Gradient flows of the entropy for finite Markov chains

By Jan Maas. 2011

346 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/193fb5ffb7a16e0b5a996b9315e3c43ecaf99b34

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

Entropic Approximation of Wasserstein Gradient Flows (2015, background): “, graphs) have been recently proposed [29, 51, 54] and could lead to structure preserving discretization schemes.”
On quotients of spaces with Ricci curvature bounded below (2017, methodology): “Definition A.6 (Maas-Mielke Ricci curvature).”
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties (2018, neutral): “(P2(X), dW ), where X is an RCD(K,∞) metric-measure space and φ is the relative entropy functional [88, 69], and there is an intensive research to construct EVI gradient flows in ad-hoc geometric settings for reaction-diffusion equations [54, 25, 57, 55] and systems [65, 48, 56], nonlinear viscoelas”
Curvature and transport inequalities for Markov chains in discrete spaces (2015, background): “Erbar and Maas recovered some of those functional inequality results with the notion of entropic Ricci curvature on graphs, and we refer the reader to [14,32] for more details.”
Displacement convexity of entropy and related inequalities on graphs (2012, background): “[12] M. Erbar and J. Maas.”
Gradient flow structures for discrete porous medium equations (2012, methodology): “Basic properties of W have been studied in [11].”
An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker–Planck Equation is Gradient Flow for the Entropy (2012, background): “A similar gradient flow structure in the space of probability measures has meanwhile been developed and exploited in many different settings [1, 2, 6, 9, 10, 12, 14, 15, 20, 22, 23, 25, 26].”
Thermodynamic Unification of Optimal Transport: Thermodynamic Uncertainty Relation, Minimum Dissipation, and Thermodynamic Speed Limits (2022, background): “For this reason, many studies have generalized the Wasserstein distance based on the geometric aspect of the Benamou–Brenier formula for discrete cases [95].”
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy (2011, methodology): “This possibility has already been indicated in [28].”
Gradient flows of the entropy for jump processes (2012, background): “Very recently, Maas [22] (see also for independent related work by Mielke [24] and Chow et al.”
Jump processes as generalized gradient flows (2020, background): “It is interesting to observe that the concavity and symmetry conditions, that one has to naturally assume to ensure the aforementioned properties of R, were singled out for the analog of the function α in the construction of the distance yielding the quadratic gradient structure of [Maa11].”
Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance (2016, background): “Related work in the commutative setting can be found in [22, 23, 27, 44, 46, 48].”