Multiscale dynamics and symmetries: multifractals and stochastic Lie algebra

Multifractal concepts and techniques had been an important breakthrough in the mid-1980’s for multiscale analyses and simulations of complex systems. Multifractals went indeed well beyond uni-scaling approaches (e.g. spectral analysis or the fractal geometry) because track the full hierarchy of the scaling singularities of nonlinear systems. However, this development has been rather limited to scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. This has created an unfortunate gap between the theory and its applications.
We first show that in a general manner the extension of multifractals to these fields requires to define their infinitesimal generators on a stochastic Lie algebra, whose vector structure is much more tractable than the manifold structure of the corresponding symmetry groups.
We then demonstrate that the combination of the statistical properties of stability and attractivity of the evy processes and that of Clifford algebra yields a quite generic and robust framework. This is illustrated with the help of applications to the fundamental problem of turbulence.