Pairwise correlations of global times in one-dimensional Brownian motion under stochastic resetting

TL;DR

This paper analytically investigates the correlations among three global times in one-dimensional Brownian motion with stochastic resetting, revealing complex dependence on the resetting rate and providing insights into their joint distributions.

Contribution

It provides the first analytical computation of joint distributions and correlations among occupation, maximum, and last-passage times in resetting Brownian motion.

Findings

Occupation and last-passage times are uncorrelated for any integer m.

Occupation and maximum times are positively correlated, decaying with increasing resetting rate.

Correlation between maximum and last-passage times changes from positive to negative as resetting rate increases.

Abstract

Brownian motion with stochastic resetting-a process combining standard diffusion with random returns to a fixed position-has emerged as a powerful framework with applications spanning statistical physics, chemical kinetics, biology, and finance. In this study, we investigate the mutual correlations among three global characteristic times for one-dimensional resetting Brownian motion $x(\tau)$ over the interval $\tau \in \left[ 0, t\right] $: the occupation time $t_o$ spent on the positive semi-axis, the time $t_m$ at which $x(\tau)$ attains its global maximum, and the last-passage time $t_{\ell}$ when the process crosses the origin. For the process starting from the origin and undergoing Poissonian resetting back to the origin, we analytically compute the pairwise joint distributions of these three times (in the Laplace domain) and derive their pairwise correlation coefficients. Our results reveal that these global times display rich correlations, with a non-trivial dependence on the resetting rate $r$. Specifically, we find that (i) While $t_{o}$ and $t_{\ell}^m$ are uncorrelated for any positive integer $m$, $t_{o}^2$ and $t_{\ell}^m$ display anti-correlation; (ii) A positive correlation exists between $t_{o}$ and $t_{m}$, which decays toward zero following a logarithmically corrected power-law way with an exponent of $-2$ as $r \to \infty$; (iii) The correlation between $t_{m}$ and $t_{\ell}$ shifts from positive to negative as $r$ increases. All analytical predictions are validated by extensive numerical simulations.