Diffusion with preferential relocation in a confining potential

We study the relaxation of a diffusive particle confined in an arbitrary external potential and subject to a non-Markovian resetting protocol. With a constant rate r, a previous time τ between the initial time and the present time t is chosen from a given probability distribution K ( τ , t ) , and the particle is reset to the position that it occupied at time τ. Depending on the shape of K ( τ , t ) , the particle either relaxes toward the Gibbs-Boltzmann distribution or toward a non-trivial stationary distribution that breaks ergodicity and depends on the initial position and the resetting protocol. From a general asymptotic theory, we find that if the kernel K ( τ , t ) is sufficiently localized near τ = 0, i.e. mostly the initial part of the trajectory is remembered and revisited, the steady state is non-Gibbs-Boltzmann. Conversely, if K ( τ , t ) decays slowly enough or increases with τ, i.e. recent positions are more likely to be revisited, the probability distribution of the particle tends toward the Gibbs-Boltzmann state at large times. In the latter case, however, the temporal approach to the stationary state is generally anomalously slow, following for instance an inverse power law or a stretched exponential, if K ( τ , t ) is not too strongly peaked at the current time t. These findings are verified by the analysis of several exactly solvable cases and by numerical simulations.