Coupling methods provide a powerful toolbox for the

quantitative analysis of the long-time behaviour of Markov

processes. In particular, coupling by reflection allows to

establish sharp exponential convergence results in

Wasserstein distance for the Fokker-Planck equation without

having to rely on pointwise assumptions on the confinement

potential. The purpose of this talk is to illustrate the

construction of a variant of coupling by reflection that applies

to optimally controlled diffusion processes, including

controlled McKean-Vlasov processes. Such construction

opens the door for a precise study of the long-time behaviour

of optimizers: in particular it provides with uniform in time

gradient (and Hessian) estimates for the solution of Hamilton-

Jacobi-Bellman equations that enable to prove various kind of

exponential turnpike properties for the optimal processes

and controls. This talk is partially based on joint work with

Katharina Eichinger, Alain Durmus, and Alekos Cecchin.