Systems of Brownian particles with asymmetric collisions

TL;DR

This paper introduces and analyzes systems of Brownian particles with asymmetric collision interactions, proving their existence, uniqueness, and connections to market models and lattice jump processes, with implications for stochastic processes and mathematical finance.

Contribution

It establishes the strong existence and uniqueness of asymmetric collision Brownian systems and links them to rank-based models, scaling limits, and determinantal structures.

Findings

Proved strong existence and uniqueness of asymmetric collision Brownian systems.

Identified these systems as universal scaling limits of lattice jump processes.

Connected the models to determinantal point processes.

Abstract

We study systems of Brownian particles on the real line, which interact by splitting the local times of collisions among themselves in an asymmetric manner. We prove the strong existence and uniqueness of such processes and identify them with the collections of ordered processes in a Brownian particle system, in which the drift coefficients, the diffusion coefficients, and the collision local times for the individual particles are assigned according to their ranks. These Brownian systems can be viewed as generalizations of those arising in first-order models for equity markets in the context of stochastic portfolio theory, and are able to correct for several shortcomings of such models while being equally amenable to computations. We also show that, in addition to being of interest in their own right, such systems of Brownian particles arise as universal scaling limits of systems of jump processes on the integer lattice with local interactions. A key step in the proof is the analysis of a generalization of Skorokhod maps which include `local times' at the intersection of faces of the nonnegative orthant. The result extends the convergence of TASEP to its continuous analogue. Finally, we identify those among the Brownian particle systems which have a probabilistic structure of determinantal type.