First-passage statistics of random walks: a general approach via Riemann-Hilbert problems

TL;DR

This paper develops a universal method using Riemann-Hilbert problems to derive exact formulas for first-passage statistics of one-dimensional random walks, applicable to various jump distributions.

Contribution

It introduces a general approach to compute first-passage distributions via Riemann-Hilbert problems, valid for both continuous and discrete, symmetric and asymmetric jumps.

Findings

Derived explicit formulas for first-passage time and position distributions.

Applicable to a wide class of jump distributions, both symmetric and asymmetric.

Validated approach with explicit examples.

Abstract

We study first-passage statistics for one-dimensional random walks $S_n$ with independent and identically distributed jumps starting from the origin. We focus on the joint distribution of the first-passage time $\tau_b$ and first-passage position $S_{\tau_b}$ beyond a threshold $b\geq0$, as well as the distribution of $S_n$ for the walks that do not cross $b$ up to step $n$. By solving suitable Riemann-Hilbert problems, we are able to obtain exact and semi-explicit general formulae for the quantities of interest. Notably, such formulae are written solely in terms of the characteristic function of the jumps. In contrast with previous results, our approach is universally valid, applicable to both continuous and discrete, symmetric and asymmetric jump distributions. We complement our theoretical findings with explicit examples.