  The Ornstein-Uhlenbeck (OU) process describes the dynamics of Brownian
particles in a confining harmonic potential, thereby constituting the
paradigmatic model of overdamped, mean-reverting Langevin dynamics. Despite its
widespread applicability, this model falls short when describing physical
systems where the confining potential is itself subjected to stochastic
fluctuations. However, such stochastic fluctuations generically emerge in
numerous situations, including in the context of colloidal manipulation by
optical tweezers, leading to inherently out-of-equilibrium trapped dynamics. To
explore the consequences of stochasticity at this level, we introduce a natural
extension of the OU process, in which the stiffness of the harmonic potential
is itself subjected to OU-like fluctuations. We call this model the OU$^2$
process. We examine its statistical, dynamic, and thermodynamic properties
through a combination of analytical and numerical methods. Importantly, we show
that the probability density for the particle position presents power-law
tails, in contrast to the Gaussian decay of the standard OU process. In turn,
this causes the trapping behavior, extreme value statistics, first passage
statistics, and entropy production of the OU$^2$ process to differ
qualitatively from their standard OU counterpart. Due to the wide applicability
of the standard OU process and of the proposed OU$^2$ generalisation, our study
sheds light on the peculiar properties of stochastic dynamics in random
potentials and lays the foundation for the refined analysis of the dynamics and
thermodynamics of numerous experimental systems.
