Open Access
2023 Evolution of a passive particle in a one-dimensional diffusive environment
François Huveneers, François Simenhaus
Author Affiliations +
Electron. J. Probab. 28: 1-31 (2023). DOI: 10.1214/22-EJP896


We study the behavior of a tracer particle driven by a one-dimensional fluctuating potential, defined initially as a Brownian motion, and evolving in time according to the heat equation. We obtain two main results. First, in the short time limit, we show that the fluctuations of the particle become Gaussian and sub-diffusive, with dynamical exponent 34. Second, in the long time limit, we show that the particle is trapped by the local minima of the potential and evolves diffusively i.e. with exponent 12.

Funding Statement

F. H. and F. S. are supported by the ANR-15-CE40-0020-01 grant LSD. F. S. is supported by the ANR/FNS-16-CE93-0003 grant MALIN.


We thank M. Jara for useful discussions on the process X evolving in a rough environment discussed in the Introduction.


Download Citation

François Huveneers. François Simenhaus. "Evolution of a passive particle in a one-dimensional diffusive environment." Electron. J. Probab. 28 1 - 31, 2023.


Received: 16 July 2021; Accepted: 22 December 2022; Published: 2023
First available in Project Euclid: 15 January 2023

MathSciNet: MR4533739
arXiv: 2012.08394
Digital Object Identifier: 10.1214/22-EJP896

Primary: 60F17 , 60G15 , 60G50

Keywords: limit theorems , random walks in dynamical random environment , scaling limits

Vol.28 • 2023
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