Mixed principal eigenvalues in dimension one

doi:10.1007/s11464-013-0229-6

Front Math Chin

0, Vol.

Issue () : 317-343 https://doi.org/10.1007/s11464-013-0229-6

RESEARCH ARTICLE

Mixed principal eigenvalues in dimension one

Mu-Fa CHEN, Lingdi WANG(

), Yuhui ZHANG

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, China

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Abstract

This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the L²-Poincaré inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.

Keywords Eigenvalue variational formula explicit estimate positivity criterion approximating procedure

Corresponding Author(s): WANG Lingdi,Email:wanglingdi@mail.bnu.edu.cn

Issue Date: 01 April 2013

Cite this article:

Yuhui ZHANG,Mu-Fa CHEN,Lingdi WANG. Mixed principal eigenvalues in dimension one[J]. Front Math Chin, 0, (): 317-343.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-013-0229-6
https://academic.hep.com.cn/fmc/EN/Y0/V/I/317

1	Chen M F. Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci China Ser A , 1999, 42(8): 805-815 doi: 10.1007/BF02884267
2	Chen M F. Explicit bounds of the first eigenvalue. Sci China Ser A , 2000, 43(10): 1051-1059 doi: 10.1007/BF02898239
3	Chen M F. Variational formulas and approximation theorems for the first eigenvalue in dimension one. Sci China Ser A , 2001, 44(4): 409-418 doi: 10.1007/BF02881877
4	Chen M F. Eigenvalues, Inequalities, and Ergodic Theory. New York: Springer, 2005
5	Chen M F. Speed of stability for birth-death process. Front Math China , 2010, 5(3): 379-516 doi: 10.1007/s11464-010-0068-7
6	Chen M F. General estimate of the first eigenvalue on manifolds. Front Math China , 2011, 6(6): 1025-1043 doi: 10.1007/s11464-011-0164-3
7	Chen M F. Basic estimates and stability rate for one-dimensional diffusion. In: Probability Approximations and Beyond . Lecture Notes in Statistics, Vol 205. 2012, 75-99 doi: 10.1007/978-1-4614-1966-2_6
8	Chen M F. Lower bounds of principle eigenvalue in dimension one. Front Math China , 2012, 7(4): 645-668 doi: 10.1007/s11464-012-0223-4
9	Chen M F, Wang F Y. Estimation of spectral gap for elliptic operators. Trans Amer Math Soc , 1997, 349(3): 1239-1267 . See also book [4] for some supplement: http://math.bnu.edu.cn/~chenmf/main_eng.htm
10	Wang J. First eigenvalue of one-dimensional diffusion processes. Elect Commun Probab , 2009, 14: 232-244 doi: 10.1214/ECP.v14-1464
11	Zettl A. Sturm-Liouville Theory. Providence: Amer Math Soc, 2005

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