  We study the relaxation of a Brownian particle with long range memory under
confinement in one dimension. The particle diffuses in an arbitrary confining
potential and resets at random times to previously visited positions, chosen
with a probability proportional to the local time spent there by the particle
since the initial time. This model mimics an animal which moves erratically in
its home range and returns preferentially to familiar places from time to time,
as observed in nature. The steady state density of the position is given by the
equilibrium Gibbs-Boltzmann distribution, as in standard diffusion, while the
transient part of the density can be obtained through a mapping of the
Fokker-Planck equation of the process to a Schr\"odinger eigenvalue problem.
Due to memory, the approach at late times toward the steady state is critically
self-organised, in the sense that it always follows a sluggish power-law form,
in contrast to the exponential decay that characterises Markov processes. The
exponent of this power-law depends in a simple way on the resetting rate and on
the leading relaxation rate of the Brownian particle in the absence of
resetting. We apply these findings to several exactly solvable examples, such
as the harmonic, V-shaped and box potentials.
