# Non-renewal resetting of scaled Brownian motion

## TL;DR

This paper explores a stochastic process combining scaled Brownian motion with stochastic resetting, revealing complex behaviors like stagnation and non-stationarity depending on resetting parameters, with implications for understanding non-renewal dynamics.

## Contribution

It introduces and analyzes a non-renewal resetting model for scaled Brownian motion, highlighting how resetting influences non-stationary behaviors and probability distributions.

## Key findings

- Mean squared displacement can stagnate without reaching a steady state.
- Resetting can either have negligible effects or significantly slow down the motion.
- The process exhibits rich, parameter-dependent behaviors distinct from fully renewal processes.

## Abstract

We investigate an intermittent stochastic process, in which the diffusive motion with time-dependent diffusion coefficient $D(t)\sim t^{\alpha-1}$, $\alpha>0$ (scaled Brownian motion), is stochastically reset to its initial position and starts anew. The resetting follows a renewal process with either exponential or power-law distribution of the waiting times between successive renewals. The resetting events, however, do not affect the time dependence of the diffusion coefficient, so that the whole process appears to be a non-renewal one. We discuss the mean squared displacement of a particle and probability density function of its positions in such a process. We show that scaled Brownian motion with resetting demonstrates a rich behavior whose properties essentially depend on the interplay of the parameters of the resetting process and the particle's displacement in a free motion. The motion of particles can remain either almost unaffected by resetting, but can also get slowed down or even be completely suppressed. Especially interesting are the nonstationary situations in which the mean squared displacement stagnates but the distribution of positions does not tend to any steady state. \color{black} This behavior is compared to the situation (discussed in the other paper of this series) in which the memory on the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. We show that the properties of the probability densities in such processes (erazing or retaining the memory on the diffusion coefficient) are vastly different. \color{black}