Geodesic
Probabilistic models of vortex filaments
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 713-731 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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A model of vortex filaments based on stochastic processes is presented. In contrast to previous models based on semimartingales, here processes with fractal properties between $1/2$ and $1$ are used, which include fractional Brownian motion and similar non-Gaussian examples. Stochastic integration for these processes is employed to give a meaning to the kinetic energy.
A model of vortex filaments based on stochastic processes is presented. In contrast to previous models based on semimartingales, here processes with fractal properties between $1/2$ and $1$ are used, which include fractional Brownian motion and similar non-Gaussian examples. Stochastic integration for these processes is employed to give a meaning to the kinetic energy.
Classification : 60H05, 60H30, 76F55, 76M35
Keywords: stochastic integration; fractional Brownian motion; $p$-variation; vortex filaments; statistical fluid mechanics 
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Flandoli, Franco; Minelli, Ida. Probabilistic models of vortex filaments. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 713-731. http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a4/

[1] E.  Alòs and D.  Nualart: Stochastic calculus with respect to the fractional Brownian motion. Preprint.

[2] E.  Alòs, O.  Mazet and D.  Nualart: Stochastic calculus with respect to Gaussian processes. (to appear). | MR

[3] J.  Bell and D.  Marcus: Vorticity intensification and the transition to turbulence in the three-dimensional Euler equation. Comm. Math. Phys. 147 (1992), 371–394. | DOI | MR

[4] J.  Bertoin: Sur une integrale pour les processus à $\alpha $-variatione bornée. Ann. Probab. 17 (1989), 1521–1535. | DOI | MR

[5] H.  Bessaih: Mean field theory for 3-D vortex filaments. In preparation.

[6] E.  Bolthausen: On the construction of the three dimensional polymer measure. Probab. Theory Related Fields 97 (1993), 81–101. | DOI | MR | Zbl

[7] A.  Chorin: Vorticity and Turbulence. Springer-Verlag, New York, 1994. | MR | Zbl

[8] A.  Colesanti and M.  Romito: Some remarks on a probabilistic model of the vorticity field of a 3D fluid. Preprint.

[9] F.  Flandoli: On a probabilistic description of small scale structures in 3D fluids. (to appear). | MR | Zbl

[10] F. Flandoli and M.  Gubinelli: The Gibbs ensamble of a vortex filament. (to appear). | MR

[11] U.  Frisch: Turbulence. Cambridge University Press, Cambridge, 1995. | MR | Zbl

[12] G.  Gallavotti: Foundations of Fluid Mechanics. Quaderni FM2000–5, http://ipparco.roma1.infn.it/

[13] A. N.  Kolmogorov: The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. C. R. (Dokl.) Acad. Sci. USSR (N.S.) 30 (1941), 299–303. | MR

[14] N. S.  Landkof: Foundations of Modern Potential Theory. Springer-Verlag, New York, 1972. | MR | Zbl

[15] P. L.  Lions and A.  Majda: Equilibrium statistical theory for nearly parallel vortex filaments. Comm. Pure Appl. Math. 53 (2000), 76–142. | DOI | MR

[16] C.  Marchioro, M.  Pulvirenti: Mathematical Theory of Incompressible Nonviscous Fluids. Springer-Verlag, Berlin, 1994. | MR

[17] I.  Minelli: In preparation.

[18] D.  Nualart, C.  Rovira and S.  Tindel: Probabilistic models for vortex filaments based on fractional Brownian motion. In preparation.

[19] L.  Onsager: Statistical hydrodynamics. Nuovo Cimento (9) 6 (1949), Supplemento no. 2, 279–287. | DOI | MR

[20] Z. S.  She, E.  Jackson and S. A.  Orszag: Structure and dynamics of homogeneous turbulence: models and simulations. Proc. Roy. Soc. Lond. Ser. A 434 (1991), 101–124. | DOI

[21] B.  Toth and W.  Werner: The true self-repelling motion. Probab. Theory Relat. Fields 111 (1998), 375–452. | DOI | MR

[22] A.  Vincent and M.  Meneguzzi: The spatial structure and statistical properties of homogeneous turbulence. J.  Fluid Mech. 225 (1991), 1–25. | DOI

[23] J.  Westwater: On Edwards model for long polymer chains. Comm. Math. Phys. 72 (1980), 131–174. | DOI | MR | Zbl

[24] L. C.  Young: An inequality of Hölder type, connected with Stieltjes integration. Acta Math. 67 (1936), 251–282. | DOI | MR

[25] M.  Zähle: Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111 (1998), 333–374. | DOI | MR