Characteristic Functionals for Incompressible Turbulent Flows

The present approach to analyze turbulent flows is based on the notion of the characteristic functional; this is justified in this section. The Bochner-Minlos and Minlos-Sazonov theorems, stated without proofs, provide the conditions for a functional defined on a nuclear space as the domain of definition to be characteristic functional. Transformations of vector fields are a useful tool for the construction of bases for argument and phase spaces; the Plancherel theorem provides a relation between scalar products in the spaces involved. Relation of Pdfs and characteristic functions to statistical moments are briefly discussed including the extension to infinitely many variables in preparation of the next chapter. Variational derivatives of the characteristic functional are reviewed.