On asymptotic behavior of the modified Arratia flow

Abstract

We study asymptotic properties of the system of interacting diffusion particles on the real line which transfer a mass [arXiv:1408.0628]. The system is a natural generalization of the coalescing Brownian motions. The main difference is that diffusion particles coalesce summing their mass and changing their diffusion rate inversely proportional to the mass. First we construct the system in the case where the initial mass distribution has the moment of the order greater then two as an $L_2$-valued martingale with a suitable quadratic variation. Then we find the relationship between the asymptotic behavior of the particles and local properties of the mass distribution at the initial time

A new approach for the construction of a Wasserstein diffusion

“…This statement is Proposition 5.11 for the limit process that we constructed in this paper, or in [11,Prop. 4.3] for the process constructed by Konarovskyi.…”

“…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 < ∞.…”

“…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:…”

“…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 [11]), 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 [3] and subject of many interest, among others in [7], [14], [16], [17].…”

Modified Massive Arratia Flow and Wasserstein Diffusion

“…In particular, µ 0 = λ in the present case, but our arguments and constructions below can be modified to the case of more general starting measure, cf. [28]. For the sake of presentation, in the sequel we stick to the µ 0 = λ case.…”