Characterisation of gradient flows on finite state Markov chains. (English) Zbl 1327.60146

Dietert, Helge

Electron. Commun. Probab. 20, Paper No. 29, 8 p. (2015).

Summary: In his work [J. Funct. Anal. 261, No. 8, 2250–2292 (2011; Zbl 1237.60058)], J. Maas has shown that the law of any time-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 a 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.

MSC:

60J27

Continuous-time Markov processes on discrete state spaces

Keywords:

gradient flows; finite state Markov chains; time-reversibility

Citations:

Zbl 1237.60058