Brownian motion in the quadrant with oblique repulsion from the sides

TL;DR

This paper investigates the strong existence and uniqueness of a Brownian motion constrained within a quadrant by oblique electrostatic repulsion, extending classical reflection models and identifying conditions for stationary distributions.

Contribution

It introduces a novel model of Brownian motion with oblique electrostatic repulsion and establishes conditions for its strong existence, uniqueness, and stationary distribution in a quadrant.

Findings

Existence and uniqueness of the process under certain conditions

Stationary distribution in product form with gamma distributions

Connection to oblique reflection in wedges

Abstract

We consider the problem of strong existence and uniqueness of a Brownian motion forced to stay in the quadrant by an electrostatic repulsion from the sides that works obliquely. The results are reminiscent of the study of a Brownian motion with oblique reflection in a wedge. Actually, the same skew symmetry condition is involved when looking for a stationary distribution in product form. the terms of the product are now gamma distributions in place of exponential ones. An associate purely deterministic problem is also considered.