Convergence to equilibrium for density dependent Markov jump processes

2025/05/19 by Barbour, Andrew, Brightwell, Graham, Luczak, Malwina
#60C05 #60F15 #60J75 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2505.12926

Abstract

We investigate the convergence to (quasi--)equilibrium of a density dependent Markov chain in~\mathbb Z^d, whose drift satisfies a system of ordinary differential equations having an attractive fixed point. For a sequence of such processes~\mathbb X^N, indexed by a size parameter~N, the time taken until the distribution of~\mathbb X^N, started in some given state, approaches its equilibrium distribution~π^N typically increases with~N. To first order, it corresponds to the time~t_N at which the solution to the drift equations reaches a distance of~√ N from their fixed point. However, the length of the time interval over which the total variation distance between \mathcal L (\mathbb X^N(t)) and its equilibrium distribution~π^N changes from being close to~1 to being close to zero is asymptotically of smaller order than~t_N. In this sense, the chains exhibit `cut--off', and we are able to prove that the cut-off window is of (optimal) constant size.

Citations

• Cutoff for the logistic SIS epidemic model with self-infection
• Cutoff for non-negatively curved Markov chains
• Long-term concentration of measure and cut-off
• Multivariate approximation in total variation, II: discrete normal approximation
• Multivariate approximation in total variation, I: Equilibrium distributions of Markov jump processes
• Escape from the boundary in Markov population processes
• Total variation approximation for quasi-equilibrium distributions, II

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