Two Brownian Particles with Rank-Based Characteristics and Skew-Elastic Collisions

TL;DR

This paper develops a two-dimensional diffusion model for two particles with rank-dependent behavior and various collision types, using novel stochastic differential equations involving local times, and analyzes its properties including transition probabilities and time reversal.

Contribution

It introduces a new class of stochastic differential equations with local times to model rank-based particle interactions with diverse collision behaviors.

Findings

Constructed well-posed SDE systems with local times

Analyzed properties of associated skew Brownian motion

Computed transition probabilities and studied time reversal

Abstract

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coefficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems. The analysis depends crucially on properties of a skew Brownian motion with two-valued drift of the bang-bang type, which we also study in some detail. These properties allow us to compute the transition probabilities of the original planar diffusion, and to study its behavior under time reversal.