On the stationary distribution of reflected Brownian motion in a non-convex wedge

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

1 SPECFUN - Symbolic Special Functions : Fast and Certified
2 RITS - Robotics & Intelligent Transportation Systems
3 LMO - Laboratoire de Mathématiques d'Orsay
4 CNRS - Centre National de la Recherche Scientifique
5 IDP - Institut Denis Poisson

We study the stationary reflected Brownian motion in a non-convex wedge, which, compared to its convex analogue model, has been much rarely analyzed in the probabilistic literature. We prove that its stationary distribution can be found by solving a two dimensional vector boundary value problem (BVP) on a single curve for the associated Laplace transforms. The reduction to this kind of vector BVP seems to be new in the literature. As a matter of comparison, one single boundary condition is sufficient in the convex case. When the parameters of the model (drift, reflection angles and covariance matrix) are symmetric with respect to the bisector line of the cone, the model is reducible to a standard reflected Brownian motion in a convex cone. Finally, we construct a one-parameter family of distributions, which surprisingly provides, for any wedge (convex or not), one particular example of stationary distribution of a reflected Brownian motion.

Mots clés: Obliquely reflected Brownian motion in a wedge; Non-convex cone; Stationarydistribution; Laplace transform; Boundary value problem

Domaines: Probabilités [math.PR]

Guy Fayolle, Sandro Franceschi, Kilian Raschel. On the stationary distribution of reflected Brownian motion in a non-convex wedge. 2021. ⟨hal-03150317v1⟩