# The standard and the fractional Ornstein-Uhlenbeck process on a growing domain

## TL;DR

This paper investigates how diffusive and subdiffusive processes in a harmonic potential behave on growing or contracting domains, revealing that domain growth rate critically influences particle confinement and distribution, with distinct effects for Brownian and subdiffusive dynamics.

## Contribution

The study introduces a fractional Fokker-Planck framework for diffusive processes on dynamic domains, highlighting the impact of domain growth on particle confinement and distribution, and contrasting Brownian and subdiffusive behaviors.

## Key findings

- Fast domain growth can break particle confinement in a harmonic potential.
- Subdiffusive CTRWs are more sensitive to domain growth, leading to different regimes.
- Analytic and numerical results are validated by simulations.

## Abstract

We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing/contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers both the case of Brownian diffusion and the case of a subdiffusive Continuous-Time Random Walk (CTRW). We find a high sensitivity of the random walk properties to the details of the domain growth rate, which gives rise to a variety of regimes with extremely different behaviors. At the origin of this rich phenomenology is the fact that the walkers still move while they wait to jump, since they are dragged by the deterministic drift arising from the domain growth. Thus, the increasingly long waiting times associated with the ageing of the subdiffusive CTRW imply that, in the time interval between two consecutive jumps, the walkers might travel over much longer distances than in the normal diffusive case. This gives rise to seemingly counterintuitive effects. For example, on a static domain, both Brownian diffusion and subdiffusive CTRWs yield a stationary particle distribution with finite width when a harmonic potential is at play, thus indicating a confinement of the diffusing particle. However, for a sufficiently fast growing/contracting domain, this qualitative behavior breaks down, and differences between the Brownian case and the subdiffusive case are found. In the case of Brownian particles, a sufficiently fast exponential domain growth is needed to break the confinement induced by the harmonic force; in contrast, for subdiffusive particles such a breakdown may already take place for a sufficiently fast power-law domain growth. Our analytic and numerical results for both types of diffusion are fully confirmed by random walk simulations.