Geometric fluid approximation for general continuous-time Markov chains

Michalis Michaelides, Jane Hillston, Guido Sanguinetti

Research output: Contribution to journalArticlepeer-review


Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous statespace endowed with a dynamics for the approximating process.

We construct here a general method based on spectral analysis of the transition matrix of the CTMC, without the need for a population structure. Specifically, we use the popular manifold learning method of diffusion maps to analyse the transition matrix as the operator of a hidden continuous process. An embedding of states in a continuous space is recovered, and the space is endowed with a drift vector field inferred via Gaussian process regression. In this manner, we construct an ODE whose solution approximates the evolution of the CTMC mean, mapped onto the continuous space (known as the fluid limit).
Original languageEnglish
Article number20190100
Number of pages25
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2229
Early online date25 Sep 2019
Publication statusPublished - 27 Sep 2019


  • Continuous-time Markov chains
  • Markov jump processes
  • Fluid approximation
  • Diffusion maps
  • Gaussian processes

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