# Span observables - "When is a foraging rabbit no longer hungry?"

## TL;DR

This paper analyzes the span of a random walk, deriving analytical formulas for the probability of reaching a span of 1, including effects of drift and boundaries, validated by simulations.

## Contribution

It provides new analytical results for the span distribution of random walks with boundaries and drift, including joint max-min probabilities.

## Key findings

- Derived formulas for span reaching probability and density.
- Extended analysis to include drift and reflecting boundaries.
- Validated analytical results with numerical simulations.

## Abstract

Be $X_t$ a random walk. We study its span $S$, i.e. the size of the domain visited up to time $t$. We want to know the probability that $S$ reaches $1$ for the first time, as well as the density of the span given $t$. Analytical results are presented, and checked against numerical simulations. We then generalize this to include drift, and one or two reflecting boundaries. We also derive the joint probability of the maximum and minimum of a process. Our results are based on the diffusion propagator with reflecting or absorbing boundaries, for which a set of useful formulas is derived.