Manifolds evolving under a geometric flow are currently a very active field of research. We develop basic notions of stochastic differential geometry in the framework of evolving manifolds. This includes in particular the notion of a canonical Brownian motion in this setting.

We review recent work on characterizing Ricci curvature bounds in terms of functional inequalities for heat semigroups. Our discussion includes extensions of these methods to geometric flows on manifolds, as well as to the path space of Riemannian manifolds evolving under a geometric flow.

Of particular interest in the theory are entropy formulas for positive solutions of the heat equation under a geometric flow. We discuss how tools from Stochastic Analysis can be used to study the evolution of entropies, analogous to Perelman’s entropy functionals.

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