Counting statistics: a Feynman-Kac perspective

TL;DR

This paper develops a Feynman-Kac framework to analyze the distribution of hit counts in discrete-time random walks with scattering and absorption, providing new insights into classical problems like gambler's ruin and the arcsine law.

Contribution

It introduces a novel Feynman-Kac formalism for counting hits in random walks with absorption, extending classical results and deriving new distributional properties.

Findings

Derived evolution equation for generating functions of hit counts.

Connected moments of hit distribution to equilibrium density.

Generalized arcsine law for walks with absorption.

Abstract

By building upon a Feynman-Kac formalism, we assess the distribution of the number of hits in a given region for a broad class of discrete-time random walks with scattering and absorption. We derive the evolution equation for the generating function of the number of hits, and complete our analysis by examining the moments of the distribution, and their relation to the walker equilibrium density. Some significant applications are discussed in detail: in particular, we revisit the gambler's ruin problem and generalize to random walks with absorption the arcsine law for the number of hits on the half-line.