Competing particle systems evolving by interacting L\'{e}vy processes

TL;DR

This paper studies systems of particles driven by Lévy processes with rank-dependent parameters, proving the existence of invariant distributions and convergence properties, with applications in finance, queueing, and physics.

Contribution

It introduces a framework for analyzing particle systems with Lévy process dynamics, establishing invariant measures and convergence results for both finite and infinite systems.

Findings

Finite systems have unique invariant gap distributions.

Gap processes converge to these distributions in total variation.

Infinite systems exhibit tightness under certain conditions.

Abstract

We consider finite and infinite systems of particles on the real line and half-line evolving in continuous time. Hereby, the particles are driven by i.i.d. L\'{e}vy processes endowed with rank-dependent drift and diffusion coefficients. In the finite systems we show that the processes of gaps in the respective particle configurations possess unique invariant distributions and prove the convergence of the gap processes to the latter in the total variation distance, assuming a bound on the jumps of the L\'{e}vy processes. In the infinite case we show that the gap process of the particle system on the half-line is tight for appropriate initial conditions and same drift and diffusion coefficients for all particles. Applications of such processes include the modeling of capital distributions among the ranked participants in a financial market, the stability of certain stochastic queueing and storage networks and the study of the Sherrington--Kirkpatrick model of spin glasses.