Poincaré inequality and exponential integrability of hitting times for linear diffusions

Abstract

Let $X$ be a regular linear continuous positively recurrent Markov process with state space $\mathbb{R}$, scale function $S$ and speed measure $m$. For $a\in \mathbb{R}$ 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 Poincaré 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 $X$.

Cited by in corpus (1)

• Poincare inequality and exponential integrability of the hitting times of a Markov process