It is traditional to derive, diffusion equations as limits of random walks by suitably shrinking both the magnitude of the steps and the time interval between successive steps. Some examples will be provided later on. However, preliminarily, it is desirable to sketch some aspects of the random walk theory. This has a two-fold purpose in our context: it will quite naturally lead us to an intuitive illustration of some of the concepts and techniques thus far discussed and it will provide a useful tool for describing stochastic processes that can be, directly or indirectly, schematized as random walks. An example of a process that can be modeled as a random walk (r.w.) was given in Ch. 1, Sec. 1, when describing the experiment of repeated coin tossings. Also in Ch. 1 we mentioned an example of a process, discrete both in state and time, whose evolution was governed by a difference equation that was not of the random walk type, i.e., the size, S , of the n-th generation of a population characterized by discrete, non-overlapping reproductive times.
KeywordsRandom Walk Sample Path Wiener Process Diffusion Approximation Random Environment
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