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.
Highlights
In the paper we study local properties of the modified Arratia flow
The flow is a variant of the Arratia flow [3, 11, 25] for a system of Brownian motions on the real line which move independently up to their meeting and coalesce
The modified Arratia flow was first constructed in [19], as a physical generalization of the system of coalescing Brownian motions, in the case where particles start from integer points with unit masses
Summary
In the paper we study local properties of the modified Arratia flow. The flow is a variant of the Arratia flow [3, 11, 25] for a system of Brownian motions on the real line which move independently up to their meeting and coalesce. The modified Arratia flow was first constructed in [19] (see [22, 18, 24, 23]), as a physical generalization of the system of coalescing Brownian motions, in the case where particles start from integer points with unit masses. Later in [20] the modified Arratia flow for a system of particles which start from all points of the interval [0, 1] with zero mass (the distribution of the mass of particles at the initial time is the Lebesgue measure on [0, 1]) was constructed as a scaling limit. We see that interpreting Z(u, t) as the position of the particle at time t starting from u, the family of processes {Z(u, ·), u ∈ A} is a description of the system of particles which start from almost all points of supp μ with the mass distribution μ.
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