Abstract
We construct a non-linear Markov process connected with a biological model of a bacterial genome recombination. The description of invariant measures of this process gives us the solution of one problem in elementary probability theory.
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Notes
Indeed, for any finite continuous-time Markov chain with invariant measure \(\mu \) and a initial measure \(\mu ^0\), with \({{\mathrm{supp}}}\mu ^0 \subset {{\mathrm{supp}}}\mu \), and for a strictly convex function \(G\) from Jensen’s inequality it follows that the time-dependent value \(\sum {G\left( \frac{\mu ^t(x)}{\mu (x)}\right) \mu (x)}\) is strictly decreasing unless \(\mu ^0\) being the invariant measure. Moreover, this value has the negative time-derivative. Applying the above argument to \(G(z)=z\log z\) we get that the time-derivative of the Kullback–Leibler divergence is negative.
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Acknowledgments
The authors are grateful to M. S. Gelfand and A. S. Kalinina for fruitful discussions. We also thanks G. A. Kabatiansky for useful advice and F. V. Petrov for the idea of the new proof of Theorem 3. The research of A.V. Akopyan is supported by the Dynasty foundation, the President’s of Russian Federation Grant MK-3138.2014.1 and funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n\(^\circ \)[291734]. The research of S. A. Pyrogov is supported by the RFBR Grants 13-01-12410, 13-07-00224, and 14-07-00035. The research of A. N. Rybko is supported by the RFBR Grants 13-01-12410, 14-01-00319, and 14-01-00379.
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Akopyan, A.V., Pirogov, S.A. & Rybko, A.N. Invariant Measures of Genetic Recombination Processes. J Stat Phys 160, 163–167 (2015). https://doi.org/10.1007/s10955-015-1238-5
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DOI: https://doi.org/10.1007/s10955-015-1238-5