Basic Estimates of Stability Rate for One-dimensional Diffusions

TL;DR

This paper provides tight bounds for Poincaré-type inequalities in one-dimensional diffusions, offering new insights into their ergodic and decay rates with refined estimates and illustrative examples.

Contribution

It introduces new bounds for inequalities in one-dimensional diffusions, resolving open problems and refining previous estimates.

Findings

Bounds have a factor of 4 difference, providing tight estimates.

Bounds give exponential ergodic and decay rates.

Refined bounds are illustrated with typical examples.

Abstract

In the context of one-dimensional diffusions, we present basic estimates (having the same lower and upper bounds with a factor of 4 only) for four Poincar\'e-type (or Hardy-type) inequalities. The derivation of two estimates have been open problems for quite some time. The bounds provide exponentially ergodic or decay rates. We refine the bounds and illustrate them with typical examples.