Time-dependent density of diffusion with stochastic resetting is invariant to return speed

Abstract

The canonical Evans-Majumdar model for diffusion with stochastic resetting to the origin assumes that resetting takes zero time: upon resetting the diffusing particle is teleported back to the origin to start its motion anew. However, in reality getting from one place to another takes a finite amount of time which must be accounted for as diffusion with resetting already serves as a model for a myriad of processes in physics and beyond. Here we consider a situation where upon resetting the diffusing particle returns to the origin at a finite (rather than infinite) speed. This creates a coupling between the particle's random position at the moment of resetting and its return time, and further gives rise to a non-trivial cross-talk between two separate phases of motion: the diffusive phase and the return phase. We show that each of these phases relaxes to the stead-state in a unique manner; and while this could have also rendered the total relaxation dynamics extremely non-trivial our analysis surprisingly reveals otherwise. Indeed, the time-dependent distribution describing the particle's position in our model is completely invariant to the speed of return. Thus, whether returns are slow or fast, we always recover the result originally obtained for diffusion with instantaneous returns to the origin.

Invariants of motion with stochastic resetting and space-time coupled returns

“…The results presented in this work complement those recently presented in [63,64] and significantly extend our knowledge on motion with stochastic resetting. All current formulations of such motion suffered from the same problem: when considering resetting they neglected the inherent spatio-temporal coupling that governs motion in our world.…”

“…Equations (6) and (8) constitute the principal set of equations that should be solved to obtain a full, time dependent, description of stochastic motion with resetting and space time coupled returns to the origin. A detailed account on how this can be done for simple Brownian motion is given in [64], but extending results there for an arbitrary stochastic processes obeying equation (1) currently seems very challenging. Thus, going forward, we focus our attention on steady-state properties of our process.…”

“…Consider our model for simple diffusion with a diffusion coefficient D, stochastic resetting rate r, and a constant return speed v r . We have recently shown that the total density, and the densities of the diffusive and return phases, can be computed for this case study at all times by solving equations (6) and (8) (see [64] for details). In particular, the steady-state solution can be obtained in this way, but in this subsection we will take an alternative approach and utilize the formalism prescribed above to get the same result a bit more directly.…”

“…Surprisingly, one can moreover show that for diffusion this invariance extends beyond the steady state, i.e. ρ(x, t)= x t , r ¥ ( )for any finite time t 0  , and we refer the reader to [64] for details.…”

“…Importantly, the analysis presented there does not make any assumptions on the underlying stochastic law of motion and is furthermore suited to treat generic space-time coupled returns and home-stay strategies. Later on, we have also shown that the time evolution of the position density of diffusion with resetting is invariant to the speed with which the particle returns to the origin [64]. Here, we study the steady state of a general stochastic motion with resetting and space-time coupled returns (figure 1).…”