Differential equation approximations of stochastic network processes: an operator semigroup approach

TL;DR

This paper rigorously connects stochastic network models to mean-field approximations using operator semigroup theory, proving convergence of solutions and rates, and extending results to more general cases and sign-conditioned transition rates.

Contribution

It introduces an operator semigroup approach to establish convergence of stochastic models to mean-field limits, including new methods for sign-conditioned transition rates.

Findings

Convergence of stochastic models to mean-field equations with rate $\\mathcal{O}(1/N)$.

Extension of convergence results beyond density-dependent cases.

A new approach for Markov chains with sign conditions on transition rates.

Abstract

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size ($N$) and can be written as $\dot u_N=A_N u_N$. Our results rely on the convergence of the transition matrices $A_N$ to an operator $A$. This convergence also implies that the solutions $u_N$ converge to the solution $u$ of $\dot u=Au$. The limiting ODE can be easily used to derive simpler mean-field-type models such that the moments of the stochastic process will converge uniformly to the solution of appropriately chosen mean-field equations. A bi-product of this method is the proof that the rate of convergence is $\mathcal{O}(1/N)$. In addition, it turns out that the proof holds for cases that are slightly more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach for proving convergence to the mean-field limit is proposed. The starting point in this case is the derivation of a countable system of ordinary differential equations for all the moments. This is followed by the proof of a perturbation theorem for this infinite system, which in turn leads to an estimate for the difference between the moments and the corresponding quantities derived from the solution of the mean-field ODE.