Entrance boundary for standard processes with no negative jumps and its application to exponential convergence to the Yaglom limit

TL;DR

This paper analyzes standard processes with no negative jumps at the entrance boundary, showing they can be extended to Feller processes and establishing exponential convergence to the Yaglom limit under certain conditions.

Contribution

It demonstrates how to extend such processes to Feller processes and characterizes the spectrum of their generators, linking it to exponential convergence to the Yaglom limit.

Findings

Process can be extended to a Feller process by boundary attachment

Spectrum characterized as zeros of an entire function

Convergence to Yaglom limit is exponentially fast under strong Feller property

Abstract

We study standard processes with no negative jumps under the entrance boundary condition. Similarly to one-dimensional diffusions, we show that the process can be made into a Feller process by attaching the boundary point to the state space. We investigate the spectrum of the infinitesimal generator in detail via the scale function, characterizing it as the zeros of an entire function. As an application, we prove that under the strong Feller property, the convergence to the Yaglom limit of the process killed on hitting the boundary is exponentially fast.