  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.
