# Exact enumeration approach to first-passage time distribution of non-Markov random walks

## TL;DR

This paper introduces an exact enumeration method to analytically derive the first-passage time distribution for non-Markov random walks, including fractional Brownian motion, applicable to various scientific fields.

## Contribution

It presents a novel exact enumeration approach for calculating FPT distributions in non-Markov processes, extending analytical tools beyond Markovian assumptions.

## Key findings

- Derived exact FPT distribution for non-Markov walks
- Applied method to fractional Brownian motion with Hurst exponent H
- Demonstrated applicability to physics, biology, and geology phenomena

## Abstract

We propose an analytical approach to study non-Markov random walks by employing an exact enumeration method. Using the method, we derive an exact expansion for the first-passage time (FPT) distribution for any continuous, differentiable non-Markov random walk with Gaussian or non-Gaussian multivariate distribution. As an example, we study the FPT distribution of a fractional Brownian motion with a Hurst exponent $H\in(1/2,1)$ that describes numerous non-Markov stochastic phenomena in physics, biology and geology, and for which the limit $H=1/2$ represents a Markov process.