First-passage and extreme value statistics for overdamped Brownian motion in a linear potential

2025/06/16 by Huang, Feng, Chen, Hanshuang · 1 citation
#FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2506.13112

Abstract

We investigate the first-passage properties and extreme-value statistics of an overdamped Brownian particle confined by an external linear potential V(x)=μ|x-x₀|, where μ>0 is the strength of the potential and x₀>0 is the position of the lowest point of the potential, which coincides with the starting position of the particle. The Brownian motion terminates whenever the particle passes through the origin at a random time t_f. Our study reveals that the mean first-passage time ⟨ t_f ⟩ exhibits a nonmonotonic behavior with respect to μ, with a unique minimum occurring at an optimal value of μ≃ 1.24468D/x₀, where D is the diffusion constant of the Brownian particle. Moreover, we examine the distribution P(M|x₀) of the maximum displacement M during the first-passage process, as well as the statistics of the time t_m at which M is reached. Intriguingly, there exists another optimal μ≃ 1.24011 D/x₀ that minimizes the mean time ⟨ t_m ⟩. All our analytical findings are corroborated through numerical simulations.

Citations

• Optimal Time to Sell a Stock in Black-Scholes Model: Comment on "Thou shall buy and hold", by A. Shiryaev, Z. Xu and X.Y. Zhou

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• The Thermodynamic Limit of Extreme First-Passage Times

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