Lower bounds of principal eigenvalue in dimension one

Abstract

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case (one-dimensional diffusions) and ends at the discrete one (birth-death processes). Possible generalization of the results studied here is discussed at the end of the paper.

Keywords

Principal eigenvalue / lower estimate / variational formula / one-dimensional diffusion / birth-death process

References

[1] Chen M F. Explicit bounds of the first eigenvalue. Sci China, Ser A, 2000, 43(10): 1051-1059

[2] Chen M F. Eigenvalues, Inequalities, and Ergodic Theory. London: Springer, 2005

[3] Chen M F. Speed of stability for birth-death processes. Front Math China, 2010, 5(3): 379-515

[4] Chen M F. Basic estimates of stability rate for one-dimensional diffusions. In: Barbour A, Chan H P, Siegmund D, eds. Probability Approximations and Beyond. Lect Notes Stat, Vol 205. Berlin: Springer, 2012, Chapter 6

[5] Chen M F, Zhang Y H, Zhao X L. Dual variational formulas for the first Dirichlet eigenvalue on half-line. Sci China, Ser A, 2003, 46(6): 847-861

[6] Fukushima M, Uemura T. Capacitary bounds of measures and ultracontracitivity of time changed processes. J Math Pures Appl, 2003, 82(5): 553-572

[7] Maz’ya V. Sobolev Spaces with Applications to Elliptic Partial Differential Equations. 2nd ed. Berlin: Springer, 2011

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg