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 $\beta$, 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 rigourous 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 \cite{evans_majumdar_2011b}, with resetting rate $\beta_Y.$ 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 $\beta$ and $\beta_Y$, 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.