Electronic Communications in Probability

Characterisation of gradient flows on finite state Markov chains

Helge Dietert

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In his 2011 work, Maas has shown that the law of anytime-reversible continuous-time Markov chain with finite state space evolves like a gradient flow of the relative entropy with respect to its stationary distribution. In this work we show the converse to the above by showing that if the relative law of a Markov chain with finite state space evolves like a gradient flow of the relative entropy functional, it must be time-reversible. When we allow general functionals in place of the relative entropy, we show that the law of a Markov chain evolves as gradient flow if and only if the generator of the Markov chain is real diagonalisable. Finally, we discuss what aspects of the functional are uniquely determined by the Markov chain.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 29, 8 pp.

Accepted: 29 March 2015
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: MSC 60J27: Continuous-time Markov processes on discrete state spaces

Gradient flows Finite state Markov chains Time-reversibility

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Dietert, Helge. Characterisation of gradient flows on finite state Markov chains. Electron. Commun. Probab. 20 (2015), paper no. 29, 8 pp. doi:10.1214/ECP.v20-3521. https://projecteuclid.org/euclid.ecp/1465320956

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