  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.
