First-passage properties of the jump process with a drift. The general case

TL;DR

This paper analyzes the first-passage behavior of a drifted jump process with arbitrary light-tailed distributions, identifying regimes and deriving explicit decay rates and asymptotic properties.

Contribution

It introduces a mapping to a discrete-time random walk to analyze first-passage properties across different drift regimes, providing explicit decay rates and asymptotic formulas.

Findings

Explicit exponential decay rates in survival and absorption regimes

Algebraic decay characterization at the critical point

Asymptotic behavior of mean first-passage time and jump counts

Abstract

We study the first-passage properties of a jump process with constant drift where jump amplitudes and inter-arrival times follow arbitrary light-tailed distributions with smooth densities. Using a mapping to an effective discrete-time random walk, we identify three regimes determined by the drift strength: survival (weak drift), absorption (strong drift), and critical. We derive explicit expressions for exponential decay rates in the survival and absorption regimes, and characterize algebraic decay at the critical point. We also obtain asymptotic behavior of the mean first-passage time, number of jumps, and their variances for processes starting either close to the origin or far from it.