  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.

Statistical Mechanics