Some quantitative results about bridges and Entropic interpolations of probability measures
In the first part of the talk I will discuss quantitative results about the bridges of the Langevin dynamics and the associated reciprocal processes.They include an equivalence between gradient estimates for bridge semigroups and couplings, comparison principles, and a Logarithmic Sobolev inequality. All results are based on a seemingly new expression of the drift of a bridge in terms of the reciprocal characteristic, which, roughly speaking, is the ``acceleration" of a bridge. The analogy between reciprocal characteristic and acceleration also allows to view our results as a probabilistic version of well known facts about ODEs.
Bridges can be used to construct interpolations of probability measures. The resulting curve, called entropic interpolation, is a stochastic counterpart to the displacement interpolation. If time allows, I will discuss connections with gradient flows on Wasserstein space and an estimate of the marginal entropy along entropic interpolations which in the small noise limit gives back the well known convexity of the entropy along displacement interpolations.
The talk is partially based on joint work with Max von Renesse.