Controlling inertial active Brownian motion via stochastic resetting

Abstract: Inertia is intrinsic to many living and synthetic active systems, from animals and robotic agents to colloidal swimmers, and it strongly shapes transport. Many such systems employ intermittent restart protocols to regulate exploration. Stochastic resetting provides a theoretical framework for these strategies and a route to control nonequilibrium steady states, yet the role of inertia in reset-controlled active dynamics remains poorly understood. Here we study an inertial active Brownian particle subject to complete stochastic resetting of position, velocity, and orientation in two dimensions. Using a moment-generating framework together with the Final-Value Theorem, we derive closed-form steady-state moments up to fourth order as functions of inertia, activity, and reset rate. We show that inertia fundamentally modifies reset-controlled transport: at large reset rates the steady-state mean-squared displacement is suppressed much more strongly than in the overdamped limit, yielding enhanced localization near the reset point. At the same time, position excess-kurtosis phase diagrams reveal strongly non-Gaussian steady states characterized by a sharp central peak coexisting with heavy tails in the position distribution, indicating rare long excursions enabled by inertial persistence. The tail weight varies non-monotonically with reset rate, reflecting a competition between inertial momentum relaxation and resetting that selects an optimal regime maximizing rare excursions. Our results provide experimentally testable signatures of inertial effects in reset-controlled active systems.