  We consider a class of L\'evy-type processes derived via a Doob-transform
from L\'evy processes conditioned by a control function called potential. These
processes have position-dependent and generally unbounded components, with
stationary distributions given by the ground states of the L\'evy generators
perturbed by the potential. We derive precise lower and upper envelopes for the
almost sure long time behaviour of these ground state-transformed L\'evy
processes, characterized through escape rates and integral tests. We also
highlight the role of the parameters by specific examples.
