On the stationary distribution of reflected Brownian motion in a wedge: differential properties

TL;DR

This work fully characterizes when the Laplace transform of the stationary distribution of a two-dimensional semimartingale reflected Brownian motion in a wedge is rational, algebraic, D-finite, or D-algebraic, tying these properties to simple linear relationships among the wedge reflection angles. Central to the analysis are Tutte's invariant theory and a difference-Galois approach, which yield a canonical invariant $w$ that rationally encodes all invariants and enables explicit, integral-free expressions for the Laplace transform in the simple-angle and double-angle regimes. The paper unifies and extends classical special cases (skew symmetry, orthogonal reflections, Dieker–Moriarty) and provides explicit densities and moment recurrences in several instances, including algebraic and D-finite examples. A key outcome is a complete dichotomy: when the simple or double angle conditions fail and $\beta/\pi$ is irrational, the Laplace transform is differentially transcendental; when $\beta/\pi$ is rational, the log-derivative is D-finite and the transform is D-finite with algebraicity dictated by the angle data. Overall, the results give a comprehensive map of the algebraic and differential structure of stationary densities for SRBMs in wedges, with explicit tools for computing or approximating the transforms in the most relevant cases.

Abstract

We consider the classical problem of determining the stationary distribution of the semimartingale reflected Brownian motion (SRBM) in a two-dimensional wedge. Under standard assumptions on the parameters of the model (opening of the wedge, angles of the reflections, drift), we study the algebraic and differential nature of the Laplace transform of this stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be rational, algebraic, differentially finite or more generally differentially algebraic. These conditions are explicit linear dependencies between the angles of the model. A complicated integral expression for this Laplace transform has recently been obtained by two authors of this paper. In the differentially algebraic case, we provide a simple, explicit integral-free expression in terms of a hypergeometric function. It specializes to earlier expressions in several classical cases: the skew-symmetric case, the orthogonal reflections case and the sum-of-exponential densities case (corresponding to the so-called Dieker-Moriarty conditions on the parameters). This paper thus closes, in a sense, the quest of all ``simple'' cases. To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply tools from diverse horizons. To establish differential algebraicity, a key ingredient is Tutte's invariant approach, which originates in enumerative combinatorics. It allows us to express the Laplace transform (or its square) as a rational function of a certain canonical invariant, a hypergeometric function in our context. To establish differential transcendence, we turn the functional equation into a difference equation and apply Galoisian results on the nature of the solutions to such equations.

Figures (13)

• Figure 1: A trajectory of the reflected Brownian motion in a wedge, and the parameters $\beta$, $\widetilde{\mu}$, $\theta$, $\delta$ and $\varepsilon$.
• Figure 2: Properties of obliquely reflected Brownian motion in terms of $\alpha=\frac{\delta+\varepsilon-\pi}{\beta}$. Here are some references: semimartingale property Williams-85reiman_boundary_1988taylor_existence_1993; Skorokhod problem HaRe-81bwilliams_semimartingale_1995; submartingale problem varadhan_brownian_1985; amount of time spent at the corner, accessibility of the corner and absorption varadhan_brownian_1985; Dirichlet process and extended Skorokhod problem lakner_dirichlet_2016kang_dirichlet_2010; skew symmetry HaRe-81harrison_multidimensional_1987; sum-of-exponential stationary density DiMo-09.
• Figure 3: Transformation of the quadrant into a wedge of opening angle $\beta$. The new parameters $\beta$, $\widetilde{\mu}$, $\delta$ and $\varepsilon$ are given by \ref{['eq:beta']}, \ref{['eq:mu']} and \ref{['eq:expression_delta_epsilon']}, respectively.
• Figure 4: Three interesting cases where the Laplace transform $\varphi_1$ is D-algebraic. From left to right: skew symmetry, Dieker and Moriarty condition (for $\alpha=-1$) and orthogonal reflections.
• Figure 5: The complex $s$-plane, from which the uniformization \ref{['eq:uniformization']} of $\mathcal{S}$ is expressed.

Theorems & Definitions (84)

• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 3.1
• Remark 3.2
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 3.5