Statistical fluctuations under resetting: rigorous results

Abstract

In this paper we investigate the normal and the large fluctuations of additive functionals of the Brownian motion under a general non-Poissonian resetting mechanism. Cumulative functionals of regenerative processes are very close to renewal-reward processes and inherit most of the properties of the latter. Here we review and use the classical law of large numbers and central limit theorem for renewal-reward processes to obtain same theorems for additive functionals of the reset Brownian motion. Then, we establish large deviation principles for these functionals by illustrating and applying a large deviation theory for renewal-reward processes that has been recently developed by the author. We discuss applications of the general results to the positive occupation time, the area, and the absolute area. While introducing advanced tools from renewal theory, we demonstrate that a rich phenomenology accounting for dynamical phase transitions emerges when one goes beyond Poissonian resetting.