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

@article{Kulik2013PoincareIA, title={Poincare inequality and exponential integrability of the hitting times of a Markov process}, author={Alexei M. Kulik}, journal={arXiv: Probability}, year={2013} }

Extending the approach of the paper [Mathieu, P. (1997) Hitting times and spectral gap inequalities, Ann. Inst. Henri Poincare 33, 4, 437 -- 465], we prove that the Poincare inequality for a (possibly non-symmetric) Markov process yields the exponential integrability of the hitting times of this process. For symmetric elliptic diffusions, this provides a criterion for the Poincare inequality in the terms of hitting times.

## 6 Citations

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## References

SHOWING 1-10 OF 20 REFERENCES

Poincaré inequalities and hitting times

- Mathematics
- 2010

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions are well known. We give here the correspondance (with quantitative results) for reversible…

Hitting times and spectral gap inequalities

- Mathematics
- 1997

Abstract The aim of this paper is to relate estimates on the hitting times of closed sets by a Markov process and a special class of inequalities involving the L p ( p ≤ 1 ) norm of a function and…

Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré

- Mathematics
- 2007

We study the relationship between two classical approaches for quantitative ergodic properties : the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one…

Lyapunov conditions for Super Poincaré inequalities

- Mathematics
- 2009

We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincare inequality (for instance logarithmic Sobolev or F-Sobolev). The case of Poincare and weak…

Asymptotic and spectral properties of exponentially ϕ-ergodic Markov processes

- Mathematics
- 2009

Abstract For L p convergence rates of a time homogeneous Markov process, sufficient conditions are given in terms of an exponential ϕ -coupling. This provides sufficient conditions for L p…

A CERTAIN PROPERTY OF SOLUTIONS OF PARABOLIC EQUATIONS WITH MEASURABLE COEFFICIENTS

- Mathematics
- 1981

In this paper Harnack's inequality is proved and the Holder exponent is estimated for solutions of parabolic equations in nondivergence form with measurable coefficients. No assumptions are imposed…

Poincaré inequality and exponential integrability of hitting times for linear diffusions

- Mathematics
- 2009

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}…

Introduction to the theory of (non-symmetric) Dirichlet forms

- Mathematics
- 1992

0 Introduction.- I Functional Analytic Background.- 1 Resolvents, semigroups, generators.- 2 Coercive bilinear forms.- 3 Closability.- 4 Contraction properties.- 5 Notes/References.- II Examples.- 1…

Lower bounds for covering times for reversible Markov chains and random walks on graphs

- Mathematics, Computer Science
- 1989

For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant. This is deduced from a more general result about stationary…