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Probabilistic models for vortex filaments based on fractional Brownian motion. (English) Zbl 1011.60032

Experiments indicate that the vorticity field of turbulent fluids is concentrated along thin structures called vortex filaments. The authors describe a vortex filament by a trajectory of a three-dimensional fractional Brownian motion (fBM) \(B\) with Hurst parameter \(>1/2\) and assume that the vorticity field is concentrated along the trajectory. For similar models see A. J. Chorin [“Vorticity and turbulence” (1994; Zbl 0795.76002)] and F. Flandoli [Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 2, 207-228 (2002; Zbl 1017.76074)]. A basic problem is the computation of the kinetic energy \(H\) of the fluid. The authors derive for \(H\) the formal expression \[ H =\int\int H_{x,y} \rho(dx)\rho(dy), \] where the interaction energy \(H_{x,y}\) is given by \[ H_{x,y} = \Gamma^2(8\Pi)^{-1}\sum_{i=1}^3 \int_0^T\int_0^T |x+B_t-y-B_s|^{-1} dB_s^idB_t^i. \] \(\Gamma\) is the circuitation, \(T>0\) and \(\rho\) is a probability measure on Euclidean three space, which controls the spread of the vorticity around the filaments. The aim of the paper is to show that under a suitable integrability condition on \(\rho\) the kinetic energy \(H\) is a well-defined nonnegative random variable with moments of all orders. The proof is based on the stochastic calculus of variations. Another argument is to concentrate in a first step the vorticity field along a thin tube centred in a trajectory of the fBM \(B\) and letting in a second step the diameter of this tube converge to zero.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60H05 Stochastic integrals
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