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The study of diffusion with preferential returns to places visited in thepast has attracted an increased attention in recent years. In these highlynon-Markov processes, a standard diffusive particle intermittently resets at agiven rate to previously visited positions. At each reset, a position to berevisited is randomly chosen with a probability proportional to the accumulatedamount of time spent by the particle at that position. These preferentialrevisits typically generate a very slow diffusion, logarithmic in time, butstill with a Gaussian position distribution at late times. Here we consider anactive version of this model, where between resets the particle isself-propelled with constant speed and switches direction in one dimensionaccording to a telegraphic noise. Hence there are two sources ofnon-Markovianity in the problem. We exactly derive the position distribution inFourier space, as well as the variance of the position at all times. Thecrossover from the short-time ballistic regime, dominated by activity, to thelarge-time anomalous logarithmic growth induced by memory is studied. We alsoanalytically derive a large deviation principle for the position, whichexhibits a logarithmic time-scaling instead of the usual algebraic form.Interestingly, at large distances, the large deviations become independent oftime and match the non-equilibrium steady state of a particle under resettingto its starting position only.

An active particle in one dimension subjected to resetting with memory