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Differential equations with random perturbations are the mathematical models of real-world processes that cannot be described via deterministic laws, and their evolution depends on the random factors. The modern theory of differential equations with random perturbations is on the edge of two mathematical disciplines: random processes and ordinary differential equations. Consequently, the sources of these methods come both from the theory of random processes and from the classic theory of differential equations. This work focuses on the approach to stochastic equations from the perspective of ordinary differential equations. For this purpose, both asymptotic and qualitative methods which appeared in the classical theory of differential equations and nonlinear mechanics are developed.
Sommario
Differential Equations with Random Right Hand Side and Random Impulse Action; Invariant Sets of Systems with Random Perturbations; Stability of Invariant Sets and the Reduction Principle for Ito Systems, Linear and Quasilinear Stochastic Ito Systems; Exponential Dichotomy in the Quadratic Mean; Asymptotic Equivalence of Linear; Extension of Ito Systems on Torus; Stability of Invariant Tori; Stochastic Invariant Tori of Nonlinear Analysis of the Equations with Random Perturbations Using Averaging.
