A two-dimensional oblique extension of Bessel processes

TL;DR

This paper studies a Brownian motion constrained in a quadrant with oblique electrostatic repulsion, analyzing its boundary hitting behavior and deriving stationary measures under specific conditions.

Contribution

It introduces a two-dimensional oblique extension of Bessel processes, providing new insights into boundary behavior and stationary distributions under electrostatic repulsion.

Findings

Identifies conditions for hitting the corner or edges.

Derives product-form stationary measures.

Establishes parallels with skew-symmetry in reflected Brownian motion.

Abstract

We consider a Brownian motion forced to stay in the quadrant by an electrostatic oblique repulsion from the sides. We tackle the question of hitting the corner or an edge, and find product-form stationary measures under a certain condition, which is reminiscent of the skew-symmetry condition for an obliquely reflected Brownian motion.