Span Observables: “When is a Foraging Rabbit No Longer Hungry?”
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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.
KeywordsSpan Random walk Diffusion First passage
After completion of this work we learned that for the drift-free case the time that the span first reaches one was already calculated in Phys. Rev. E 94, 062131 (2016).
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