Stochastic Ornstein-Uhlenbeck capacitors

We introduce and study a class of random capacitor systems which are both charged and discharged stochastically. A capacitor is ‘fed’ by a random inflow with stationary and independent increments. Discharging occurs according to a Markovian rate which is linear in the capacitor’s level. The resulting capacitor dynamics are Markovian, stochastically cyclic, and regenerative. We coin these systems “Lévy-charged Ornstein-Uhlenbeck capacitors”. Various random quantities associated with these systems are analyzed, including: the time-todischarge; the duration of the charging cycle; the trajectory and the peak height of the capacitor level during a charging cycle; and, the capacitor’s stationary equilibrium level. Furthermore, we show that there are sharp distinctions between these capacitor systems and corresponding ‘standard’ Lévy-driven Ornstein-Uhlenbeck systems.