q-Heat flow and the gradient flow of the Renyi entropy in the p-Wasserstein space

Abstract

Based on the idea of a recent paper by Ambrosio-Gigli-Savaré in Invent. Math. (2013), we show that flow of the q-Cheeger energy, called q-heat flow, solves the gradient flow problem of the Renyi entropy functional in the p-Wasserstein. For that, a further study of the q-heat flow is presented including a condition for its mass preservation. Under a convexity assumption on the upper gradient, which holds for all q ≥ 2, one gets uniqueness of the gradient flow and the two flows can be identified. Smooth solution of the q-heat flow are solution the parabolic q-Laplace equation, i.e. ∂tft = ∆qft.

$q$-Moment Estimates for the Singular $p$-Laplace Equation and Applications

“…Our result only requires finite q-moment instead of compact support for initial data, but for any weak solution constructed through mollification. As for the connection to the flow in the q-Wasserstein space, the work of Kell [12] is the closest to the present one. There, the author shows that a smooth solution of Cauchy problem (1.1) solves a generalized gradient flow problem of Rényi entropy functional in the q-Wasserstein space based on gradient flow and functional analysis methods.…”

“…Based on these ideas, it seems natural to regularize empirical measures with p-Laplacian semigroup. And further, to connect the Cauchy problem (1.1) to flows in the q-Wasserstein space, as done in Kell [12]. A key point in this connection concerns the behavior of qmoment of weak solutions u(x, t) of Cauchy problem (1.1), namely,…”

Entropy Associated with Information Storage and Its Retrieval

“…Figure 6 shows that the production of entropy is accounted for by the entropy reduction for the first coupling. The first coupling approaches its asymptotic value of relative entropy quicker than that of other fields, showing the high gradation flow [44] of its entropy.…”

The normal contraction property for non-bilinear Dirichlet forms

“…We refer to [2,4,5] for this theory, which originates from [16,35]. See also [28,30] for more estimates and contraction properties of Cheeger's energies.…”