Stationary brownian motion in a 3/4-plane: Reduction to a riemann-hilbert problem via fourier transforms

Guy Fayolle (1, 2) , Sandro Franceschi (3, 4, 5) , Kilian Raschel (6, 7)

1 ASTRA - Systèmes de transport automatisés et sécurisés
2 MATHEXP - Calcul formel, mathématiques expérimentales et interactions
3 IP Paris - Institut Polytechnique de Paris
4 TSP - CITI - Communications, Images et Traitement de l'Information
5 SOP - SAMOVAR - Statistiques, Optimisation, Probabilités
6 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
7 MATHSTIC - SFR UA MathSTIC

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.

Mots clés: Obliquely reflected Brownian motion in a wedge; Stationary distribution; Laplace transform; Boundary value problem

Domaines: Probabilités [math.PR]

Guy Fayolle, Sandro Franceschi, Kilian Raschel. Stationary brownian motion in a 3/4-plane: Reduction to a riemann-hilbert problem via fourier transforms. Indagationes Mathematicae, 2023, 34 (5), pp.17 (874-890). ⟨10.1016/j.indag.2022.10.008⟩. ⟨hal-03832431v2⟩