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Rough analysis of two scale systems
[Analyse d’un système multi-échelles par la théorie des chemins rugueux]
Arnaud Debussche1 ; Martina Hofmanová2
1Univ Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France
2Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 2, pp. 223-270

We address a slow-fast system of coupled three dimensional Navier–Stokes equations where the fast component is perturbed by an additive Brownian noise. By means of the rough path theory, we establish the convergence in law of the slow component towards a Navier–Stokes system with an Itô–Stokes drift and a rough path driven transport noise. This gives an alternative, more general and direct proof to [9]. Notably, the limiting rough path is identified as a geometric rough path, which does not necessarily coincide with the Stratonovich lift of the Brownian motion.

Nous nous intéressons à un système multi-échelles d’équations de Navier–Stokes en dimension 3. La composante rapide est perturbée par un bruit blanc additif. Nous utilisons la théorie des chemins rugueux et établissons la convergence en loi de la composante lente vers la solution d’équations de Navier–Stokes contenant une dérive de Itô–Stokes et un bruit transport. Ceci donne une preuve alternative, plus générale et directe par rapport à [9]. Notamment, le chemin rugueux limite est identifié comme un chemin rugueux géométrique, qui ne coïncide pas nécessairement avec le chemin rugueux correspondant au produit Stratonovich.

Reçu le :
Accepté le :
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DOI : 10.5802/afst.1844
Classification : 35Q30, 60H15, 60L20
Keywords: Navier–Stokes equations, transport noise, Itô–Stokes drift, rough paths

Arnaud Debussche  1   ; Martina Hofmanová  2

1 Univ Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France
2 Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Arnaud Debussche; Martina Hofmanová. Rough analysis of two scale systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 2, pp. 223-270. doi: 10.5802/afst.1844
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