A localized coupling approach to interacting continuous-state branching processes

TL;DR

This paper introduces a new class of continuous-state branching processes with multiple interactions, constructed via stochastic differential equations, and establishes conditions for their ergodicity using a novel localized coupling method.

Contribution

It presents a new model combining branching, immigration, predation, and competition, and develops a localized coupling approach for proving ergodicity.

Findings

Established sharp conditions for uniform ergodicity in total variation.

Constructed the model as a unique strong solution to SDEs with jumps.

Developed a novel localized Markovian coupling technique.

Abstract

We introduce a class of continuous-state branching processes with immigration, predation and competition, which can be viewed as a combination of the classical Lotka-Volterra model and continuous-state branching processes with competition that were introduced by Berestycki, Fittipaldi, and Fontbona (Probab. Theory Relat. Fields, 2018). This model can be constructed as a unique strong solution to a class of two-dimensional stochastic differential equations with jumps. We establish sharp conditions for the uniform ergodicity in the total variation of this model. Our proof relies on a novel, localized Markovian coupling approach, which is of its own interest in the ergodicity theory of Markov processes with interactions.