Quasilimiting behavior for one-dimensional diffusions with killing

TL;DR

This paper investigates the conditions under which one-dimensional diffusions with killing converge to quasistationary distributions, extending previous results by analyzing the role of the killing rate at infinity and the eigenfunction's integrability.

Contribution

It provides a complete characterization of convergence to quasistationarity based on the comparison between killing at infinity and the principal eigenvalue, clarifying when the eigenfunction is integrable.

Findings

Convergence occurs if and only if the principal eigenfunction is integrable when the killing at infinity differs from the eigenvalue.

If killing at infinity exceeds the principal eigenvalue, the eigenfunction is always integrable.

When killing at infinity is smaller, integrability depends on the recurrence of the process.

Abstract

This paper extends and clarifies results of Steinsaltz and Evans [Trans. Amer. Math. Soc. 359 (2007) 1285-1234], which found conditions for convergence of a killed one-dimensional diffusion conditioned on survival, to a quasistationary distribution whose density is given by the principal eigenfunction of the generator. Under the assumption that the limit of the killing at infinity differs from the principal eigenvalue we prove that convergence to quasistationarity occurs if and only if the principal eigenfunction is integrable. When the killing at infinity is larger than the principal eigenvalue, then the eigenfunction is always integrable. When the killing at infinity is smaller, the eigenfunction is integrable only when the unkilled process is recurrent; otherwise, the process conditioned on survival converges to 0 density on any bounded interval.