# Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes

## TL;DR

This paper develops Lyapunov criteria to analyze the uniform convergence of conditional distributions in absorbed Markov processes, including Lotka-Volterra and Feller diffusions, advancing understanding of their quasi-stationary behavior.

## Contribution

It introduces novel non-linear Lyapunov criteria involving two functions, applicable to a broad class of Markov processes with absorption.

## Key findings

- Criteria apply to Lotka-Volterra birth and death processes
- Criteria extend to Feller diffusions with interactions
- Results establish conditions for uniform convergence of conditional distributions

## Abstract

We study the quasi-stationary behavior of multidimensional processes absorbed when one of the coordinates vanishes. Our results cover competitive or weakly cooperative Lotka-Volterra birth and death processes and Feller diffusions with competitive Lotka-Volterra interaction. To this aim, we develop original non-linear Lyapunov criteria involving two Lyapunov functions, which apply to general Markov processes.