Two fundamental properties of the heat flow are the preservation of log-concavity and the creation of log-semiconvexity through the Li–Yau–Hamilton inequality. These results have numerous consequences, both for the regularity of solutions and for their qualitative and quantitative long-time behavior. In view of their importance, it is natural to seek extensions of these estimates to more general diffusion equations.

In this talk, I will present two results in this direction. First, I will show how, on the torus, these inequalities can be extended to equations of granular-media type, combining diffusion, a Fokker–Planck drift, and a nonlocal interaction drift. Second, I will discuss ongoing work on the propagation of certain moduli of convexity along generalized heat flows, using stochastic control and variational arguments.

If time permits, I will explain how such estimates can also be established at the level of the JKO scheme, an optimal transport-based time discretization of diffusion equations, and how they can be used to improve known convergence results for the scheme.