The Annals of Probability

The evolution of a random vortex filament

Hakima Bessaih, Massimiliano Gubinelli, and Francesco Russo

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We study an evolution problem in the space of continuous loops in a three-dimensional Euclidean space modeled upon the dynamics of vortex lines in 3d incompressible and inviscid fluids. We establish existence of a local solution starting from Hölder regular loops with index greater than 1/3. When the Hölder regularity of the initial condition X is smaller or equal to 1/2, we require X to be a rough path in the sense of Lyons [Rev. Mat. Iberoamericana 14 (1998) 215–310, System Control and Rough Paths (2002). Oxford Univ. Press]. The solution will then live in an appropriate space of rough paths. In particular, we can construct (local) solution starting from almost every Brownian loop.

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Ann. Probab. Volume 33, Number 5 (2005), 1825-1855.

First available in Project Euclid: 22 September 2005

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Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals 76B47: Vortex flows

Vortex filaments rough path theory path-wise stochastic integration


Bessaih, Hakima; Gubinelli, Massimiliano; Russo, Francesco. The evolution of a random vortex filament. Ann. Probab. 33 (2005), no. 5, 1825--1855. doi:10.1214/009117905000000323.

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