On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval

TL;DR

This paper investigates the weak convergence rate of the fractional step method for the Arratia flow, using Wasserstein distance and finite-dimensional densities to analyze discretization effects on point measures.

Contribution

It introduces explicit finite-dimensional densities for collision sequences and analyzes convergence of discretized approximations of the Arratia flow's point measure.

Findings

Established convergence rate in Wasserstein distance.

Derived explicit formulas for collision densities.

Discussed discretization effects on point measure approximation.

Abstract

The rate of the weak convergence in the fractional step method for the Arratia flow is established in terms of the Wasserstein distance between the images of the Lebesque measure under the action of the flow. We introduce finite-dimensional densities describing sequences of collisions in the Arratia flow and derive an explicit expression for them. With the initial interval discretized, the convergence of the corresponding approximations of the point measure associated with the Arratia flow is discussed in terms of such densities.