On higher-order affine distributions

The aim of the paper is to provide a discussion on the notion of nonlinear interaction between scales. This notion is extremely important in the study of nonlinear physical phenomena, such as the study of the energy cascade in fully developed turbulence in fluids. The discussion is centered on the temptative definition of higher-order time-multiscale distributions. A simple answer which we call the biscalogram enlightens the main difficulty arising in our attempt to define nonlinear interactions between scales. Some simulations illustrate that point. Furthermore, two different classes of higher-order time-multiscale distributions are proposed, the first one based on a set of affine covariance properties, the other on a set of invariance-covariance properties