Strong existence, pathwise uniqueness and chains of collisions in infinite Brownian particle systems

TL;DR

The paper analyzes an infinite-dimensional stochastic system of competing Brownian particles with local-time-based collisions, governed by $X_j(t)= x_j + g_j t + B_j(t) + p L_{(j-1,j)}(t) - q L_{(j,j+1)}(t)$ with $q=1-p$. It establishes strong existence and pathwise uniqueness for all $p\in[0,1)$ under precise initial-data and local-time-growth conditions, and it distinguishes three regimes: (i) $p<q$ where uniqueness holds broadly, (ii) $p\ge q$ with a mild initial-growth requirement and a local-time growth condition cn:cstar, and (iii) $p>q$ with a stronger condition cn:cstar2. A central technical tool is the finiteness of collision-chains $K^*(i,[u,v])$, which enables decoupling of finite subsystems from the infinite tail; this is established using Brownian last-passage percolation, large-deviation bounds for random-matrix eigenvalues, and Gaussian process concentration. The work also connects the infinite-model analysis to approximative (finite-N) solutions and leverages Girsanov’s theorem to extend existence results to nonzero drifts, with implications for rank-based diffusions and KPZ-type scaling limits.

Abstract

We study strong existence and pathwise uniqueness for a class of infinite-dimensional singular stochastic differential equations (SDE), with state space as the cone $\{x \in \mathbb{R}^{\mathbb{N}}: -\infty < x_1 \leq x_2 \leq \cdots\}$, referred to as an infinite system of competing Brownian particles. A `mass' parameter $p \in [0,1]$ governs the splitting proportions of the singular collision local time between adjacent state coordinates. Solutions in the case $p=1/2$ correspond to the well-studied rank-based diffusions, while the general case arises from scaling limits of interacting particle systems on the lattice with asymmetric interactions and the study of the KPZ equation. Under conditions on the initial configuration, the drift vector, and the growth of the local time terms, we establish pathwise uniqueness and strong existence of solutions to the SDE. A key observation is the connection between pathwise uniqueness and the finiteness of `chains of collisions' between adjacent particles influencing a tagged particle in the system. Ingredients in the proofs include classical comparison and monotonicity arguments for reflected Brownian motions, techniques from Brownian last-passage percolation, large deviation bounds for random matrix eigenvalues, and concentration estimates for extrema of Gaussian processes.

Theorems & Definitions (42)

• Definition 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Theorem 2.7
• Corollary 2.8
• Remark 2.9
• Lemma 3.1