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logarithmic factors) of the trajectories Indeed, FBM is a good candidate for modeling phenomena which, using a statistical processing, are found to have C H trajectories and are supposed, for theoretical reasons, to be self-similar The importance of the identi cation of H, starting from the samples of the phenomenon necessarily taken in discrete time, is thus crucial At the same time, it is necessary to remember that FBMs are processes with stationary increments, which simpli es the spectral study of the process but is too restrictive for certain applications Indeed, in many elds (when we want to simulate textures in an image), it is expected, a priori, that order H depends on the point at which the process is observed For example, if, using a random eld, we want to model the aerial photographing of a forest, we would like to have a model where the parameter of self-similarity around the point x, noted by h(x), depends on the geological nature of the ground in the vicinity of x However, a spatial modulation of the eld law is generally incompatible with the property of self-similarity, which is the overall property Consequently, the problem consists of arriving at a concept of a suf ciently exible, locally self-similar eld, so that the parameters which de ne the eld law could vary with the position, yet be simple enough to enable the identi cation of these parameters Unfortunately, the simple approach, which consists of reproducing H by a function h(x) in the formula giving the covariance of a FBM, is not satisfactory: generally, we can show that there does not exist any Gaussian eld having this generalized covariance We will thus have to introduce the mathematical tools which will make it possible to build models generalizing FBMs and also to identify the functional parameters of these models These theoretical recaps will be dealt with in section 62, where we will consider each time the relevance of the concept introduced for an example of fractional Brownian motion In this context, we discuss traditional techniques for the study of Gaussian elds and also the tools of analysis in wavelets Using this theoretical framework, in section 63 we will formally de ne the property of local self-similarity and present two examples which form the base of all the later models Having established that these models are not suf ciently general for the applications, we will penetrate into the multifractional world in section 64 In each preceding model, speci c attention will be given to the regularity of the trajectories and, in section 65, we shall develop the statistical methods which make it possible to estimate this regularity This is what we call model identi ability At this point, it is necessary to clarify that the term elds is used for families 1 In applications, the most of random variables indexed by groups of d for d interesting cases correspond to d > 1 (for example, d = 2, for the images) However, certain statements, particularly those concerning identi cation, will relate only to processes (ie, elds where we have d = 1).

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62 Recap of two representations of fractional Brownian motion We begin with the presentation of tools for the study of locally self-similar elds based on the concept of reproducing kernel Hilbert space We derive from this a Karhunen-Loeve expansion regarding FBM We nd that there is a spectral representation of fractional Brownian motion called harmonizable It will be an occasion to recall some concepts of the multiresolution analysis 621 Reproducing kernel Hilbert space The study of Gaussian elds is largely facilitated by a tool of analysis which is traditionally associated with these elds: reproducing kernel Hilbert space From a physical point of view, the reproducing kernel Hilbert space can be regarded as a space which describes the energy associated with a Gaussian eld within the meaning of a spectral energy On the other hand, mathematically, a reproducing kernel Hilbert space is a Hilbert space of deterministic functions whose standards characterize all the properties of the eld See [NEV 68] for a detailed study Let us now recall its formal de nition DEFINITION 61 Let (Xx )x d be a centered Gaussian eld (ie, E(Xx ) = 0) We will call Gaussian space associated with X the space of square integrable random variables (noted by L2 ( , A, P )) and made up with the help of linear combinations of the variables Xx and their limits, that is:

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(61)

where adh means that we take the closure of the set for the topology de ned by L2 ( , A, P ) The space:

HX = hZ , such that Z HX / hZ (x) = E(ZXx )