What is it about?
For biochemical modelling it is important to be able to compute the long-term stationary state of continuous-time Markov chains. We tried to compute the stationary distribution on a state space that was put together from two smaller ones. Unfortunately, it's not simple and one needs more info than the stationary distributions of the two parts. However, we derived nice, intuitive inequalities.
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Why is it important?
Mélykúti, Hespanha, Khammash (Royal Society Interface, 2014) initiated a programme to catalogue all chemical reaction systems where it is possible to compute the long-term stationary (equilibrium) distribution. The current paper proposes a way to compute stationary distributions via a recursion on the state space. The paper is a nice theoretical study based on a regenerative structure and the ergodic theorem. We arrived at a nice special case and nice estimates in the general case, the general case itself turned out to be demanding and our result is not easily applicable.
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This page is a summary of: The Stationary Distribution of a Markov Jump Process Glued Together from Two State Spaces at Two Vertices, Stochastic Models, July 2015, Taylor & Francis,
DOI: 10.1080/15326349.2015.1055769.
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Resources
Excursions of Markov processes: An approach via Markov additive processes
Prof Haya Kaspi (Technion) told me that our proof technique reminded her of her work around 1983 on excursions in much more general settings. I think she meant Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 64, Issue 2 , pp 251-268 (DOI:10.1007/BF01844609) and possibly following papers.
Free version on arXiv
The paper is also available for free at this address.
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