Exact joint distributions of three global characteristic times for Brownian motion

Abstract

We consider three global characteristic times for a one-dimensional Brownian motion $x(\tau )$ in the interval $\tau \in [0,t]$: the occupation time $t_{\mathrm{o}}$ denoting the cumulative time where $x(\tau )>0$, the time $t_{\mathrm{m}}$ at which the process achieves its global maximum in $[0,t]$ and the last-passage time $t_l$ through the origin before $t$. All three random variables have the same marginal distribution given by Lévy's arcsine law. We compute exactly the pairwise joint distributions of these three times and show that they are quite different from each other. The joint distributions display rather rich and nontrivial correlations between these times. Our analytical results are verified by numerical simulations.

arXiv commentnew version merges paper and supplementary material, more discussion of results and physical behaviour, 19 pages with 10 figures

References in corpus (54)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Statistical Aging and Non Ergodicity in the Fluorescence of Single Nanocrystals
• Stochastic Ergodicity Breaking: a Random Walk Approach
• Extreme value statistics of correlated random variables: a pedagogical review
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals
• Exact Maximal Height Distribution of Fluctuating Interfaces
• 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
• The Local and the Occupation Time of a Particle Diffusing in a Random Medium
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• Spatial extent of an outbreak in animal epidemics
• On the time to reach maximum for a variety of constrained Brownian motions
• Safe Leads and Lead Changes in Competitive Team Sports
• Generalized arcsine laws for fractional Brownian motion
• Properties of additive functionals of Brownian motion with resetting
• Hitting probability for anomalous diffusion processes
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• Generalised `Arcsine' laws for run-and-tumble particle in one dimension
• Time to reach the maximum for a random acceleration process
• The convex hull of the run-and-tumble particle in a plane
• Diffusive persistence and the `sign-time' distribution
• Large-Deviation Functions for Nonlinear Functionals of a Gaussian Stationary Markov Process
• Distribution of the time at which N vicious walkers reach their maximal height
• Residence Time Statistics for Normal and Fractional Diffusion in a Force Field
• Residence Time Distribution for a Class of Gaussian Markov Processes
• Statistical Properties of Functionals of the Paths of a Particle Diffusing in a One-Dimensional Random Potential
• Extremal statistics for stochastic resetting systems
• Extreme-Value Statistics of Fractional Brownian Motion Bridges
• Occupation time of a run-and-tumble particle with resetting
• Generalized persistence exponents: an exactly soluble model
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Time between the maximum and the minimum of a stochastic process
• Sign-time distributions for interface growth
• Near-extreme statistics of Brownian motion
• Distribution of the Time Between Maximum and Minimum of Random Walks
• Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
• The convex hull for a random acceleration process in two dimensions
• Extreme value statistics and arcsine laws for heterogeneous diffusion processes
• Time to reach the maximum for a stationary stochastic process
• Mean area of the convex hull of a run and tumble particle in two dimensions
• Stationary definition of persistence for finite temperature phase ordering
• Arcsine Laws in Stochastic Thermodynamics
• On certain functionals of the maximum of Brownian motion and their applications
• Residence time statistics for $N$ blinking quantum dots and other stochastic processes
• Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models
• Statistics of occupation time and connection to local properties of non-homogeneous random walks
• Statistics of the occupation time for a class of Gaussian Markov processes
• Dynamical phase transition in the occupation fraction statistics for non-crossing Brownian particles
• Sampling first-passage times of fractional Brownian Motion using adaptive bisections
• Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier
• Last-passage time for linear diffusions and application to the emptying time of a box
• Occupation time of a system of Brownian particles on the line with steplike initial condition
• Generalized arcsine laws for a sluggish random walker with subdiffusive growth

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