Poincar\'e inequality and exponential integrability of hitting times for linear diffusions

TL;DR

This paper investigates the relationships between hitting times, inequalities, and spectral properties of linear diffusions, establishing conditions for exponential integrability and spectral gaps through probabilistic and functional analytic methods.

Contribution

It introduces new relations connecting hitting times, Hardy and Poincaré inequalities, and spectral gaps for linear diffusions, providing a unified framework.

Findings

Exponential moments of hitting times are characterized by spectral gap conditions.

Established equivalence between exponential integrability of hitting times and spectral gap existence.

Derived relations linking scale functions, speed measures, and functional inequalities.

Abstract

Let $X$ be a regular linear continuous positively recurrent Markov process with state space $\R$, scale function $S$ and speed measure $m$. For $a\in \R$ denote B^+_a&=\sup_{x\geq a} \m(]x,+\infty[)(S(x)-S(a)) B^-_a&=\sup_{x\leq a} \m(]-\infty;x[)(S(a)-S(x)) We study some characteristic relations between $B^+_a$, $B^-_a$, the exponential moments of the hitting times $T_a$ of $X$, the Hardy and Poincar\'e inequalities for the Dirichlet form associated with $X$. As a corollary, we establish the equivalence between the existence of exponential moments of the hitting times and the spectral gap of the generator of $X$.