Abstract
The displacement λ-convexity of a non-standard entropy with respect to a non-local transportation metric in finite state spaces is shown using a gradient flow approach. The constant λ is computed explicitly in terms ofa prioriestimates of the solution to a finite-difference approximation of a non-linear Fokker–Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation.
Highlights
Displacement convexity, which was introduced by McCann [20], describes the geodesic convexity of functionals on the space of probability measures endowed with a transportation metric
As one step in this direction, we show in this paper that a certain entropy functional, related to the finite-difference approximation of non-linear Fokker–Planck equations, is displacement convex
Maas [18] and Mielke [21] introduced non-local transportation distances on probability spaces such that continuous-time Markov chains can be formulated as gradient flows of the entropy, and they explored geodesic convexity properties of the functionals
Summary
Displacement convexity, which was introduced by McCann [20], describes the geodesic convexity of functionals on the space of probability measures endowed with a transportation metric. As one step in this direction, we show in this paper that a certain entropy functional, related to the finite-difference approximation of non-linear Fokker–Planck equations, is displacement convex. Before making this statement more precise, let us review the state of the art of the literature. Maas [18] and Mielke [21] introduced non-local transportation distances on probability spaces such that continuous-time Markov chains can be formulated as gradient flows of the entropy, and they explored geodesic convexity properties of the functionals. Showed that a discrete one-dimensional porous-medium equation is a gradient flow of the Renyi entropy function f(s) = sα with respect to a suitable non-local transportation metric induced by the mean function: Λα(s, t) α.
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