On asymptotic behavior of iterates of piecewise constant monotone maps

TL;DR

This paper investigates the convergence rate of iterates of iid random piecewise constant monotone maps to the coalescing Brownian motions' transport map, establishing a power law rate and exponential stability of the fixed point.

Contribution

It provides the first analysis of the convergence rate and stability properties of these maps in the context of coalescing Brownian motions.

Findings

Convergence rate follows a power law.

The fixed point is exponentially stable.

Provides insights into the dynamics of monotone maps.

Abstract

In this paper we study the rate of convergence of the iterates of \iid random piecewise constant monotone maps to the time-$1$ transport map for the process of coalescing Brownian motions. We prove that the rate of convergence is given by a power law. The time-1 map for the coalescing Brownian motions can be viewed as a fixed point for a natural renormalization transformation acting in the space of probability laws for random piecewise constant monotone maps. Our result implies that this fixed point is exponentially stable.