Nualart, D.; Rovira, C.; Tindel, S. Probabilistic models for vortex filaments based on fractional Brownian motion. (English) Zbl 1011.60032 RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 95, No. 2, 213-218 (2001). 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. Reviewer: Andreas Martin (Neuherberg/ München) Cited in 3 Documents MSC: 60H07 Stochastic calculus of variations and the Malliavin calculus 60H05 Stochastic integrals Keywords:vortex filaments; kinetic energy of fluid; fractional Brownian motion Citations:Zbl 0795.76002; Zbl 1017.76074 PDFBibTeX XMLCite \textit{D. Nualart} et al., RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 95, No. 2, 213--218 (2001; Zbl 1011.60032) Full Text: EuDML