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–Savaré (2014) [5], we show that the L2L2-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≥2q≥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. ∂tft=Δqft∂tft=Δqft.