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The scaled Brownian motion (SBM) is regarded as one of the paradigmatic random processes, featuring the anomalous diffusion property characterized by the diffusion exponent. It is a Gaussian, self-similar process with independent…

Probability · Mathematics 2024-04-29 Hubert Woszczek , Aleksei Chechkin , Agnieszka Wylomanska

We consider the one-dimensional target search process that involves an immobile target located at the origin and $N$ searchers performing independent Brownian motions starting at the initial positions $\vec x = (x_1,x_2,..., x_N)$ all on…

Statistical Mechanics · Physics 2010-09-16 P. L. Krapivsky , Satya N. Majumdar , Alberto Rosso

We calculate analytically the probability density $P(t_m)$ of the time $t_m$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the…

Statistical Mechanics · Physics 2008-02-25 Julien Randon-Furling , Satya N. Majumdar

We study the non-equilibrium steady states and first passage properties of a Brownian particle with position $X$ subject to an external confining potential of the form $V(X)=\mu|X|$, and that is switched on and off stochastically. Applying…

Statistical Mechanics · Physics 2020-11-11 Gabriel Mercado-Vásquez , Denis Boyer , Satya N. Majumdar , Grégory Schehr

We provide an exact formula for the mean first-passage time (MFPT) to a target at the origin for a single particle diffusing on a $d$-dimensional hypercubic {\em lattice} starting from a fixed initial position $\vec R_0$ and resetting to…

Statistical Mechanics · Physics 2026-04-30 Alexander K. Hartmann , Satya N. Majumdar

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…

Statistical Mechanics · Physics 2025-08-18 Yihao Wang , Hanshuang Chen

We combine the processes of resetting and first-passage to define \emph{first-passage resetting}, where the resetting of a random walk to a fixed position is triggered by a first-passage event of the walk itself. In an infinite domain,…

Statistical Mechanics · Physics 2021-06-22 B. De Bruyne , J. Randon-Furling , S. Redner

We study the statistical properties of the time-averaged mean-square displacements (TAMSD). This is a standard non-local quadratic functional for inferring the diffusion coefficient from an individual random trajectory of a diffusing tracer…

Statistical Mechanics · Physics 2020-01-03 Alexei Andreanov , Denis Grebenkov

We consider Brownian motion under resetting in higher dimensions for the case when the return of the particle to the origin occurs at a constant speed. We investigate the behavior of the probability density function (PDF) and of the…

Statistical Mechanics · Physics 2020-09-23 Anna S. Bodrova , Igor M. Sokolov

The strategy of stochastic resetting is known to expedite the first passage to a target, in diffusive systems. Consequently, the mean first passage time is minimized at an optimal resetting parameter. With Poisson resetting, vanishing…

Soft Condensed Matter · Physics 2023-03-08 Saeed Ahmad , Dibyendu Das

Diffusion with an incorporated resetting mechanism provides a reference framework for modeling a wide range of natural phenomena. Within this framework, the optimal resetting rate is a key quantity that arises from the optimization of the…

Statistical Mechanics · Physics 2026-05-12 Pedro Julián-Salgado , Pavel Castro-Villarreal , Leonardo Dagdug , Denis Boyer

We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time $t_f$ is finite and the searcher returns to its starting point at $t_f$. This is simply a Brownian motion with a Poissonian…

Statistical Mechanics · Physics 2022-05-23 Benjamin De Bruyne , Satya N. Majumdar , Gregory Schehr

We investigate an intermittent stochastic process, in which the diffusive motion with time-dependent diffusion coefficient $D(t)\sim t^{\alpha-1}$, $\alpha>0$ (scaled Brownian motion), is stochastically reset to its initial position and…

Statistical Mechanics · Physics 2019-07-24 Anna S. Bodrova , Aleksei V. Chechkin , Igor M. Sokolov

We study a one-dimensional gas of $N$ Brownian particles that diffuse independently, but are {\it simultaneously} reset to the origin at a constant rate $r$. The system approaches a non-equilibrium stationary state (NESS) with long-range…

Statistical Mechanics · Physics 2025-11-11 Marco Biroli , Hernan Larralde , Satya N. Majumdar , Gregory Schehr

We consider the dynamical evolution of a Brownian particle undergoing stochastic resetting, meaning that after random periods of time it is forced to return to the starting position. The intervals after which the random motion is stopped…

Statistical Mechanics · Physics 2022-07-19 Mattia Radice

We study the correlations between the maxima $m$ and $M$ of a Brownian motion (BM) on the time intervals $[0,t_1]$ and $[0,t_2]$, with $t_2>t_1$. We determine exact forms of the distribution functions $P(m,M)$ and $P(G = M - m)$, and…

Statistical Mechanics · Physics 2016-08-23 O. Benichou , P. L. Krapivsky , C. Mejia-Monasterio , G. Oshanin

Diffusion in a confining potential offers a minimal setting to understand the interplay between random motion and deterministic forces driving a particle towards a focal point or potential minimum. In continuous space and time, two…

Statistical Mechanics · Physics 2026-05-12 Debraj Das , Luca Giuggioli

We consider branching Brownian motion in which initially there is one particle at $x$, particles produce a random number of offspring with mean $m+1$ at the time of branching events, and each particle branches at rate $\beta = 1/2m$.…

Probability · Mathematics 2023-10-03 Pascal Maillard , Jason Schweinsberg

We study the maximum of Branching Brownian motion (BBM) with branching rates that vary in space, via a periodic function of a particle's location. This corresponds to a variant of the F-KPP equation in a periodic medium, extensively studied…

Probability · Mathematics 2020-05-22 Eyal Lubetzky , Chris Thornett , Ofer Zeitouni

We study the position distribution of an active Brownian particle (ABP) in the presence of stochastic resetting in two spatial dimensions. We consider three different resetting protocols : (I) where both position and orientation of the…

Statistical Mechanics · Physics 2021-04-20 Vijay Kumar , Onkar Sadekar , Urna Basu