High-dimensional limits for reflected Brownian motion in the orthant

TL;DR

This paper analyzes high-dimensional limits of interacting reflected Brownian motions in the orthant, establishing well-posedness and propagation of chaos, and deriving a McKean--Vlasov limit for both homogeneous and heterogeneous cases.

Contribution

It proves propagation of chaos and characterizes the limiting nonlinear reflected Brownian motion for a broad class of boundary interaction coefficients.

Findings

Global well-posedness under the condition a > -1

Propagation of chaos in the homogeneous case

Convergence to a McKean--Vlasov limit for heterogeneous coefficients

Abstract

We study interacting Brownian particles on the half-line whose interaction occurs through boundary local times at the origin. The particle system is given by \[ X_i^n(t)=X^n_{0,i}+W_i^n(t)+L_i^n(t) +\frac{1}{n-1}\sum_{j\ne i}\rho^n_{ij}L_j^n(t), \qquad i\in[n],\ t\ge0, \] where the initial conditions are exchangeable, the driving Brownian motions $W_i^n$ are i.i.d., and $L_i^n$ denotes the boundary local time of $X_i^n$ at zero. For each fixed coefficient array $\{\rho^n_{ij}\}$, the system can be viewed as a semimartingale reflected Brownian motion in the orthant. We first consider the homogeneous case $\rho^n_{ij}=a$. In this case, global well-posedness holds under the completely-$\mathcal S$ condition $a>-1$. We prove propagation of chaos under this condition; the subregime $a\in(-1,0]$, in the homogeneous setting, was previously covered as part of the results of \cite{baker2025particle}. The limiting process is the nonlinear reflected Brownian motion \[ \bar X(t)=\bar X_0+\bar W(t)+\bar L(t)+a\mathbb E[\bar L(t)], \qquad t\ge0. \] We also treat heterogeneous random coefficients $\rho^n_{ij}$, assumed to have mean $a$, support in a compact subset of $(-1,1)$, and to be independent across $j$ for each $i$. In both the quenched and annealed settings, the particle system converges to the same McKean--Vlasov limit as in the homogeneous case. The model is motivated by large Jackson networks in heavy traffic.