arXiv subjects

Jules Flin

Publications and source records attributed to Jules Flin.

4 recordsLinked to original sources

arXiv2026

Stationary Distribution of Brownian Motion in the Half-Plane with Two-sided Reflections

We investigate the unique stationary measure of a positive recurrent reflecting Brownian motion in the upper half-plane, where the direction of reflection is constant on each half-axis. The Laplace transform of the stationary distribution is characterized by a functional equation, whose resolution is reduced to solving a discontinuous Riemann boundary value problem. By applying the Sokhotski-Plemelj formulas, we derive an explicit expression for the Laplace transform. Finally, we establish the local behavior of the stationary density at the origin and its asymptotics along the boundary axes using Tauberian theorems and asymptotic analysis.

math.PR↗

arXiv2025

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

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 $\gamma$ (along with two additional parameters $\gamma_1$ and $\gamma_2$) which governs many properties of the degenerate process. This parameter is reminiscent of the famous parameter $\alpha$ introduced by Varadhan and Williams (and the two parameters $\alpha_1$ and $\alpha_2$ recently introduced by Bousquet-M{\'e}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.

math.PR↗

arXiv2024

Reflected Brownian Motion in a wedge: sum-of-exponential absorption probability at the vertex and differential properties

We study a Brownian motion with drift in a wedge of angle $\beta$ which is obliquely reflected on each edge along angles $\varepsilon$ and $\delta$. We assume that the classical parameter $\alpha=\frac{\delta+\varepsilon - \pi}{\beta}$ is greater than $1$ and we focus on transient cases where the process can either be absorbed at the vertex or escape to infinity. We show that $\alpha\in\mathbb{N}^*$ is a necessary and sufficient condition for the absorption probability, seen as a function of the starting point, to be written as a finite sum of terms of exponential product form. In such cases, we give expressions for the absorption probability and its Laplace transform. When $\alpha\in\mathbb{Z}+\frac{\pi}{\beta}\mathbb{Z}$ we find explicit D-algebraic expression for the Laplace transform. Our results rely on Tutte's invariant method and on a recursive compensation approach.

math.PR↗

arXiv2022

Pascal's formulas and vector fields

We consider four examples of combinatorial triangles $\left(T(n,k)\right)_{0\le k\le n}$ (Pascal, Stirling of both types, Euler) : through saddle-point asymptotics, their \emph{Pascal's formulas} define four vector fields, together with their field lines that turn out to be the conjectured limit of sample paths of four well known Markov chains. We prove this asymptotic behaviour in three of the four cases.

math.CO↗