Entrance boundary for standard processes with no negative jumps and its application to exponential convergence to the Yaglom limit
Kosuke Yamato
TL;DR
This work extends the entrance boundary theory to standard processes with no negative jumps by attaching the left boundary and forming a Feller process on an augmented state space. It then develops a scale-function framework that yields a meromorphic resolvent and a spectrum for the infinitesimal generator, established via the zeros of an entire function $Z^{(q)}(0)$. Under the instantaneous entrance and strong Feller conditions, the resolvent is compact and a spectral gap exists, enabling exponential convergence to the Yaglom limit with an explicit projection operator. These results unify boundary classifications, spectral theory, and quasi-stationary behavior, with implications for rates of convergence and stability of conditioned processes in non-compact settings.
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.