Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann–Hilbert problem via Fourier transforms

Guy Fayolle; Sandro Franceschi; Kilian Raschel

doi:10.1016/j.indag.2022.10.008

Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann–Hilbert problem via Fourier transforms

Guy Fayolle
, Sandro Franceschi
, Kilian Raschel

Research output: Contribution to journal › Article › peer-review

Abstract

The stationary reflected Brownian motion in a three-quarter plane has been rarely analyzed in the probabilistic literature, in comparison with the quarter plane analogue model. In this context, our main result is to prove that the stationary distribution can indeed be found by solving a boundary value problem of the same kind as the one encountered in the quarter plane, up to various dualities and symmetries. The main idea is to start from Fourier (and not Laplace) transforms, allowing to get a functional equation for a single function of two complex variables.

Original language	English
Pages (from-to)	874-890
Number of pages	17
Journal	Indagationes Mathematicae
Volume	34
Issue number	5
DOIs	https://doi.org/10.1016/j.indag.2022.10.008
Publication status	Published - 1 Sept 2023

Keywords

Boundary value problem.
Laplace transform
Non-convex cone
Obliquely reflected Brownian motion in a wedge
Stationary distribution

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10.1016/j.indag.2022.10.008

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title = "Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann–Hilbert problem via Fourier transforms",

abstract = "The stationary reflected Brownian motion in a three-quarter plane has been rarely analyzed in the probabilistic literature, in comparison with the quarter plane analogue model. In this context, our main result is to prove that the stationary distribution can indeed be found by solving a boundary value problem of the same kind as the one encountered in the quarter plane, up to various dualities and symmetries. The main idea is to start from Fourier (and not Laplace) transforms, allowing to get a functional equation for a single function of two complex variables.",

keywords = "Boundary value problem., Laplace transform, Non-convex cone, Obliquely reflected Brownian motion in a wedge, Stationary distribution",

author = "Guy Fayolle and Sandro Franceschi and Kilian Raschel",

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AU - Fayolle, Guy

AU - Franceschi, Sandro

AU - Raschel, Kilian

PY - 2023/9/1

Y1 - 2023/9/1

N2 - The stationary reflected Brownian motion in a three-quarter plane has been rarely analyzed in the probabilistic literature, in comparison with the quarter plane analogue model. In this context, our main result is to prove that the stationary distribution can indeed be found by solving a boundary value problem of the same kind as the one encountered in the quarter plane, up to various dualities and symmetries. The main idea is to start from Fourier (and not Laplace) transforms, allowing to get a functional equation for a single function of two complex variables.

AB - The stationary reflected Brownian motion in a three-quarter plane has been rarely analyzed in the probabilistic literature, in comparison with the quarter plane analogue model. In this context, our main result is to prove that the stationary distribution can indeed be found by solving a boundary value problem of the same kind as the one encountered in the quarter plane, up to various dualities and symmetries. The main idea is to start from Fourier (and not Laplace) transforms, allowing to get a functional equation for a single function of two complex variables.

KW - Boundary value problem.

KW - Laplace transform

KW - Non-convex cone

KW - Obliquely reflected Brownian motion in a wedge

KW - Stationary distribution

UR - https://www.scopus.com/pages/publications/85141254750

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JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 5

ER -

Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann–Hilbert problem via Fourier transforms

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Keywords

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