Explicit and sharp two-sided estimates for the killed Langevin process

TL;DR

This paper derives precise two-sided estimates for the transition density of a killed Langevin process with quadratic potential, revealing its long-term behavior and conditional ergodicity.

Contribution

It provides explicit sharp bounds and long-time asymptotics for the transition density of the killed Langevin process, a novel contribution in stochastic process analysis.

Findings

Explicit two-sided estimates for the transition density.

Long-time asymptotic behavior characterized.

Killed semigroup shown to be uniformly conditionally ergodic.

Abstract

We prove explicit and sharp two-sided estimates for the transition density of the Langevin process with quadratic potential, killed outside of the position interval (0,1). The long-time asymptotics of this transition density are also obtained. In particular, this allows us to show that the killed semigroup is uniformly conditionally ergodic.