Abstract
Coupling techniques provide a powerful method for analyzing the long-time behavior of Markov processes and associated PDEs. The aim of this talk is to show how it is possible to construct and analyze different couplings between optimal trajectories, within the framework of stochastic control, which allow us to obtain quantitative estimates on the optimal processes and the corresponding Hamilton-Jacobi-Bellman equations. In particular, we consider two examples. In the first, we will see how a controlled version of reflection coupling allows us to establish exponential turnpike estimates within the framework of McKean-Vlasov stochastic control. Then, we'll consider the problem of obtaining quantitative bounds on the Hessian of Schrödinger potentials in the context of entropic optimal transport, and discuss their implications in terms of functional inequalities and measurement concentration for optimal couplings.
