Ornstein-Uhlenbeck pinball: I. Poincaré inequalities in a punctured domain.

Emmanuel Boissard (1) , Patrick Cattiaux (1) , Arnaud Guillin (2) , Laurent Miclo (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

In this paper we study the Poincaré constant for the Gaussian measure restricted to $D=\R^d - B(y,r)$ where $B(y,r)$ denotes the Euclidean ball with center $y$ and radius $r$, and $d\geq 2$. We also study the case of the $l^\infty$ ball (the hypercube). This is the first step in the study of the asymptotic behavior of a $d$-dimensional Ornstein-Uhlenbeck process in the presence of obstacles with elastic normal reflections (the Ornstein-Uhlenbeck pinball) we shall study in a companion paper.

Mots clés: Poincaré inequalities; Lyapunov functions; hitting times; obstacles

Domaines: Probabilités [math.PR]

Emmanuel Boissard, Patrick Cattiaux, Arnaud Guillin, Laurent Miclo. Ornstein-Uhlenbeck pinball: I. Poincaré inequalities in a punctured domain.. Séminaire de Probabilités, 2018, XLIX (2215). ⟨hal-00857842⟩