At the end of the 1990s it was discovered by Jordan/Kinderlehrer/Otto that the diffusion equation is a gradient flow in the space of probability measures. The driving functional is the Boltzmann-Shannon entropy, and the dissipation mechanism is given by the 2-Wasserstein metric from optimal transport. This result has been the starting point for striking developments at the interface of analysis, probability, and metric geometry.
In this survey talk I will present dynamical optimal transport metrics that yield gradient flow structures for discrete stochastic dynamics and dissipative quantum systems. This naturally leads to discrete and non-commutative notions of Ricci curvature, entropy inequalities, and sharp rates of convergence to equilibrium for (quantum) Markov semigroups.
The talk is based on joint works with Matthias Erbar and with Eric Carlen.