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Extra info for A Bound on the Ratio between the Packing and Covering Densities of a Convex Body

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It is characterized by the fact that the elementary outcomes coincide with the sample paths. 42 2. 2 Separability Two stochastic processes X^ and Xf are said to be equivalent, if for every t eO: P{Xf = Xf] = 1. In this case Xf is called a version of Xf, and vice versa, and the two processes have an identical hierarchy of distribution functions. This does not however imply that the realizations of the two processes are identical. Especially their analytical properties, such as continuity and differentiability, can be quite different.

A + C^, where A corresponds to the average state of the environment, and ^^ describes the fluctuations around it. Obviously we have E{Q = 0. 12) In a very large class of phenomenological equations, encountered in applications, the external parameter appears in a linear way. (The nonlinear case is discussed in Sect. 7 and Chap. ) We consider for the time being only one fluctuating external parameter. 12) is then of the form Xt=h{X,) + Xg{X,) + i:,g{Xt). 13) The next step in the modehng procedure is to specify explicitly the probabilistic characteristics of the random process in terms of the physical properties of the environment.

2. Elements of Probability Theory Probability theory is the adequate mathematical framework within which to tackle the problem of the effect of external noise on nonlinear systems. To be able to discuss the modehng procedure of these phenomena in a clear way, it is necessary to recall in this chapter, with a certain degree of mathematical precision, some basic notions of probability theory. At the same time it will serve to establish our notations. We have made every effort to make this monograph as far as possible self-contained, especially the mathematical aspects of our approach.

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