Typical long time behaviour of ground state-transformed jump processes

TL;DR

This paper studies the long-term behavior of a class of ground state-transformed Le9vy processes, providing precise bounds and characterizations of their almost sure escape rates influenced by potential functions.

Contribution

It introduces a detailed analysis of the long-time behavior of ground state-transformed Le9vy processes, including explicit envelope bounds and the impact of parameters.

Findings

Derived precise lower and upper envelopes for process behavior

Characterized escape rates through integral tests

Highlighted parameter roles with specific examples

Abstract

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.