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

2025/07/10 by Fernando Cordero, Cordero, Fernando, Grégoire Véchambre +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Gene Regulatory Network Analysis #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2507.07587

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 Λ-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 Λ-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.

Citations

• Bernstein duality revisited: frequency-dependent selection, coordinated mutation and opposing environments
• The comonotone flow of a stochastically monotone Feller process on the real line
• Bernoulli factories and duality in Wright-Fisher and Allen-Cahn models of population genetics
• Local explosions and extinction in continuous-state branching processes with logistic competition
• The effective strength of selection in random environment

Related

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• Stationarity and ergodic properties for some observation-driven models in random environments
• Markov Renewal Theory for Transfer Operators and Point Processes on the Line