Multifractal vector fields and stochastic Clifford algebra

D Schertzer (1) , Ioulia Tchiguirinskaia (1)

1 HM&Co - Hydrologie, Météorologie et Complexité

In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Levy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Levy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Levy stability grants a given statistical universality

Domaines: Sciences de l'environnement; Ecologie, Environnement

D Schertzer, Ioulia Tchiguirinskaia. Multifractal vector fields and stochastic Clifford algebra. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015, 25 (12), ⟨10.1063/1.4937364⟩. ⟨hal-01299360⟩