In this talk I will review the 1998 Jordan-Kinderlehrer-Otto interpretation of Fokker-Planck equations as a Wasserstein gradient flow in the space of probability measures, based on optimal transport theory. I will also discuss the corresponding derivation of microscopic-to-macroscopic dissipation, in the language of large deviation and short time limits. This provides a somehow canonical recipee to study vatiational structures for diffusion processes.

If time permits I will also discuss a counter-example to this general idea, based on the so-called (reflected) Sticky Brownian Motion SBM.

Based on joint works with JB Casteras, L. Nenna and F. Santambrogio.