Séminaire de Probabilités XLIX pp 1-55 | Cite as

# Ornstein-Uhlenbeck Pinball and the Poincaré Inequality in a Punctured Domain

## Abstract

In this paper we study the Poincaré constant for the Gaussian measure restricted to \(D={\mathbb R}^d - \mathbb {B}\) where \(\mathbb {B}\) is the disjoint union of bounded open sets. We will mainly look at the case where the obstacles are Euclidean balls *B*(*x*_{i}, *r*_{i}) with radii *r*_{i}, or hypercubes with vertices of length 2*r*_{i}, and *d* ≥ 2. This will explain the asymptotic behavior of a *d*-dimensional Ornstein-Uhlenbeck process in the presence of obstacles with elastic normal reflections (the Ornstein-Uhlenbeck pinball).

## Keywords

Poincaré inequalities Lyapunov functions Hitting times Obstacles## MSC 2010

26D10 39B62 47D07 60G10 60J60## Notes

### Acknowledgements

We want to heartily thank an anonymous referee for an amazing and so accurate work on the paper, correcting many typos, minor and not so minor mistakes. It is a real pleasure nowadays to receive such a report.

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