Bi-infinite systems of singularly interacting Brownian particles and the KPZ equation

Abstract

We study a bi-infinite system of interacting Brownian particles on the real line with singular asymmetric interactions mediated by the collision local times. Particles perform Brownian motions, and when neighboring particles collide, the associated local time is split in proportions $p$ and $q=1-p$. We first develop well-posedness theory for the particle system, proving pathwise uniqueness and strong existence under natural growth assumptions on the initial configuration and local times. We also identify a family of stationary distributions for the infinite-dimensional process of gaps between successive particles: for every $λ>0$, the product measure with i.i.d. Exp$(λ)$ gaps is invariant. Our main result concerns the equilibrium fluctuations of the associated particle-count (height) function in a weakly asymmetric regime. Taking $p=p_\varepsilon$ so that $p_\varepsilon^{-1}-1=\exp\{σ\varepsilon^{1/4}\}$, initializing the interparticle gaps with i.i.d. Exp$(1)$ random variables, and applying a microscopic Hopf-Cole transform to the diffusively rescaled count function, we prove convergence, as $\varepsilon \rightarrow 0$, to the multiplicative stochastic heat equation (SHE) with Brownian exponential initial data. Equivalently, the logarithm of the limit is the Hopf-Cole solution of the KPZ equation with two-sided Brownian initial data. The proof combines localization to finite particle subsystems via chains of collisions, Brownian last-passage percolation estimates, a key local time cancellation property, and a martingale problem for a scale-adapted mollification of the Hopf-Cole field, whose space-time regularity is tuned to match that of the limiting SHE. The resulting fluctuation theorem places these Brownian particle systems with asymmetric singular collision dynamics within the KPZ universality class.