Coalescing Brownian flows: A new approach

Abstract

The coalescing Brownian flow on R is a process which was introduced by Arratia , the topology and statespace required a moment of order 3 − ε for this convergence to hold. The proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work-in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the noncrossing property.

The Brownian Web, the Brownian Net, and their Universality

“…To characterize a probability measure on H, it turns out to be sufficient to determine the probability of the joint crossing of any finite collection of tubes chosen from a deterministic countable dense subset of T . To prove the weak convergence of a sequence of probability measures on H, it is sufficient to find a large enough set of tubesT ⊂ T such that the the probability of the joint crossing of any finite subset ofT converges to a limit, see [BGS15,Prop. 2.12].…”

“…the tube topology. For coalescing Brownian flow on the Sierpinski gasket, this was carried out in [BGS15]. Can one construct the Lévy web in the paths topology, and what type of special points may arise in the spirit of Theorem 2.11?…”

A construction of the Stable Web

“…And based on this, our main motivation here is to build a stable version of the Brownian Web or simply a "Stable Web" and also prove an invariance principle for it. We point out that an alternative weak topology was introduced in Berestycki et al (2015) to deal with the convergence of other systems of random coalescing paths that do not have the non-crossing property.…”

Stationary points in coalescing stochastic flows on R

“…Main distinction of a theorem 1.1 modification is that it combines perfect cocycle property of ϕ with the measurability of the group of shifts θ. It must be noted that a number of various modifications of the Arratia flow that do not deal with the group of shifts of underlying probability space appeared in [8,9,10,11,12].…”