×

Cheeger’s inequalities for general symmetric forms and existence criteria for spectral gap. (English) Zbl 1044.60078

Summary: Some new forms of J. Cheeger’s inequalities [in: Probl. Anal., Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 195–199 (1970; Zbl 0212.44903)] are established for general (maybe unbounded) symmetric forms (Theorems 1.1 and 1.2): the resulting estimates improve and extend the ones obtained by G. F. Lawler and A. D. Sokal [Trans. Am. Math. Soc. 309, No. 2, 557–580 (1988; Zbl 0716.60073)] for bounded jump processes. Furthermore, some existence criteria for spectral gap of general symmetric forms or general reversible Markov processes are presented (Theorems 1.4 and 3.1), based on Cheeger’s inequalities and a relationship between the spectral gap and the first Dirichlet and Neumann eigenvalues on local region.

MSC:

60J75 Jump processes (MSC2010)
60J25 Continuous-time Markov processes on general state spaces
47A75 Eigenvalue problems for linear operators
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Bobkov, S. and Ledoux, M. (1997). PoincarĂ© inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107 383- 400 · Zbl 0878.60014
[2] Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis. A Symposium in Honor of S. Bochner (R. C. Gunning, ed.) 195-199. Princeton Univ. Press. · Zbl 0212.44903
[3] Chen, M.F.(1992). From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore · Zbl 0753.60055
[4] Chen, M.F.(1996). Estimation of spectral gap for Markov chains. Acta Math. Sinica. (N. S.) 12 337-360 · Zbl 0867.60038
[5] Chen, M.F.and Wang, F.Y.(1997). Estimation of spectral gap for elliptic operators. Trans. Amer. Math. Soc. 349 1209-1237 JSTOR: · Zbl 0872.35072
[6] Chung, F.R.K.(1997). Spectral Graph Theory. Amer. Math. Soc., Providence, RI. · Zbl 0867.05046
[7] Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin. · Zbl 0838.31001
[8] Lawler, G.F.and Sokal, A.D.(1988). Bounds on the L2 spectrum for Markov chain and Markov processes: a generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309 557-580 JSTOR: · Zbl 0716.60073
[9] Saloff-Coste, L. (1997). Lectures on finite Markov chains. Lecture Notes in Math. 1665 301-413. Springer, Berlin. · Zbl 0885.60061
[10] Wang, F.Y.(1999). Existence of spectral gap for elliptic operators. Ark. Mat. 37 395-407. · Zbl 1075.35540
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.