Infinite Dyson Brownian Motion as a Gradient Flow

The Dyson Brownian motion (DMB) is a system of interacting Brownian motions with logarithmic interaction potential, which was introduced by Freeman Dyson '62 in relation to the random matrix theory. In this talk, we discuss the case where the number of particles is infinite and show that the DBM induces a diffusion structure on the configuration space having the Bakry-Émery lower Ricci curvature bound. As an application, we show that the DBM can be realised as the unique Benamou-Brenier-type gradient flow of the Boltzmann-Shannon entropy associated with the sine_beta point process.