Interpolation process between standard diffusion and fractional diffusion
- Others:
- COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)
- Instituto Superior Técnico, Universidade Técnica de Lisboa (IST)
- Instituto Nacional de Matemática Pura e Aplicada (IMPA)
- Méthodes quantitatives pour les modèles aléatoires de la physique (MEPHYSTO-POST) ; Inria Lille - Nord Europe ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
- This work benefited from the support of the project EDNHS ANR-14-CE25-0011 of the French National Research Agency (ANR) and of the PHC Pessoa Project 37854WM. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovative programme (grant agreement No 715734). C.B. thanks the French National Research Agency (ANR) for its support through the grant ANR-15-CE40-0020-01 (LSD). P.G. thanks FCT/Portugal for support through the project UID/MAT/04459/2013. M.J. thanks CNPq for its support through the grant 401628/2012-4 and FAPERJ for its support through the grant JCNE E17/2012. M.J. was partially supported by NWO Gravitation Grant 024.002.003-NETWORKS and by MathAmSud grant LSBS-2014. M.S. thanks CAPES (Brazil) and IMPA (Instituto de Matematica Pura e Aplicada, Rio de Janeiro) for the post-doctoral fellowship, and also the Labex CEMPI (ANR-11-LABX0007-01) for its partial support.
- ANR-14-CE25-0011,EDNHS,Diffusion de l'énergie dans des systèmes hamiltoniens bruitésés(2014)
- ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)
- ANR-15-CE40-0020,LSD,Modèles stochastiques en grande dimension pour la physique statistique hors équilibre(2015)
Description
We consider a Hamiltonian lattice field model with two conserved quantities, energy and volume, perturbed by stochastic noise preserving the two previous quantities. It is known that this model displays anomalous diffusion of energy of fractional type due to the conservation of the volume [5, 3]. We superpose to this system a second stochastic noise conserving energy but not volume. If the intensity of this noise is of order one, normal diffusion of energy is restored while it is without effect if intensity is sufficiently small. In this paper we investigate the nature of the energy fluctuations for a critical value of the intensity. We show that the latter are described by an Ornstein-Uhlenbeck process driven by a Lévy process which interpolates between Brownian motion and the maximally asymmetric 3/2-stable Lévy process. This result extends and solves a problem left open in [4].
Abstract
International audience
Additional details
- URL
- https://hal.science/hal-01348503
- URN
- urn:oai:HAL:hal-01348503v2
- Origin repository
- UNICA