Local time of diffusion with stochastic resetting

Abstract

Diffusion with stochastic resetting has recently emerged as a powerful modeling tool with a myriad of potential applications. Here, we study local time in this model, covering situations of free and biased diffusion with, and without, the presence of an absorbing boundary. Given a Brownian trajectory that evolved for $t$ units of time, the local time is simply defined as the total time the trajectory spent in a small vicinity of its initial position. However, as Brownian trajectories are stochastic --- the local time itself is a random variable which fluctuates round and about its mean value. In the past, the statistics of these fluctuations has been quantified in detail; but not in the presence of resetting which biases the particle to spend more time near its starting point. Here, we extend past results to include the possibility of stochastic resetting with, and without, the presence of an absorbing boundary and/or drift. We obtain exact results for the moments and distribution of the local time and these reveal that its statistics usually admits a simple form in the long-time limit. And yet, while fluctuations in the absence of stochastic resetting are typically non-Gaussian --- resetting gives rise to Gaussian fluctuations. The analytical findings presented herein are in excellent agreement with numerical simulations.

Selected article for the special issue "New Trends in First-Passage Methods and Applications in the Life Sciences and Engineering" https://iopscience.iop.org/journal/1751-8121/page/New-trends-in-first-passage-methods

References in corpus (12)

• Physics of Microswimmers - Single Particle Motion and Collective Behavior
• First Passage Under Restart
• First order transition for the optimal search time of Lévy flights with resetting
• Diffusion in a potential landscape with stochastic resetting
• Dynamical transition in the temporal relaxation of stochastic processes under resetting
• First passage time statistics of Brownian motion with purely time dependent drift and diffusion
• Stochastic Search with Poisson and Deterministic Resetting
• Monotonous continuous-time random walks with drift and stochastic reset events
• Localization transition induced by learning in random searches
• Integral Fluctuation Theorems for Stochastic Resetting Systems
• Stochastic resetting in underdamped Brownian motion
• Statistical Properties of Functionals of the Paths of a Particle Diffusing in a One-Dimensional Random Potential