Lower bounds of principal eigenvalue in dimension one

TL;DR

This paper provides a new, more direct proof for lower bounds of the principal eigenvalue in one-dimensional diffusion and birth-death processes, improving existing estimates and exploring dual boundary conditions.

Contribution

It introduces an alternative proof method for basic eigenvalue estimates and extends the results to Neumann boundary conditions and discrete processes.

Findings

Improved lower bounds for principal eigenvalues in Dirichlet case

Extended estimates to Neumann boundary conditions

Application to both continuous diffusions and birth-death processes

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.