On quasi-ergodic distribution for one-dimensional diffusions

TL;DR

This paper investigates the existence and uniqueness of quasi-ergodic distributions for one-dimensional diffusions killed at zero, utilizing spectral theory and ultracontractivity properties of the semigroup.

Contribution

It establishes conditions under which a unique quasi-ergodic distribution exists for such diffusions, particularly when the semigroup is intrinsically ultracontractive.

Findings

Existence of a unique quasi-ergodic distribution under ultracontractivity.

Characterization of ultracontractivity for the killed semigroup.

Illustrative example demonstrating the theoretical results.

Abstract

In this paper, we study quasi-ergodicity for one-dimensional diffusion $X$ killed at 0, when 0 is an exit boundary and $+\infty$ is an entrance boundary. Using the spectral theory tool, we show that if the killed semigroup is intrinsically ultracontractive, then there exists a unique quasi-ergodic distribution for $X$. An example is given to illustrate the result. Moreover, the ultracontractivity of the killed semigroup is also studied.