Geodesic
Poincar\'e inequality and exponential integrability of the hitting times of a Markov process
Teoriâ slučajnyh processov, Tome 17 (2011) no. 2, pp. 71-80.

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Extending the approach of the paper [Mathieu, P. (1997) Hitting times and spectral gap inequalities, Ann. Inst. Henri Poincaré 33, 4, 437 – 465], we prove that the Poincaré 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 Poincaré inequality in the terms of hitting times.
Keywords: Markov process, exponential $\phi$-coupling, Poincaré inequality, hitting time. 
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Alexey M. Kulik. Poincar\'e inequality and exponential integrability of the hitting times of a Markov process. Teoriâ slučajnyh processov, Tome 17 (2011) no. 2, pp. 71-80. http://geodesic.mathdoc.fr/item/THSP_2011_17_2_a5/

[1] F. Aldous, J. Fill, Reversible Markov chains and random walks on graphs, http://www.stat.berkeley.edu/users/aldous/RWG/book.html

[2] D. Bakry, P. Cattiaux, A. Guillin, “Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré”, J. Func. Anal., 254 (2008), 727 – 759

[3] E. B. Dynkin, A. A. Yushkevich, Markov processes: theorems and problems, Plenum Press, 1969

[4] P. Cattiaux, A. Guillin, F-Y. Wang, L. Wu, “Lyapunov conditions for super Poincaré inequalities”, J. Funct. Anal., 256:6 (2009), 1821–1841

[5] P. Cattiaux, A. Guillin, P.-A. Zitt, Poincaré inequalities and hitting times, 2010, arXiv: 1012.5274v1

[6] A. M. Il’in, A. S. Kalashnikov, O. A. Oleinik, “Linear second order parabolic equations”, Uspekhi Mat. Nauk, 17:3 (1962), 3–-143

[7] N. V. Krylov, M. V. Safonov, “A certain property of solutions of parabolic equations with measurable coefficients”, Math. USSR Izvestija, 16 (1981), 151–164

[8] A. M. Kulik, “Asymptotic and spectral properties of exponentially $\phi$-ergodic Markov processes”, Stochastic Processes and Applications, 121 (2011), 1044–1075

[9] D. Loukianova, O. Loukianov, Sh. Song, Poincaré inequality and exponential integrability of hitting times for linear diffusions, 2009, arXiv: 0907.0762v1

[10] Z.-M. Ma, M. Röckner, Introduction to the theory of (non-symmetric) Dirichlet forms, Springer-Verlag, London, Ltd., 1992

[11] P. Mathieu, “Hitting times and spectral gap inequalities”, Ann. Inst. Henri Poincaré, 33:4 (1997), 437–465

[12] W. Rudin, Functional Analysis, McGraw-Hill, New-York, 1973