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

## TL;DR

This paper extends stochastic resetting theory to include realistic space-time coupling, revealing shape invariance of steady-state distributions regardless of return protocols and speeds, with broad implications across physics and related fields.

## Contribution

It introduces a new model of stochastic resetting that accounts for finite, coupled space-time returns, and demonstrates shape invariance of the steady-state distribution.

## Key findings

- Steady-state distribution shape is independent of return protocol.
- A simple recipe for computing the steady-state distribution is provided.
- Certain processes exhibit invariant steady-states regardless of return speed.

## Abstract

Motion under stochastic resetting serves to model a myriad of processes in physics and beyond, but in most cases studied to date resetting to the origin was assumed to take zero time or a time decoupled from the spatial position at the resetting moment. However, in our world, getting from one place to another always takes time and places that are further away take more time to be reached. We thus set off to extend the theory of stochastic resetting such that it would account for this inherent spatio-temporal coupling. We consider a particle that starts at the origin and follows a certain law of stochastic motion until it is interrupted at some random time. The particle then returns to the origin via a prescribed protocol. We study this model and surprisingly discover that the shape of the steady-state distribution which governs the stochastic motion phase does not depend on the return protocol. This shape invariance then gives rise to a simple, and generic, recipe for the computation of the full steady-state distribution. Several case studies are analyzed and a class of processes whose steady-state is completely invariant with respect to the speed of return is highlighted. For processes in this class we recover the same steady-state obtained for resetting with instantaneous returns---irrespective of whether the actual return speed is high or low. Our work significantly extends previous results on motion with stochastic resetting and is expected to find various applications in statistical, chemical, and biological physics.