Date: 05 Nov 2013

Noise and Stochastic Processes

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Abstract

This chapter presents an overview of the noise models that will be used throughout the book. In the area of magnetic hysteresis, most of the research employed Gaussian white noise as noise model, which is mathematically described by independent and identically distributed random variables following a Gaussian distribution. Once the interest in hysteresis was extended beyond the traditional area of magnetism, various noise models appeared naturally in the hysteresis analysis. For example, pink noise is ubiquitous in multi-stable electronic systems, Brown noise is very common in physical and chemical diffusion processes, and even white noise is often encountered in its impulsive form in economics and biological hysteretic systems, so Cauchy or Laplace probability distributions emerge as better model choices in these systems than Gaussian white noise. Various noise models and the numerical methods used in this study to simulate them are discussed in the first part of this chapter. The second part is devoted to introducing the theory of stochastic processes defined on graphs recently developed by Freidlin and Wentzell, which proved to be naturally suited to the stochastic analysis of hysteretic systems. First, several definitions and general properties of stochastic processes are discussed, stressing the link between transition probability of Markov processes and the semigroups of contractions. This relationship provides the characterization tool for the diffusion processes that can be defined on a graph, subject that is elaborated in the final sections of this chapter.