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Applications of the continuous-time ballot theorem to Brownian motion and related processes. (English) Zbl 1060.60046

Summary: Motivated by questions related to a fragmentation process which has been studied by D. Aldous, J. Pitman and J. Bertoin, we use the continuous-time ballot theorem to establish some results regarding the lengths of the excursions of Brownian motion and related processes. We show that the distribution of the lengths of the excursions below the maximum for Brownian motion conditioned to first hit \(\lambda >0\) at time \(t\) is not affected by conditioning the Brownian motion to stay below a line segment from \((0,c)\) to \((t,\lambda )\). We extend a result of J. Bertoin by showing that the length of the first excursion below the maximum for a negative Brownian excursion plus drift is a size-biased pick from all of the excursion lengths, and we describe the law of a negative Brownian excursion plus drift after this first excursion. We then use the same methods to prove similar results for the excursions of more general Markov processes.

MSC:

60G51 Processes with independent increments; Lévy processes
60C05 Combinatorial probability
60J75 Jump processes (MSC2010)
60J25 Continuous-time Markov processes on general state spaces
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