Stochastic analysis on a Riemannian manifold is a well developed area of research in probability theory.
We will discuss some recent developments on stochastic analysis on a manifold whose Riemannian metric evolves with time, a typical case of which is the Ricci flow. Familiar results such as stochastic parallel transport, integration by parts formula, martingale representation theorem, and functional inequalities have interesting extensions from
time independent metrics to time dependent ones. In particular, we will discuss an extension of Beckner’s inequality on the path space over a Riemannian manifold with time-dependent metrics. The classical version of this inequality includes the Poincare inequality and the logarithmic Sobolev inequality as special cases.