Local time of diffusion with stochastic resetting. (5th June 2019)

• Record Type: Journal Article
• Authors: Pal, Arnab; Chatterjee, Rakesh; Reuveni, Shlomi; Kundu, Anupam
• Is Part Of: Journal of physics. Volume 52:Number 26(2019)
• Publication Date: 2019-06-05
• Subjects: local time -- diffusion -- Brownian motion -- stochastic resetting -- first-passage; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ab2069
• Languages: English
• ISSNs: 1751-8113

Abstract

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.