Our purpose in this book is to develop an understanding of the concepts of stochastic system theory and linear and nonlinear stochastic operator equations as well as the ability to solve the nonlinear stochastic equations arising in applications. We are interested in the relevant mathematics, of course, but also in its physical interpretation and use. Throughout physics, engineering, economics, biology, etc., the problems are generally nonlinear and stochastic and are not correctly solved by mathematics that essentially neglects these difficulties. The mathematical methods used in these problems change the problems in an essential way to problems that are mathematically tractable. Very important mathematics has been based on the study of physically unrealistic processes, and the procedures used are essentially perturbation methods that cannot treat strong nonlinearities or randomness.
The methods developed here do not linearize the nonlinearities nor replace stochastic quantities by averages. They are applicable to a wide range of phenomena in the sciences and to the models developed for them. Further applications will be found in works planned specifically related to the different fields.
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