First-Passage Observables of $d$-dimensional Confined Jump Processes

TL;DR

This paper develops a comprehensive framework for calculating first-passage observables of confined $d$-dimensional jump processes, bridging the gap between discrete stochastic dynamics and continuous approximations in complex geometries.

Contribution

It introduces a systematic method to evaluate first-passage observables for $d$-dimensional confined jump processes, addressing limitations of continuous models and providing explicit asymptotic expressions.

Findings

Unified approach for edge and bulk regimes

Explicit asymptotic expressions for key observables

Illustrations with 2D disk and heavy-tailed processes

Abstract

First-passage observables (FPO) are central to understanding stochastic processes in confined domains, with applications spanning chemical reaction kinetics, foraging behavior, and molecular transport. While extensive analytical results exist for continuous processes, discrete jump processes -- crucial for describing empirically observed dynamics -- remain largely unexplored in this context. This paper presents a comprehensive framework to systematically evaluate FPO for $d$-dimensional confined isotropic jump processes, capturing both geometric and dynamical observables. Leveraging the connection between jump processes and their continuous counterparts, we address the limitations of continuous approximations in capturing discrete effects, particularly near absorbing boundaries. Our method unifies FPO calculations across edge and bulk regimes, providing explicit asymptotic expressions for key observables, such as splitting probabilities, harmonic measures, and first-passage time distributions in complex geometries. We illustrate our approach with paradigmatic examples, including eccentric splitting probabilities in two-dimensional disks and mean exit times for heavy-tailed processes. These results underscore the broad applicability of our framework to diverse physical systems, offering novel insights into the interplay between discrete dynamics and confining geometries.