Abstract
We consider finite-dimensional, time-continuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λ-convexity of the entropy and treat several examples including some discretizations of one-dimensional Fokker–Planck equations.
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Mielke, A. Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. 48, 1–31 (2013). https://doi.org/10.1007/s00526-012-0538-8
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DOI: https://doi.org/10.1007/s00526-012-0538-8
Keywords
- Time-continuous
- Markov chain
- Detailed balance
- Relative entropy
- Onsager matrix
- Logarithmic mean
- Geodesic convexity