Aggregation of Markov flows I: theory.

A Markov flow is a stationary measure, with the associated flows and mean first passage times, for a continuous-time regular jump homogeneous semi-Markov process on a discrete state space. Nodes in the state space can be eliminated to produce a smaller Markov flow which is a factor of the original one. Some improvements to the elimination methods of Wales are given. The main contribution of the paper is to present an alternative, namely a method to aggregate groups of nodes to produce a factor. The method can be iterated to make hierarchical aggregation schemes. The potential benefits are efficient computation, including recomputation to take into account local changes, and insights into the macroscopic behaviour.This article is part of the theme issue 'Hilbert's sixth problem'.