Explicit expression for the stationary distribution of reflected brownian motion in a wedge

Sandro Franceschi (1, 2) , Kilian Raschel (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polyno-mials. To that purpose we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.

Mots clés: Stationary distribution; Carleman-type boundary value problem; Reflected Brownian motion in a wedge; Laplace transform; Boundary value problem with shift; Conformal mapping

Domaines: Probabilités [math.PR]

Sandro Franceschi, Kilian Raschel. Explicit expression for the stationary distribution of reflected brownian motion in a wedge. 2018. ⟨hal-01495788v2⟩