Ornstein-Uhlenbeck pinball: I. Poincar\'e inequalities in a punctured domain

TL;DR

This paper investigates the Poincaré inequality constants for Gaussian measures restricted to domains with obstacles, laying groundwork for analyzing Ornstein-Uhlenbeck processes with reflections in complex environments.

Contribution

It introduces the first analysis of Poincaré constants in punctured domains for Gaussian measures, advancing understanding of Ornstein-Uhlenbeck processes with obstacles.

Findings

Poincaré constants are characterized for Gaussian measures in punctured Euclidean domains.

Results extend to hypercube domains, providing new bounds.

Foundation for future study of Ornstein-Uhlenbeck processes with obstacle reflections.

Abstract

In this paper we study the Poincar\'e 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.