Flandoli, Franco On a probabilistic description of small scale structures in 3D fluids. (English) Zbl 1017.76074 Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 2, 207-228 (2002). Summary: We introduce a vortex structure based on three-dimensional Brownian motion. The interaction energy \(H_{xy}\) between different vortex filaments \((x+W_t)\) and \((y+W_t)\) is rigorously defined and proved to be finite. The divergence of \(H_{xy}\) as \(|x-y|\to 0\) is analyzed and used to prove that the total energy of vortex structure is finite, under suitable assumptions. We also establish a relation with the intersection local time. Cited in 1 ReviewCited in 12 Documents MSC: 76M35 Stochastic analysis applied to problems in fluid mechanics 76F55 Statistical turbulence modeling 60K40 Other physical applications of random processes Keywords:small scale structures; vortex structure; three-dimensional Brownian motion; interaction energy; total energy; intersection local time PDF BibTeX XML Cite \textit{F. Flandoli}, Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 2, 207--228 (2002; Zbl 1017.76074) Full Text: DOI Numdam EuDML OpenURL