Poincaré inequality and exponential integrability of hitting times for linear diffusions

Details Poincaré inequality and exponential integrability of hitting times for linear diffusions Poincaré inequality and exponential integrability of hitting times for linear diffusions

2009-07-04

arxiv.org/abs/0907.0762v1

Mathematics

Probability

Scientific paper

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 $X$.

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