Sign-changes as a universal concept in first-passage time calculations

TL;DR

This paper introduces an analytical method based on sign changes to calculate first-passage time distributions for non-differentiable Gaussian processes, overcoming previous limitations of boundary conditions and noise strengths.

Contribution

It generalizes crossing counting to sign changes, enabling analytical solutions for a broad class of first-passage problems in Gaussian processes.

Findings

Applicable to non-differentiable Gaussian processes

Works with various time-dependent boundaries

Handles different noise strengths effectively

Abstract

First-passage time problems are ubiquitous across many fields of study including transport processes in semiconductors and biological synapses, evolutionary game theory and percolation. Despite their prominence, first-passage time calculations have proven to be particularly challenging. Analytical results to date have often been obtained under strong conditions, leaving most of the exploration of first-passage time problems to direct numerical computations. Here we present an analytical approach that allows the derivation of first-passage time distributions for the wide class of non-differentiable Gaussian processes. We demonstrate that the concept of sign changes naturally generalises the common practice of counting crossings to determine first-passage events. Our method works across a wide range of time-dependent boundaries and noise strengths thus alleviating common hurdles in first-passage time calculations.