Comparison Techniques for Competing Brownian Particles

TL;DR

This paper investigates comparison results for finite systems of rank-dependent Brownian particles, showing how removing top particles affects the system's gap sizes, collision local times, and particle positions in stochastic order.

Contribution

It provides new comparison theorems for rank-dependent Brownian systems, illustrating how modifications like removing particles influence system properties.

Findings

Removing top particles increases gaps between remaining particles.

Collision local times decrease when top particles are removed.

Remaining particles shift upward in stochastic order.

Abstract

Consider a finite system of Brownian particles on the real line. Each particle has drift and diffusion coefficients depending on its current rank relative to other particles, as in Karatzas, Pal and Shkolnikov (2012). We prove some comparison results for these systems. As an example, we show that if we remove a few particles from the top, then the gaps between adjacent particles become stochastically larger, the local times of collision between adjacent particles become stochastically smaller, and the remaining particles shift upward, in the sense of stochastic ordering.