Convergence of Imprecise Continuous-Time Markov Chains

TL;DR

This paper investigates the convergence behavior of a class of non-linear differential equations related to imprecise continuous-time Markov chains, establishing conditions for their ergodicity and convergence to a unique limit.

Contribution

It provides necessary and sufficient conditions for the ergodicity and convergence of imprecise continuous-time Markov chains modeled by non-linear differential equations.

Findings

Established conditions for ergodicity of the non-linear differential equation.

Proved convergence criteria for the set of non-homogeneous Markov chains.

Linked the solution of the differential equation to the transition operator of imprecise Markov chains.

Abstract

We study the limit behaviour of a generally non-linear ordinary differential equation whose solution is a superadditive generalisation of a stochastic matrix, and provide necessary and sufficient conditions for this solution to be ergodic, in the sense that it converges to an operator that, essentially, maps functions to constants. In the linear case, the solution of our differential equation is equal to the matrix exponential of an intensity matrix and can then be interpreted as the transition operator of a homogeneous continuous-time Markov chain. Similarly, in the generalised non-linear case that we consider, the solution can be interpreted as the lower transition operator of a specific set of non-homogeneous continuous-time Markov chains, called an imprecise continuous-time Markov chain. In this context, our main result provides a necessary and sufficient condition for such an imprecise continuous-time Markov chain to converge to a unique limiting distribution.