# On the Duration of the Longest Excursion

• F. B. Knight
Part of the Progress in Probability and Statistics book series (PRPR, volume 12)

## Abstract

Let Xt be a standard Markov process, and let 0 be a point regular of 0, so that there exists a continuous additive functional A(t) whose points of increase are contained in {t: Xt = 0}, called the local time at 0. Then A(t) is unique up to a constant factor which is of no relevance to the present paper (see [1] for a more complete discussion and references). We are concerned here with the durations of the excursions away from 0, which does not depend on the normalization of A(t). For arbitrary t ≥ 0, we set t0(t) = sup{s ≤ t: XS = 0}, tl(t) = inf{s ≥ t: XS = 0}, and d(t) = tl(t) - t0(t). Then we call d(t) the duration of the excursion from 0 containing t (we always assume X0 = 0, hence d(t) is well-defined when we permit d(t) = ∞).

## Keywords

Brownian Motion Local Time Strong Markov Property Routine Differentiation Brownian Excursion

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