inequalities (28) and (29). 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, 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 , Furthermore, up to choosing a greater n 0 , we can suppose that for all n n 0 , we also have: 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: 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: 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, by Lemma 4.6. Therefore, since n n 0 +q εn 2 n h 1 < 1 2 n 0 −1 for every q 1, which is almost surely finite. Thus the first term of the right hand side of (55) 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 (55) also converges to 0. Hence we have justified convergence (54).
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: By Burkholder-Davis-Gundy inequality, we deduce the following estimation: Corollary 5.12. For each β ∈ (0, 3 2 − 1 p ), sup t T E 1 0 (y(u, t) − g(u)) 2β du < +∞.
Recall that we suppose that t s > 0. By continuity of time of y(u, ·) and y(u , ·), equality (60) 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. is a martingale with quadratic variation process 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 [vRS09]. 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.