Strong solutions of non-colliding particle systems

TL;DR

This paper proves the existence and uniqueness of strong, non-colliding solutions for a class of stochastic differential equations modeling ordered particles with repulsive interactions, starting from colliding initial points.

Contribution

It establishes the existence of strong, pathwise unique solutions for particle systems with singular repulsive interactions under broad conditions.

Findings

Strong solutions exist for all initial configurations.

Solutions are pathwise unique and non-colliding.

Applicable to general coefficients under natural assumptions.

Abstract

We study systems of stochastic differential equations describing positions x_1,x_2,...,x_p of p ordered particles, with inter-particles repulsions of the form H_{ij}(x_i,x_j)/(x_i-x_j). We show the existence of strong and pathwise unique non-colliding solutions of the system with a colliding initial point x_1(0)\leq ...\leq x_p(0) in the whole generality, under natural assumptions on the coefficients of the equations.