# Diffusion with stochastic resetting is invariant to return speed

## TL;DR

This paper demonstrates that in diffusion with stochastic resetting, the steady-state distribution remains unchanged regardless of the return speed, contrasting with the intuitive expectation that return speed would affect the dynamics.

## Contribution

The study shows that the steady-state distribution in diffusion with stochastic resetting is invariant to the return speed, extending the classical model to finite return velocities.

## Key findings

- Steady-state distribution is unaffected by return speed.
- Relaxation to steady state occurs uniquely for diffusive and return phases.
- Total relaxation dynamics remain simple despite coupling effects.

## 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.