The Brownian marble

TL;DR

This paper introduces the $R$-marble, a stochastic process involving coalescing Brownian motions with fragmentation, and explores its behavior, phase transitions, and correlations, especially when the fragmentation rate function $R$ varies near zero.

Contribution

It constructs the $R$-marble for unbounded $R$ as a limit of bounded cases and characterizes its phase transition and correlation structure, introducing the Brownian marble with parameter $$.

Findings

For $  6$, the $R$-marble behaves like the Brownian web.

For $ < 6$, the process has large gaps and is nontrivial.

Explicit correlation structure via the Brownian vein.

Abstract

Let $R:(0,\infty) \to [0,\infty)$ be a measurable function. Consider coalescing Brownian motions started from every point in the subset $\{ (0,x) : x \in \mathbb{R} \}$ of $[0,\infty) \times \mathbb{R}$ (with $[0,\infty)$ denoting time and $\mathbb{R}$ denoting space) and proceeding according to the following rule: the interval $\{t\} \times [L_t,U_t]$ between two consecutive Brownian motions instantaneously fragments' at rate $R(U_t - L_t)$. At a fragmentation event at a time $t$, we initiate new coalescing Brownian motions from each of the points $\{ (t,x) : x \in [L_t,U_t]\}$. The resulting process, which we call the $R$-marble, is easily constructed when $R$ is bounded, and may be considered a random subset of the Brownian web. Under mild conditions, we show that it is possible to construct the $R$-marble when $R$ is unbounded as a limit as $n \to \infty$ of $R_n$-marbles where $R_n(g) = R(g) \wedge n$. The behaviour of this limiting process is mainly determined by the shape of $R$ near zero. The most interesting case occurs when the limit $\lim_{g \downarrow 0} g^2 R(g) = \lambda$ exists in $(0,\infty)$, in which case we find a phase transition. For $\lambda \geq 6$, the limiting object is indistinguishable from the Brownian web, whereas if $\lambda < 6$, then the limiting object is a nontrivial stochastic process with large gaps. When $R(g) = \lambda/g^2$, the $R$-marble is a self-similar stochastic process which we refer to as the \emph{Brownian marble with parameter $\lambda > 0$}. We give an explicit description of the spacetime correlations of the Brownian marble, which can be described in terms of an object we call the Brownian vein; a spatial version of a recurrent extension of a killed Bessel-$3$ process.