Applications of optimal transport to Dyson Brownian Motions and beyond

Abstract: We develop a new method based on Caffarelli's contraction theorem in optimal transport to obtain sharp and uniform modulus of continuity estimates for $\beta$-Dyson Brownian motions with $\beta \geq 2$. Our method extends to a large class of random curve collections, which can be viewed as log-concave perturbations of Brownian motions, including the $\beta$-Dyson Brownian motion, the Air$\text{y}_{\beta}$ line ensemble, the KPZ line ensemble, and the O'Connell-Yor line ensemble.