Typical long-time behavior of ground state-transformed jump processes

We consider a class of Lévy-type processes derived via a Doob transform from Lévy processes conditioned by a control function called potential. These ground state transformed processes (also called [Formula: see text]-processes) have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Lévy generators perturbed by the potential. We derive precise upper envelopes for the almost sure long-time behavior of these ground state-transformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.