Partial versus total resetting for L\'evy flights in d dimensions: similarities and discrepancies

TL;DR

This paper compares partial and total stochastic resetting in Le9vy flights and Brownian motion across multiple dimensions, revealing key similarities and differences in their stationary states and dynamical behaviors.

Contribution

It provides explicit formulas for propagators and stationary measures under PSR, and uncovers a dynamical phase transition unique to Brownian motion.

Findings

Stationary measures derived for both Le9vy flights and Brownian motion with PSR.

A dynamical phase transition occurs in Brownian motion but not in Le9vy flights.

Significant differences found in process behavior near the resetting position.

Abstract

While stochastic resetting (or total resetting) is less young and more established concept in stochastic processes, partial stochastic resetting (PSR) is a relatively new field. PSR means that, at random moments in time, a stochastic process gets multiplied by a factor between 0 and 1, thus approaching but not reaching the resetting position. In this paper, we present new results on PSR highlighting the main similarities and discrepancies with total resetting. Specifically, we consider both symmetric $\alpha$-stable L\'evy processes (L\'evy flights) and Brownian motion with PSR in arbitrary d dimensions. We derive explicit expressions for the propagator and its stationary measure, and discuss in detail their asymptotic behavior. Interestingly, while approaching to stationarity, a dynamical phase transition occurs for the Brownian motion, but not for L\'evy flights. We also analyze the behavior of the process around the resetting position and find significant differences between PSR and total resetting.