Abstract
Deterministic limit of a class of continuous time Markov chains is considered based purely on differential equation techniques. Starting from the linear system of master equations, ordinary differential equations for the moments and a partial differential equation, called Fokker–Planck equation, for the distribution is derived. Introducing closures at the level of the second and third moments, mean-field approximations are introduced. The accuracy of the mean-field approximations and the Fokker–Planck equation is investigated by using two differential equation-based and an operator semigroup-based approach.
Highlights
Large systems consisting of many identical particles are usually described by stochastic processes, while their deterministic limit can be modeled by differential equations
It will be assumed that the transition rate depends on the proportion of nodes in the different states, the state of the whole system can be given by the number of particles of different types
It is assumed that each transition rate depends on the proportion of nodes in the different states, the state of the whole system can be given by the number of particles of different types
Summary
Large systems consisting of many identical particles are usually described by stochastic processes, while their deterministic limit can be modeled by differential equations. The number of particles in state Q and T at time t is denoted by XQ(t) and XT(t), respectively These are considered to be random variables and our main goal is to derive differential equations yielding approximations for the expected value of these variables. The aim of the paper is to review and study the differential equation-based methods that has been developed to estimate the accuracy of mean-field ODE and PDE approximations. Such and similar questions were studied in the density dependent case by several authors, see [7, 10, 14].
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