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# Optimal transport, duality, and functional inequalities for Markov semigroups

### Mathematical Physics and Probability Seminar

Optimal transport, duality, and functional inequalities for Markov semigroups
Series: Mathematical Physics and Probability Seminar
Location: online
Presenter: Nate Eldredge, University of Northern Colorado

The transition semigroup $P_t$ of a continuous-time Markov process on a metric space $X$ acts as an operator on the space $B(X)$ of bounded measurable function on $X$, as well as on the space $\mathcal{P}(X)$ of probability measures on $X$.  Certain smoothing'' properties of $P_t$ can be expressed in terms of contractive inequalities on $\mathcal{P}(X)$ involving various distances between probability measures, such as the Kantorovich--Wasserstein optimal transport distance.  We study a duality relationship between such inequalities and reverse'' functional inequalities for $P_t$ acting on functions, such as the reverse Poincar\'e and reverse log Sobolev inequalities.  I will discuss some applications including results about rates of convergence to equilibrium and smoothness of transition densities.

This is joint work with Fabrice Baudoin (University of Connecticut).  Our preprint is at https://arxiv.org/abs/2004.02050.