Abstract: We consider time correlation for KPZ growth in 1+1 dimensions in a neighborhood of a characteristics. We prove convergence of the covariance with droplet, flat and stationary initial profile. In particular, this provides a rigorous proof of the exact formula of the covariance for the stationary case obtained in [SIGMA 12 (2016), 074]. Furthermore, we prove the universality of the first order correction when the two observation times are close and provide a rigorous bound of the error term. This result holds also for random initial profiles which are not necessarily stationary.

Abstract: In this paper we consider the totally asymmetric simple exclusion process, with non-random initial condition having three regions of constant densities of particles. From left to right, the densities of the three regions are increasing. Consequently, there are three characteristics which meet, i.e., two shocks merge. We study the particle fluctuations at this merging point and show that they are given by a product of three (properly scaled) GOE Tracy-Widom distribution functions. We work directly in TASEP without relying on the connection to last passage percolation.

Abstract: We introduce Brownian motions on time-dependent metric measure spaces, proving their existence and uniqueness. We prove contraction estimates for their trajectories assuming that the time-dependent heat flow satisfies transport estimates with respect to every {\$}{\$}L^p{\$}{\$}Lp-Kantorovich distance, {\$}{\$}p{\backslash}in [1,{\backslash}infty ]{\$}{\$}p∈[1,∞]. These transport estimates turn out to characterize super-Ricci flows, introduced by Sturm (J Funct Anal 275(12):3504--3569, 2015.)