Fractional Brownian Motion (FBM) is a very classical continuous
self-similar Gaussian field with stationary increments. In 1940,
some works of Kolmogorov on turbulence led him to introduce this
quite natural extension of Brownian Motion, which, in contrast with
the latter, has correlated increments. However, the denomination
FBM is due to a very famous article by Mandelbrot and Van Ness,
published in 1968. Not only in it, but also in several of his
following works, Mandelbrot emphasized the importance of FBM as a
model in several applied areas, and thus he made it to be known by
a wide community. Therefore, FBM has been studied by many authors,
and used in a lot of applications.In spite of the fact that FBM is
a very useful model, it does not always fit to real data. This is
the reason why, for at least two decades, there has been an
increasing interest in the construction of new classes of random
models extending it, which offer more flexibility. A paradigmatic
example of them is the class of Multifractional Fields.
Multifractional means that fractal properties of models, typically,
roughness of paths and self-similarity of probability
distributions, are locally allowed to change from place to place.In
order to sharply determine path behavior of Multifractional Fields,
a wavelet strategy, which can be considered to be new in the
probabilistic framework, has been developed since the end of the
90's. It is somehow inspired by some rather non-standard methods,
related to the fine study of Brownian Motion roughness, through its
representation in the Faber-Schauder system. The main goal of the
book is to present the motivations behind this wavelet strategy,
and to explain how it can be applied to some classical examples of
Multifractional Fields. The book also discusses some topics
concerning them which are not directly related to the wavelet
strategy.
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