Distance to stationarity and open set recurrence for ergodic processes on the unit interval with a Bernstein or a Siegmund dual

TL;DR

This paper employs duality techniques, specifically Siegmund and Bernstein duality, to analyze ergodic and recurrence properties of Markov processes on the unit interval, providing new bounds and criteria for stationarity and recurrence.

Contribution

It introduces a novel approach using duality to derive bounds on stationarity distance and establish recurrence, extending analysis to complex population models beyond classical methods.

Findings

Derived sharp bounds on the distance to stationarity.

Established criteria for open set recurrence.

Extended methods to a broad class of population models.

Abstract

We use duality techniques - specifically Siegmund and Bernstein duality - as tools to analyse ergodic and recurrence properties of $[0,1]$-valued Markov processes. These dualities enable the derivation of sharp bounds on the distance to stationarity and allow for a novel approach to establishing topological recurrence, whereby ergodicity implies recurrent visits to neighbourhoods of points in the stationary distribution's support. We first relate Siegmund and Bernstein duality in cases where both apply, such as $\Lambda$-Wright--Fisher processes. We then exploit the dual processes to derive bounds on the distance to stationarity and simple criteria for open set recurrence. Our results apply to a broad class of $\Lambda$-Wright--Fisher processes with frequency-dependent selection, mutation, and random environmental effects. In many cases, Bernstein duals can be constructed from the same ancestral structures underlying moment duals, allowing our methods to extend and strengthen existing results based on moment duality. This framework provides a robust approach to studying the long-term behaviour of complex population models beyond the classical moment dual setting.