Tutte's invariant approach for Brownian motion reflected in the quadrant

TL;DR

This paper introduces a Tutte's invariant method to analyze a drifted Brownian motion reflected in the quadrant, deriving an explicit Laplace transform formula using generalized Chebyshev polynomials, bridging discrete and continuous stochastic models.

Contribution

It adapts Tutte's invariant approach to continuous Brownian motion with reflections, providing a new explicit formula for the stationary distribution's Laplace transform.

Findings

Explicit formula for the Laplace transform in terms of Chebyshev polynomials

Connection established between discrete walk enumeration and continuous Brownian motion

Method offers a new analytical tool for reflected stochastic processes

Abstract

We consider a Brownian motion with drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte's invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev polynomials.