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

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In this paper we study the Poincaré constant for the Gaussian measure restricted to D=ᵈ - B(y,r) where B(y,r) denotes the Euclidean ball with center y and radius r, and d≥ 2. We also study the case of the l∞ 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.