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

TL;DR

This paper establishes strong existence and pathwise uniqueness for a class of infinite-dimensional singular SDEs modeling competing Brownian particles, revealing connections to KPZ scaling limits and collision chains.

Contribution

It introduces new conditions ensuring well-posedness of infinite competing Brownian particle systems, extending known results for rank-based diffusions to more general cases.

Findings

Proved strong existence and pathwise uniqueness under specific conditions.

Connected collision chain finiteness to uniqueness in the system.

Utilized techniques from Brownian percolation, large deviations, and Gaussian analysis.

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.