# Local time of diffusion with stochastic resetting

## TL;DR

This paper investigates the statistical properties of local time in diffusion processes with stochastic resetting, providing exact results for moments and distributions, and revealing a transition to Gaussian fluctuations due to resetting.

## Contribution

It extends previous analyses of local time to include stochastic resetting, deriving exact moments and distributions under various conditions, and highlighting the Gaussian nature of fluctuations with resetting.

## Key findings

- Exact expressions for local time moments and distributions.
- Resetting induces Gaussian fluctuations in local time.
- Long-time behavior simplifies the statistics of local time.

## 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.