Stochastic Dynamical Systems
The correct way from the intuitive idea of probability to the theory of stochastic stability is long and fatiguing: Measure Theory, Probability Theory, Theory of Stochastic Processes, Theory of Stochastic Differential Equations and finally Stochastic Stability. This tour is seldom done completely by mathematicians and is unacceptable for a man in applications. On the other hand, just the results at the end of this cumbersome way — the criteria of stochastic stability — are of great interest in mechanics and engineering. Therefore, here the attempt will be made to reach the correct results on stochastic stability by a non rigorous but brief treatment based more on faith and intuition than on an extensive mathematical background and sophisticated proofs. Only some elementary knowledge on probability is supposed. Section 2.1 contains notations and basic definitions on density functions, moments and stochastic processes. In 2.2 dynamic systems corrupted by noise are introduced. Sections 2.3 to 2.6 give a hint of an outline of the theory of stochastic differential equations. Finally in 2.7 the basic definitions and results on stochastic stability are collected. As the topics of 2.3 to 2.7 are relatively new and the few books treating them are written on a very high level, the formulae and theorems most important for the applications in sections 3 to 6 are deduced by heuristic considerations. For strong proofs see , , ,  and consult previously , ,  — if necessary.
KeywordsWhite Noise Equilibrium Position Stochastic Differential Equation Stochastic System Wiener Process
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