Gradient flows of the entropy for finite Markov chains

Maas J. 2011. Gradient flows of the entropy for finite Markov chains. Journal of Functional Analysis. 261(8), 2250–2292.


Journal Article | Published
Abstract
Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wasserstein gradient flow interpretation of the heat flow in Rn by Jordan, Kinderlehrer and Otto (1998). The metric W is similar to, but different from, the L2-Wasserstein metric, and is defined via a discrete variant of the Benamou–Brenier formula.
Publishing Year
Date Published
2011-03-04
Journal Title
Journal of Functional Analysis
Acknowledgement
Supported by Rubicon subsidy 680-50-0901 of the Netherlands Organisation for Scientific Research (NWO)
Volume
261
Issue
8
Page
2250 - 2292
IST-REx-ID

Cite this

Maas J. Gradient flows of the entropy for finite Markov chains. Journal of Functional Analysis. 2011;261(8):2250-2292. doi:10.1016/j.jfa.2011.06.009
Maas, J. (2011). Gradient flows of the entropy for finite Markov chains. Journal of Functional Analysis. Academic Press. https://doi.org/10.1016/j.jfa.2011.06.009
Maas, Jan. “Gradient Flows of the Entropy for Finite Markov Chains.” Journal of Functional Analysis. Academic Press, 2011. https://doi.org/10.1016/j.jfa.2011.06.009 .
J. Maas, “Gradient flows of the entropy for finite Markov chains,” Journal of Functional Analysis, vol. 261, no. 8. Academic Press, pp. 2250–2292, 2011.
Maas J. 2011. Gradient flows of the entropy for finite Markov chains. Journal of Functional Analysis. 261(8), 2250–2292.
Maas, Jan. “Gradient Flows of the Entropy for Finite Markov Chains.” Journal of Functional Analysis, vol. 261, no. 8, Academic Press, 2011, pp. 2250–92, doi:10.1016/j.jfa.2011.06.009 .
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