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Related Experiment Video

Updated: Jan 2, 2026

Measurement & Analysis of the Temporal Discrimination Threshold Applied to Cervical Dystonia

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Published on: January 27, 2018

Time Between the Maximum and the Minimum of a Stochastic Process.

Francesco Mori¹, Satya N Majumdar¹, Grégory Schehr¹

¹LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.

Physical Review Letters

|December 7, 2019

Summary

We derived an exact solution for the time difference between extrema in Brownian motion and Brownian bridges. This finding is universal for random walks and applicable to Kardar-Parisi-Zhang interfaces.

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Area of Science:

Statistical Mechanics
Stochastic Processes
Mathematical Physics

Background:

Brownian motion is a fundamental model for random processes.
Understanding the distribution of extrema is crucial in various physical systems.

Purpose of the Study:

To derive the exact probability density function for the time difference between the minimum and maximum of a one-dimensional Brownian motion.
To generalize these findings to Brownian bridges and Kardar-Parisi-Zhang interfaces.
To investigate the universality of the Brownian motion result for discrete-time random walks and Lévy flights.

Main Methods:

Exact analytical solution for the probability density function.
Generalization to Brownian bridges and application to Kardar-Parisi-Zhang interfaces.
Numerical computation for Lévy flights.

Main Results:

An exact solution for P(τ=t_min-t_max|T) for Brownian motion and Brownian bridges.
Demonstrated applicability to (1+1)-dimensional Kardar-Parisi-Zhang interfaces.
Established universality for discrete-time random walks and highlighted deviations for Lévy flights.

Conclusions:

The derived solution provides a universal framework for analyzing time differences between extrema in stochastic processes.
The results offer insights into the behavior of fluctuating interfaces and random walks.
Lévy flights exhibit distinct behavior compared to Brownian motion in this context.