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-Savar\'e 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 $q\ge2$, 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. $\partial_{t}f_{t}=\Delta_{q}f_{t}.$