On diffusions with stochastic resettings: noisy restarts, optimal rates and interaction modelling

Abstract

In this paper we address a model of particle diffusion, introduced by Evans and Majumdar (2011 J. Phys. A: Math. Theor. 44 435001) where, at random times, at rate , a particle resets to a position drawn from a probability distribution with density . We study an invariance law for the standard error of first-passage time observed by Reuveni (2016 Phys. Rev. Lett. 116 170601) and find that, when dealing with restarts, noise around the resetting position breaches such a law. We show that the mean first-passage time is finite and can be minimized with respect to , for a rather general class of functions . Moreover, we propose a model of a particle system evolution where the particles independently diffuse and reset but also at random times interact. We exhibit the non-equilibrium stationary distribution for the case of two interacting particles.