We propose in this paper a construction of a diffusion process on the space P 2 (R) of probability measures with a second-order moment. This process was introduced in several papers by Konarovskyi (see e.g. [12]) and consists of the limit as N tends to +∞ 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 in [22]. 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. conducted by Ambrosio, Gigli, Savare, Villani, Lions and many others ([1], [5], [15], [20], [21]), which led to important improvements in optimal transport theory. Second, the transition costs of the Wasserstein diffusion are given by a Varadhan formula (see [22], Corollary 7.19). 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 [8]), it can also be obtained as a limit of finite-dimensional systems of interacting particles, see [2] and [19]. 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 [10] and [12]. (iv) for all u, u ∈ [0, 1], y(u, ·), y(u , ·) t = t 0 1 {τ u,u s} m(u, s) ds, where m(u, t) = 1 0 1 {∃s t: y(u,s)=y(v,s)} dv and τ u,u = inf{t 0 : y(u, t) = y(u , t)} ∧ T .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 function 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 [11] 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 ...

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We introduce in this paper a strategy to prove gradient estimates for some infinite-dimensional diffusions on $$L_2$$ L 2 -Wasserstein spaces. For a specific example of a diffusion on the $$L_2$$ L 2 -Wasserstein space of the torus, we get a Bismut-Elworthy-Li formula up to a remainder term and deduce a gradient estimate with a rate of blow-up of order $$\mathcal O(t^{-(2+\varepsilon )})$$ O ( t - ( 2 + ε ) ) .