KPZ equation limit of sticky Brownian motion

Abstract

We consider the motion of a particle under a continuum random environment whose distribution is given by the Howitt-Warren flow. In the moderate deviation regime, we establish that the quenched density of the motion of the particle (after appropriate centering and scaling) converges weakly to the $(1+1)$ dimensional stochastic heat equation driven by multiplicative space-time white noise. Our result confirms physics predictions and computations in [LDT17, BLD20] and is the first rigorous instance of such weak convergence in the moderate deviation regime. Our proof relies on a certain Girsanov transform and works for all Howitt-Warren flows with finite and nonzero characteristic measures. Our results capture universality in the sense that the limiting distribution depends on the flow only via the total mass of the characteristic measure. As a corollary of our results, we prove that the fluctuations of the maximum of an $N$-point sticky Brownian motion are given by the KPZ equation plus an independent Gumbel on timescales of order $(\log N)^2.$