Conditioning to avoid bounded sets for a one-dimensional L\'{e}vy processes

TL;DR

This paper investigates the limiting behavior of one-dimensional Lévy processes conditioned to avoid certain bounded sets, including finite points, bounded F_sigma-sets, and integer lattices, as the conditioning time tends to infinity.

Contribution

It introduces new results on the asymptotic behavior of Lévy processes conditioned to avoid complex bounded sets over increasing time horizons.

Findings

Limit behavior characterized for processes avoiding finite points and F_sigma-sets.

Results extend to processes avoiding integer lattices with exponential clocks.

Provides a framework for understanding conditioned Lévy processes over large times.

Abstract

For several classes of bounded sets $A$, the limit of a one-dimensional L\'{e}vy process conditioned to avoid $A$ up to a parametrized random time which tends to infinity. For $A$ we take the set of finite points with several clocks and a bounded $F_\sigma$-set with exponential clock. We also take an integer lattice with exponential clock.