Brownian motion conditioned to spend limited time below a barrier

Frank Aurzada and Dominic T. Schickentanz

Stochastic Processes and their Applications, 2022, vol. 146, issue C, 360-381

Abstract: We condition a Brownian motion with arbitrary starting point y∈R on spending at most 1 time unit below 0 and provide an explicit description of the resulting process. In particular, we provide explicit formulas for the distributions of its last zero g=gy and of its occupation time Γ=Γy below 0 as functions of y. This generalizes Theorem 4 of Benjamini and Berestycki (2011), which covers the special case y=0. Additionally, we study the behavior of the distributions of gy and Γy, respectively, for y→±∞.

Date: 2022
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DOI: 10.1016/j.spa.2022.01.007

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