^{1,a)}and Ioulia Tchiguirinskaia

^{1,b)}

### Abstract

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 Lévy 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 Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifoldstructure of symmetry groups while the Lévy stability grants a given statistical universality.

This research was partially supported by the Chair “Hydrology for Resilient Cities” endowed by Veolia (http://www.enpc.fr/node/1073) and by the RainGain project (www.raingain.eu) of the North West Europe Interreg program (https://www.nweurope.eu). D. Schertzer is grateful to L. Kavvas for his kind invitation and generous financial support to participate to the Orlob International Symposium on Theoretical Hydrology that he organized at UC Davis on August 5-6, 2013, and during which preliminary results of this paper were presented and discussed. We acknowledge stimulating and thoughtful comments and suggestions of the anonymous referees.

I. INTRODUCTION A. What is at stake? B. How is organized this paper? II. GENERALIZED SCALE INVARIANCE (GSI) III. PULLBACK AND PUSH-FORWARD TRANSFORMS IV. A GENERALIZED DEFINITION OF MULTIFRACTALS V. SCALAR-VALUED UNIVERSAL MULTIFRACTALS A. Fundamental properties of Lévy stable variables B. Lévy stable generators C. Universal multifractals VI. FROM PRODUCTS TO EXPONENTIAL: LIE ALGEBRA AND GROUPS VII. THE EXAMPLE OF PSEUDO-QUATERNIONS VIII. CLIFFORD ALGEBRA IX. CLIFFORD ALGEBRA AND UNIVERSAL MULTIFRACTALS X. CLIFFORD LAPLACE TRANSFORM AND FINITE STATISTICS XI. CONCLUSIONS

### Key Topics

- Lie algebras
- 10.0
- Fractals
- 9.0
- Manifolds
- 9.0
- Signal generators
- 9.0
- Algebraic structures
- 8.0

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### Abstract

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 Lévy 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 Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifoldstructure of symmetry groups while the Lévy stability grants a given statistical universality.

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