Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

An entropic interpolation problem for incompressible viscous fluids

Marc Arnaudon, Ana Bela Cruzeiro, Christian Léonard, and Jean-Claude Zambrini

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In view of studying incompressible inviscid fluids, Brenier introduced in the late 80’s a relaxation of a geodesic problem addressed by Arnold in 1966. Instead of inviscid fluids, the present paper is devoted to incompressible viscous fluids. A natural analogue of Brenier’s problem is introduced, where generalized flows are no more supported by absolutely continuous paths, but by Brownian sample paths. It turns out that this new variational problem is an entropy minimization problem with marginal constraints entering the class of convex minimization problems.

This paper explores the connection between this variational problem and Brenier’s original problem. Its dual problem is derived and the general form of its solution is described. Under the restrictive assumption that the pressure is a nice function, the kinematics of its solution is made explicit and its relation with viscous fluid dynamics is discussed.


Afin d’étudier les fluides incompressibles non-visqueux, Brenier a introduit à la fin des années 80 une relaxation du problème géodésique posé par Arnold en 1966. Dans le présent article, nous nous intéressons aux fluides visqueux incompressibles. Nous définissons un analogue naturel du problème de Brenier, où les flots généralisés ne sont plus supportés par des trajectoires absolument continues, mais plutôt par des trajectoires browniennes. Ce nouveau problème variationnel devient un problème de minimisation d’entropie avec des contraintes de lois marginales, et il entre dans le cadre des problèmes de minimisation convexe. Nous étudions le lien entre ce problème variationnel et le problème originel de Brenier. Nous déterminons le problème dual et nous décrivons la forme générale de sa solution. Sous l’hypothèse additionnelle que la pression est une fonction régulière, nous déterminons la cinématique de la solution et sa relation avec la dynamique des fluides visqueux.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 3 (2020), 2211-2235.

Received: 3 November 2017
Revised: 5 July 2019
Accepted: 1 November 2019
First available in Project Euclid: 26 June 2020

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Mathematical Reviews number (MathSciNet)

Primary: 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 28D20: Entropy and other invariants 49Q20: Variational problems in a geometric measure-theoretic setting 49S05: Variational principles of physics (should also be assigned at least one other classification number in section 49) 60G99: None of the above, but in this section

Incompressible viscous fluids Entropy minimization Diffusion processes Convex duality Stochastic velocities


Arnaudon, Marc; Cruzeiro, Ana Bela; Léonard, Christian; Zambrini, Jean-Claude. An entropic interpolation problem for incompressible viscous fluids. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 3, 2211--2235. doi:10.1214/19-AIHP1036.

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