Ergodic properties of Brownian motion under stochastic resetting

E. Barkai, R. Flaquer-Galmés, V. Méndez

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5 Scopus citations


We study the ergodic properties of one-dimensional Brownian motion with resetting. Using generic classes of statistics of times between resets, we find respectively for thin- or fat-tailed distributions the normalized or non-normalized invariant density of this process. The former case corresponds to known results in the resetting literature and the latter to infinite ergodic theory. Two types of ergodic transitions are found in this system. The first is when the mean waiting time between resets diverges, when standard ergodic theory switches to infinite ergodic theory. The second is when the mean of the square root of time between resets diverges and the properties of the invariant density are drastically modified. We then find a fractional integral equation describing the density of particles. This finite time tool is particularly useful close to the ergodic transition where convergence to asymptotic limits is logarithmically slow. Our study implies rich ergodic behaviors for this nonequilibrium process which should hold far beyond the case of Brownian motion analyzed here.

Original languageEnglish
Article number064102
JournalPhysical Review E
Issue number6
StatePublished - Dec 2023

Bibliographical note

Publisher Copyright:
© 2023 American Physical Society.


The support of the Israel Science Foundation (E.B.) and the Spanish government (R.F., V.M.) under Grants No. 1614/21 and No. PID2021-122893NB-C22, respectively, is acknowledged. The authors also acknowledge the helpful comments and discussions with Dr. Axel Masó-Puigdellosas.

FundersFunder number
Israel Science FoundationPID2021-122893NB-C22, 1614/21


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