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 β. 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 α. For Poissonian resetting, stochastic interruption suppresses fluctuations, driving the area to be ergodic, regardless of the value of β. In contrast, power-law resetting yields a rich spectrum of ergodic regimes. For 0 < α< 1, the system displays weak ergodicity breaking and the maximum contribution dominates the area accumulated during the waiting times. For 1+β< α< 2, we unveil an ergodic transition exclusive to the subdiffusive regime at α=1+β, 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.
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BibTeX
@misc{mendez2026absolute,
title = {{Absolute area under a scaled Brownian motion with resetting}},
author = {Vicenç Méndez and Javier Cristín},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19985},
eprint = {2610.19985},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19985},
note = {CC0},
}
RIS
TY - UNPB
TI - Absolute area under a scaled Brownian motion with resetting
AU - Vicenç Méndez
AU - Javier Cristín
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19985
UR - https://arxiv.org/abs/2610.19985
AB - 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 β. 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 α. For Poissonian resetting, stochastic interruption suppresses fluctuations, driving the area to be ergodic, regardless of the value of β. In contrast, power-law resetting yields a rich spectrum of ergodic regimes. For 0 < α< 1, the system displays weak ergodicity breaking and the maximum contribution dominates the area accumulated during the waiting times. For 1+β< α< 2, we unveil an ergodic transition exclusive to the subdiffusive regime at α=1+β, 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.
ER -