Interacting diffusions and trees of excursions: convergence and comparison

TL;DR

This paper studies systems of interacting diffusions with local regulation, establishing bounds and convergence results, and providing conditions for extinction and duality with McKean-Vlasov equations.

Contribution

It introduces a comparison framework linking total mass processes to trees of excursions and proves convergence of finite island systems to these trees.

Findings

Total mass process is bounded by a tree of excursions.

Explicit conditions for extinction of total mass.

Convergence of finite island systems to a tree of excursions.

Abstract

We consider systems of interacting diffusions with local population regulation. Our main result shows that the total mass process of such a system is bounded above by the total mass process of a tree of excursions with appropriate drift and diffusion coefficients. As a corollary, this entails a sufficient, explicit condition for extinction of the total mass as time tends to infinity. On the way to our comparison result, we establish that systems of interacting diffusions with uniform migration between finitely many islands converge to a tree of excursions as the number of islands tends to infinity. In the special case of logistic branching, this leads to a duality between the tree of excursions and the solution of a McKean-Vlasov equation.