The Annals of Probability

The evolution of a random vortex filament

Hakima Bessaih, Massimiliano Gubinelli, and Francesco Russo

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 22 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1127395875

Digital Object Identifier
doi:10.1214/009117905000000323

Mathematical Reviews number (MathSciNet)
MR2165581

Zentralblatt MATH identifier
1084.60030

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

Keywords
Vortex filaments rough path theory path-wise stochastic integration

Citation

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. https://projecteuclid.org/euclid.aop/1127395875


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