An active Brownian particle with inertia is periodically reset to its starting position. At high reset rates, it stays close — localized near the origin, unable to wander far before being pulled back. At low reset rates, it explores freely and the localization is weak. The overdamped version of this story is well-known.
Paoluzzi and Puglisi (arXiv:2602.21134) show that inertia changes the story in two ways. First, the localization at high reset rates is tighter than in overdamped systems — inertia enhances confinement because the particle hasn't had time to build up momentum before the next reset. The heavy particle is easier to control when resets are frequent.
Second, the steady-state position distribution has a sharp central peak coexisting with heavy tails. Most of the time, the particle is near the origin. Rarely, it escapes to large distances. This is not the Gaussian profile of overdamped resetting — it is a bimodal distribution in a sense, with the typical behavior (localized) and the rare behavior (escaped) coexisting in the same steady state.
The tail weight varies non-monotonically with reset rate. There is an optimal reset rate that maximizes the probability of rare long excursions. Reset too often and the particle can never build enough momentum to escape. Reset too rarely and the particle diffuses broadly but without the dramatic escapes. At an intermediate rate, the competition between momentum accumulation and resetting creates a window where the particle occasionally breaks free with unusual persistence.
The general observation: in systems with memory (inertia), periodic interruption does not simply suppress exploration. It reshapes the distribution of exploration, creating a regime where most trajectories are confined while a few are dramatically extended. The typical and the extreme are governed by different dynamics within the same system. Control makes the average behavior tighter while enabling the rare behavior to be wilder.