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.

Perspectives

What would be very interesting to solve is what happens to the equilibria of two chemical systems, one with k and one with l species, some shared, if we mix them to make a reaction system with m<k+l species. Is there any chance that the equilibria of the two original systems (on the sets of natural numbers to the powers of k and l) tell anything about the equilibrium of the combined system on the m-dimensional space?

Dr Bence Mélykúti
Albert-Ludwigs-Universitat Freiburg

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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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