Power-law relaxation of a confined diffusing particle subject to resetting with memory

Abstract

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.