p-Wasserstein空间中的q热流和Renyi熵梯度流
q-heat flow and the gradient flow of the Renyi entropy in the p-Wasserstein space
摘要：
基于Ambrosio-Gigli-Savaré最近在Invent上发表的一篇论文的想法。（2013）中，我们证明了称为$q$ -热流的$q$ -Cheeger能量流解决了$p$ -Wasserstein中Renyi熵泛函的梯度流问题。为此，进一步研究的$q$热流，包括其质量保持的条件。在对上梯度的凸性假设下，这对所有$q\ge2$都成立，人们得到了梯度流的唯一性，并且可以识别两个流。$q$ -热流的光滑解是抛物型q-拉普拉斯方程的解，即$\partial_{t}f_{t}=Δ_{q}f_{t}.$
英文摘要：
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 $q\ge2$, 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. $\partial_{t}f_{t}=Δ_{q}f_{t}.$
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