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

Abstract

We investigate the first-passage properties and extreme-value statistics of an overdamped Brownian particle confined by an external linear potential $V(x)=\mu |x-x_0|$, where $\mu >0$ is the strength of the potential and $x_0>0$ 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 $\mu$, with a unique minimum occurring at an optimal value of $\mu \simeq 1.24468D/x_0$, where $D$ is the diffusion constant of the Brownian particle. Moreover, we examine the distribution $P(M|x_0)$ 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 $\mu \simeq 1.24011D/x_0$ that minimizes the mean time $⟨t_m⟩$. All our analytical findings are corroborated through numerical simulations.

8 pages, 4 figures

References in corpus (29)

• Intermittent search strategies
• Stochastic Resetting and Applications
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme value statistics of correlated random variables: a pedagogical review
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• Random Convex Hulls and Extreme Value Statistics
• Maximal Height Scaling of Kinetically Growing Surfaces
• Spatial extent of an outbreak in animal epidemics
• On the time to reach maximum for a variety of constrained Brownian motions
• Generalized arcsine laws for fractional Brownian motion
• Hitting probability for anomalous diffusion processes
• Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem
• Time to reach the maximum for a random acceleration process
• Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries
• The first-passage area for drifted Brownian motion and the moments of the Airy distribution
• Near-extreme statistics of Brownian motion
• Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
• Temporal correlations of the running maximum of a Brownian trajectory
• Splitting Probabilities of Jump Processes
• Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
• Discrete Sampling of Extreme Events Modifies Their Statistics
• Universal survival probability for a correlated random walk and applications to records
• Statistics of the first passage area functional for an Ornstein-Uhlenbeck process
• Joint statistics of space and time exploration of $1d$ random walks
• First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion
• Mean perimeter of the convex hull of a random walk in a semi-infinite medium