Concentration inequalities for the hydrodynamic limit of a two-species stochastic particle system

TL;DR

This paper proves exponential concentration inequalities for the empirical measures of a two-species stochastic particle system, demonstrating convergence to nonlinear kinetic equations using discretization and Hoeffding-type inequalities.

Contribution

It introduces a novel concentration inequality framework for a high-dimensional PDMP modeling grain boundary coarsening, linking particle systems to kinetic equations.

Findings

Exponential concentration inequalities established for the particle system.

Method involves discretization and Hoeffding-type inequalities.

Results demonstrate high-probability convergence to kinetic equations.

Abstract

We study a stochastic particle system which is motivated from grain boundary coarsening in two-dimensional networks. Each particles lives on the positive real line and is labeled as belonging to either Species 1 or Species 2. Species 1 particles drift at unit speed toward the origin, while Species 2 particles do not move. When a particle in Species 1 hits the origin, it is removed, and a randomly selected particle mutates from Species 2 to Species 1. The process described is an example of a high-dimensional piecewise deterministic Markov process (PDMP), in which deterministic flow is punctuated with stochastic jumps. Our main result is a proof of exponential concentration inequalities of the Kolmogorov-Smirnoff distance between empirical measures of the particle system and solutions of limiting nonlinear kinetic equations. Our method of proof involves a time and space discretization of the kinetic equations, which we compare with the particle system to derive recurrence inequalities for comparing total numbers in small intervals. To show these recurrences occur with high probability, we appeal to a state dependent Hoeffding type inequality at each time increment.