hal-01299360
https://enpc.hal.science/hal-01299360
https://enpc.hal.science/hal-01299360/document
https://enpc.hal.science/hal-01299360/file/1.4937364.pdf
doi:10.1063/1.4937364
[SDE] Sciences De l'Environnement
[ENPC] Ecole des Ponts ParisTech
[PARISTECH] ParisTech
[GIP-BE] GIP Bretagne Environnement
[HMCO] Hydrologie, Météorologie et Complexité
Multifractal vector fields and stochastic Clifford algebra
Schertzer, D
Tchiguirinskaia, Ioulia
[SDE] Environmental Sciences
[SDV.EE] Life Sciences [q-bio]/Ecology, environment
ART
In the mid 1980s, the development of multifractal concepts and techniques was an importantbreakthrough for complex system analysis and simulation, in particular, in turbulence andhydrology. Multifractals indeed aimed to track and simulate the scaling singularities of theunderlying equations instead of relying on numerical, scale truncated simulations or on simplifiedconceptual 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 thispaper that the combination of stable Levy processes with Clifford algebra is a good candidate tobridge up the present gap between theory and applications. We show that it indeed defines a convenientframework to generate multifractal vector fields, possibly multifractal manifold-valuedfields, based on a few fundamental and complementary properties of Levy processes and Cliffordalgebra. In particular, the vector structure of these algebra is much more tractable than the manifoldstructure of symmetry groups while the Levy stability grants a given statistical universality
2015-12
2016-04-07
en
Chaos: An Interdisciplinary Journal of Nonlinear Science
American Institute of Physics