On a diffusion which stochastically restarts from moving random spatial positions: a non-renewal framework

Abstract

We consider a diffusive particle that at random times, exponentially distributed with parameter β, stops its motion and restarts from a moving random position Y(t) in space. The position X(t) of the particle and the restarts do not affect the dynamics of Y(t), so our framework constitutes in a non-renewal one. We exhibit the feasibility to build a rigorous general theory in this setup from the analysis of sample paths. To prove the stochastic process X(t) has a non-equilibrium steady-state, assumptions related to the confinement of Y(t) have to be imposed. In addition we design a detailed example where the random restart positions are provided by the paradigmatic Evans and Majumdar’s diffusion with stochastic resettings (Evans M and Majumdar S 2011 Phys. Rev. Lett. 106 160601), with resetting rate β Y . We show the ergodic property for the main process and for the stochastic process of jumps performed by the particle. A striking feature emerges from the examination of the jumps, since their negative covariance can be minimized with respect to both rates β and β Y , independently. Moreover we discuss the theoretical consequences that this non-renewal model entails for the analytical study of the mean first-passage time (FPT) and mean cost up to FPT.