Abstract : Lamperti transformation is a known means to connect stationary processes and
self-similar processes. We enlarge its use to deal in a general way with the
description of scale properties in physics problems or for stochastic processes.
The proper notion of scale according to this transformation is found to be
based on the Mellin intregral transform and we first use it to study the general
properties of self-similar processes by converting stationary results or
methods. Second, we generalize the Lamperti transformation to broken
scale invariance and introduce an enlarged connection for those problems with
nonstationary signal processing.
We propose results on representation, modelization and analysis for complete or
broken scale invariance, with a focus on mixed time-Mellin scale representations.
Specific kind of broken invariance are envisaged: finite size scale invariance,
local self-similarity and a major part is devoted to stochastic discrete scale
invariance that we define and study as the image in the scale domain
The scale properties are central to the problem of fluid turbulence and we
study turbulence by means of the fine-structure models of turbulence based on
collections of coherent structures or vortices. We propose some reflexions
on the characterization of coherent structures and we specifically focus on
the Lundgren strained spiral vortex model by means of the (Mellin)