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The walk is then said to be self-similar, when the equality in law (51) is valid for all > 0 A comprehensive survey of random walk on fractal media, orientated toward physicists, can be found in [HAV 87] A more mathematically grounded framework can be read in [BAR 95] Apart from the framework of random walk, reasons for which a physical quantity possesses invariance with a power law are generally very tricky to discover Indications of physical nature can be found in [HER 90], particularly in Duxburg s contribution, which elaborates a scale renormalization theory for crack dynamics In [DUB 97], a number of contributions in various elds such as nancial markets, avalanches, metallurgy, etc provide illustrations of scale invariance In particular, [DUR 97] proposes an analysis of invariance phenomena in avalanches, which is both experimental and theoretical One major source of inspiration for the de nition and study of the property of scale invariance, is that of hydrodynamic turbulence more precisely, Kolmogorov s work (see, for example, [FRI 97]) which, from the basis of Richardson s work on the energy cascades, established the famous 5 law, in 1941, based on the modeling of energy 3 transfers in turbulent ows This theory provides a powerful means to de ne the stochastic self-similar processes and to study their properties The reader is referred to [FRI 97] and references therein Scale invariance, or self-similarity, sometimes leads to a correlation property referred to as long-range dependence Generally, for processes satisfying (51), increments have a power law correlation decrease, a slow decline that indicates long-term persistence Mandelbrot and van Ness [MAN 68] popularized fractional Brownian motion, historically introduced in [KOL 40], precisely to model long-range correlation These processes had since an extraordinary success and numerous extensions provide quantities of generally identi able models Let us brie y describe the article [WIL 98], in which the authors give a physical theory of fractional.

The typical approach to the key management problem in wireline networks is based on the use of trusted third parties A trusted third party (TTP) is an entity trusted by all users of the system Three different types of TTPs can be categorized depending on the nature of their involvement These are the in-line TTP, on-line TTP, and off-line TTP, as shown in Figure 31 An in-line TTP is an active participant in the communication path between the two users, as shown in the gure Such an entity is not only involved in deciding the keying material, but is also involved during the data transfer An on-line TTP on the other hand is also an active participant, but only for the purpose of deciding the keying material Thus, once the keying material is decided, the on-line TTP no longer participates in the data communication between the users The off-line TTP communicates with the users prior to the setting up of the communication link During such a time, the keying material would be distributed to the various entities present in the system As a result, during the actual communication, the off-line TTP need not be present in the system at all

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Brownian motion A typical machine is either active, or inactive; the durations of activity and inactivity are independent An in nity of machines is then considered At the time tT , we consider all the active machines more exactly, the uctuations around the average number of active machines at time tT Then, we renormalize in T to obtain the law of this phenomenon The set of all durations of active machines at a given time is distributed in a rather similar way to that of the balls of the distribution model of non-renormalized mass which we will study later The reader is also directed to 12 The goal of this chapter is to present a set of stochastic processes which have partial self-similarity and stationarity properties Unfortunately, we cannot claim to present an exhaustive study moreover, the available space prohibits it We had to make choices We preferred to challenge the reader by asking him or her questions whose answers appeared to us as surprising The intention is to show that the concept of scale invariance remains, to a great extent, misunderstood We thus hope, with what has been said before, centered on physics, and which we will present, to have opened paths which other researchers will perhaps follow Trees will be used here as the leading path to scale invariance It seemed to us that such a simple geometric structure, with such great exibility, is well-adapted to the study of self-similarity In order to become familiarized with trees and invariance by dilation and translation, we begin our presentation with a study of purely geometric scaling, where trees and spaces are mixed From this, we present random or non-random fractals with scale invariance This leads us to the model of mass distribution This provides a convenient means to generate a quantity of processes, stochastic or not, with scale and translation invariance It is remarkable that, through a suitable wavelet decomposition, all the self-similar stochastic processes with stationary increments relate to a layer-type model, except perhaps for Takenaka s process These models of scale invariance enable us to question the difference between two concepts of equal importance: long-range correlation and sample path regularity From examples, we show that these two concepts are independent This chapter consists of four sections The rst is mostly an introduction The second clari es the Gaussian case, a quasi-solved problem The third section turns to some non-Gaussian cases, mostly that of stable processes The last section studies correlation and regularity from defective examples Generally, certain technical dif culties, sometimes even major ones, are overlooked Consequently, the results may give the impression of lack of rigor, while returning to the original work is a necessity.

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