Degenerate systems of three Brownian particles with asymmetric collisions: invariant measure of gaps

TL;DR

This work analyzes a degenerate three-particle Brownian system with asymmetric collisions, focusing on the invariant distribution of the gaps. The gap process is modeled as an obliquely reflected Brownian motion in a quadrant, and the authors derive a kernel functional equation for the Laplace transform of the invariant measure, then solve it via a uniformization to a Riemann surface and Tutte invariants, introducing a decoupling function $D(s)$. A detailed differential-hierarchy classification is obtained: the Laplace transform is rational, algebraic, or D-finite only under specific integrality conditions on the parameters $oldsymbol{ u}$, $oldsymbol{ u}_1$, and $oldsymbol{ u}_2$, while in the generic case it is D-transcendental; when possible, boundary densities on the quadrant walls are given explicitly using Jacobi theta functions and Mittag-Leffler expansions. The results connect degenerate to nondegenerate regimes and leverage difference Galois theory to establish necessary and sufficient conditions for the transform’s function class, with explicit formulas for the boundary densities that have potential applications in stochastic portfolio-like models and reflected Brownian motion in wedges.

Abstract

We consider a degenerate system of three Brownian particles undergoing asymmetric collisions. We study the gap process of this system and focus on its invariant measure. The gap process is described as an obliquely reflected degenerate Brownian motion in a quadrant. For all possible parameter cases, we compute the Laplace transform of the invariant measure, and fully characterize the conditions under which it belongs to the following classes: rational, algebraic, differentially finite, or differentially algebraic. We also derive explicit formulas for the invariant measure on the boundary of the quadrant, expressed in terms of a Theta-like function, to which we apply a polynomial differential operator. In this study, we introduce a new parameter called $γ$ (along with two additional parameters $γ_1$ and $γ_2$) which governs many properties of the degenerate process. This parameter is reminiscent of the famous parameter $α$ introduced by Varadhan and Williams (and the two parameters $α_1$ and $α_2$ recently introduced by Bousquet-M{é}lou et al.) to study nondegenerate reflected Brownian motion in a wedge. To establish our main results we start from a kernel functional equation characterizing the Laplace transform of the invariant measure. By an analytic approach, we establish a finite difference equation satisfied by the Laplace transform. Then, using certain so-called decoupling functions, we apply Tutte's invariant approach to solve the equation via conformal gluing functions. Finally, difference Galois theory and exhaustive study allows us to find necessary and sufficient conditions for the Laplace transform to belong to the specified function hierarchy.

Figures (8)

• Figure 1: paths of the interacting Brownian particles, colored by name (left), colored by rank (middle) and the gap process $({G}_1,{G}_2)$ (right).
• Figure 2: The regions of convergence of the Laplace transforms $\phi_1$ (bounded by the red curves) and $\phi_2$ (bounded by the blue curves), seen through the uniformization \ref{['eq:unif']}, and their respective Galois automorphisms.
• Figure 3: The sets to which the Laplace transforms are successively extended. Each curve is labelled with a function $h$ such that the curve is defined by $\{h(y)+iy : y\in\mathbb R\}$. Colors are chosen as in Figure \ref{['fig:fundamental']}.
• Figure 4: The strips $\mathfrak{B}_2\subset \mathfrak{B}_1$ are subsets of the domain $\Delta$. Colors correspond to those used in Figures \ref{['fig:fundamental']} and \ref{['fig:1']}.
• Figure 5: Geometric interpretation for the rational decoupling condition \ref{['eq:decouplcond']}. Every point on the parabola $K(x,y)=0$ is labelled with its unique preimage by $(\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}})$, i.e.$(\boldsymbol{\mathrm{x}}(s),\boldsymbol{\mathrm{y}}(s))$ is labelled by $s$.

Theorems & Definitions (150)

• Remark 1
• Proposition 2: Functional equation
• Lemma 3: Relation between $\pi$, $\nu_1$ and $\nu_2$
• proof
• Lemma 4: Value of Laplace transforms at $0$
• proof
• Theorem A: Explicit expression of $\phi_1$
• Theorem B: Algebraic and differential nature
• Theorem C: Explicit expression of $\nu_1$
• Remark 5: Overview of Tutte's invariant method