Mixed principal eigenvalues in dimension one

TL;DR

This paper investigates the principal eigenvalues of elliptic operators in one dimension, providing variational formulas, explicit estimates, and criteria for positivity to understand the decay rates of diffusion processes.

Contribution

It introduces new variational formulas for mixed principal eigenvalues and offers explicit estimates and an approximation method, advancing the analysis of one-dimensional stochastic processes.

Findings

Derived variational formulas for eigenvalues

Established criteria for eigenvalue positivity

Developed an approximating procedure for eigenvalues

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 $L^{2}$-Poincar\'e 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.