Multifractional Stochastic Fields: Wavelet Strategies In Multifractional Frameworks

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. Two paradigmatic examples of them are the class of Multifractional Fields and that of Anisotropic 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; while Anisotropic means that they are allowed to change from one direction to another. In order to sharply determine path behavior of Multifractional and Anisotropic 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 more or less classical examples of Multifractional and Anisotropic Fields. The book also discusses some topics concerning them which are not directly related to the wavelet strategy, namely their self-similarity property and statistical inference for them.