Absolute area under a scaled Brownian motion with resetting

Abstract

We investigate the statistical properties and ergodic behavior of the absolute area below the trajectory of a particle subject to stochastic resetting and moving according to a scaled Brownian motion with exponent $\beta$. We compare the contributions to the absolute area from the waiting times between resets and the backward recurrence time. Using the renewal formalism we derive the lower-order moments of the absolute area under both Poissonian and heavy-tailed power-law resetting waiting time distributions with exponent $\alpha$. For Poissonian resetting, stochastic interruption suppresses fluctuations, driving the area to be ergodic, regardless of the value of $\beta$. In contrast, power-law resetting yields a rich spectrum of ergodic regimes. For $0<\alpha <1$, the system displays weak ergodicity breaking and the maximum contribution dominates the area accumulated during the waiting times. For $1+\beta <\alpha <2$, we unveil an ergodic transition exclusive to the subdiffusive regime at $\alpha =1+\beta$, where ergodicity is restored. These results clarify the subtle interplay between anomalous diffusion and power-law waiting times between resettings. Our results are tested by numerical simulations exhibiting an excellent agreement.