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

Based on the idea of a recent paper by Ambrosio-Gigli-Savare (2014) [5], we show that the L-2-gradient flow of the q-Cheeger energy, called q-heat flow, solves a generalized 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 solutions of the q-heat flow are solutions to the parabolic q-Laplace equation, i.e. partial derivative(t) f(t) = Delta(q)f(t). (C) 2016 Elsevier Inc. All rights reserved.