# Integral expression for the stationary distribution of reflected brownian motion in a wedge

## TL;DR

This paper derives an explicit formula for the Laplace transform of the stationary distribution of a two-dimensional reflected Brownian motion in a wedge, using complex analysis and boundary value problem techniques.

## Contribution

It provides a novel explicit integral expression for the stationary distribution's Laplace transform in a wedge with oblique reflection.

## Key findings

- Explicit Laplace transform formula derived
- Solution to a boundary value problem on a hyperbola
- Uses Cauchy integrals and Chebyshev polynomials

## Abstract

For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polyno-mials. To that purpose we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.