The dynamics of probability density functions has been extensively studied in science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics that can be formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. The goal of this paper is to parallel the success of techniques using functional inequalities, for dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences as energy functionals. Such dynamics take the form of a nonlocal differential equation, for which existing analysis critically relies on using the explicit solution formula in special cases. We provide a comprehensive study on functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of the obtained functional inequalities is that they do not depend on the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posed) is uniform across general target distributions, making them potentially desirable dynamics for posterior sampling applications in Bayesian inference.

翻译：概率密度函数的动力学在科学与工程领域已被广泛研究，用以理解物理现象并促进算法设计。特别令人关注的是那些能够在瓦瑟斯坦度量下表述为能量泛函梯度流的动力学。函数不等式（例如对数索博列夫不等式）的发展在分析这些动力学的收敛性中起着关键作用。本文的目标在于，针对在费舍尔-拉奥度量下作为梯度流的动力学，并以其种$f$-散度作为能量泛函，建立与上述基于函数不等式的成功技术相平行的理论框架。此类动力学表现为非局部微分方程的形式，现有分析主要依赖于在特殊情况下使用显式解公式。我们在最小假设下，对费舍尔-拉奥梯度流的函数不等式及相关测地凸性进行了全面研究。所获得的函数不等式的一个显著特征是，它们不依赖于目标分布的对数凹性或对数索博列夫常数。因此，该动力学（假设适定）的收敛速率在一般目标分布上是一致的，这使得它们可能成为贝叶斯推断中后验采样应用的理想动力学。