Weak convergence of the localized disturbance flow to the coalescing Brownian flow

TL;DR

This paper introduces new mathematical frameworks for the coalescing Brownian flow on the circle, proving its weak convergence from localized disturbances and establishing properties like time-reversibility and coalescence times.

Contribution

It defines complete metric spaces for the coalescing Brownian flow, proves weak convergence from localized disturbances, and provides new proofs of key properties such as time-reversibility.

Findings

Weak convergence of localized disturbance flows to the coalescing Brownian flow.

New metric space structures for analyzing the flow.

Explicit calculation of coalescence time Laplace transform.

Abstract

We define a new state-space for the coalescing Brownian flow, also known as the Brownian web, on the circle. The elements of this space are families of order-preserving maps of the circle, depending continuously on two time parameters and having a certain weak flow property. The space is equipped with a complete separable metric. A larger state-space, allowing jumps in time, is also introduced, and equipped with a Skorokhod-type metric, also complete and separable. We prove that the coalescing Brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps, each jump arising from a small localized disturbance of the circle. A local version of this result is also obtained, in which the weak limit law is that of the coalescing Brownian flow on the line. Our set-up is well adapted to time-reversal and our weak limit result provides a new proof of time-reversibility of the coalescing Brownian flow. We also identify a martingale associated with the coalescing Brownian flow on the circle and use this to make a direct calculation of the Laplace transform of the time to complete coalescence.