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

  • Emmnuel Boissard
  • Patrick Cattiaux
  • Arnaud Guillin
  • Laurent Miclo
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)

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(xi, ri) with radii ri, or hypercubes with vertices of length 2ri, 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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Emmnuel Boissard
    • 1
  • Patrick Cattiaux
    • 1
  • Arnaud Guillin
    • 2
  • Laurent Miclo
    • 1
  1. 1.Institut de Mathématiques de Toulouse, CNRS UMR 5219Université Paul SabatierToulouse CedexFrance
  2. 2.Laboratoire de Mathématiques, CNRS UMR 6620Université Blaise PascalAubièreFrance

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