Conditioning to avoid bounded sets for a one-dimensional Lévy processes

TL;DR

The paper develops a general framework for conditioning a one-dimensional Lévy process to avoid bounded sets $A$ using various clocks that tend to infinity, and expresses the limiting conditioned process as an $h$-transform with a nonnegative harmonic function $h_A$ (or its variants $h^{(\gamma)}$). It provides explicit constructions of the relevant harmonic functions $\varphi$ for configurations ranging from two points to $n$ points, bounded $F_\sigma$-sets, and an integer lattice, along with comprehensive limit theorems under exponential clocks, hitting-time clocks, and inverse local time clocks. Theoretical results are complemented by concrete examples (Brownian motion, stable processes, spectrally negative Lévy processes) and extended to excursion-theoretic cases for boundary cases like $x=a_n$. These contributions yield Doob $h$-transforms that characterize the long-time conditioned dynamics and offer explicit formulas and domains of admissible starting points. The methods have potential applications in stochastic conditioning, potential theory, and the study of conditioned Lévy processes in various geometries.

Abstract

For several classes of bounded sets $A$, the limit of a one-dimensional Lévy 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 $F_σ$-set with exponential clock. We also take an integer lattice with exponential clock.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1: Theorem 1.1 and Theorem 1.2 of TY
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3