Stochastic invariances and Lamperti transformations for Stochastic Processes

Pierre Borgnat (1) , Pierre-Olivier Amblard (2) , Patrick Flandrin (1)

1 Phys-ENS - Laboratoire de Physique de l'ENS Lyon
2 LIS - Laboratoire des images et des signaux

Scale-invariant processes, and hereafter processes with broken versions of this symmetry, are studied by means of the Lamperti transformation, a one-to- one transformation linking stationary and self-similar processes. A general overview of the use of the transformation, and of the stationary generators it builds, is given for modelling and analysis of scale invariance. We put an emphasis on generalizations to non-strictly scale-invariant situations. The examples of discrete scale invariance and finite-size scale invariance are developed by means of the Lamperti transformation framework, and some specific examples of processes with these generalized symmetries are given.

Keywords: Lamperti transform; scale invariance; self-similarity; fractional Brownian motion; generalized Lamperti transform; broken scale invariance

Domains: Signal and Image Processing; Signal and Image processing

Pierre Borgnat, Pierre-Olivier Amblard, Patrick Flandrin. Stochastic invariances and Lamperti transformations for Stochastic Processes. Journal of Physics A: Mathematical and Theoretical, 2005, 38, pp.2081-2101. ⟨ensl-00175960⟩