Counting statistics: a Feynman-Kac perspective

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.

10 pages, 6 figures

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Cited by in corpus (6)

• Properties of branching exponential flights in bounded domains
• Branching exponential flights: travelled lengths and collision statistics
• Cauchy's formulas for random walks in bounded domains
• Strongly constrained stochastic processes: the multi-ends Brownian bridge
• Discrete Feynman-Kac formulas for branching random walks
• Surprising variants of Cauchy's formula for mean chord length