The author answers three types of questions concerning this area. First, what probabilistic tools are necessary for constructing a stochastic model and deriving the FKP equation for nonlinear stochastic dynamical systems? Secondly, what are the main results concerning the existence and uniqueness of an invariant measure and its associated stationary response? Finally, what is the class of multidimensional dynamical systems that have an explicit invariant measure and what are the fundamental examples for applications?

Contents:

• Stochastic Canonical Equation of Multidimensional Nonlinear Dissipative Hamiltonian Dynamical Systems
• Fundamental Examples of Nonlinear Dynamical Systems and Associated Second-Order Equation
• Brief Review of Probability and Random Variables
• Probabilistic Tools I. Classical Stochastic Processes
• Probabilistic Tools II. Mean-Square Theory of Linear Integral Transformations and of Linear Differential Equations
• Probabilistic Tools III. Diffusion Processes and Fokker-Planck Equation
• Probabilistic Tools IV. Stochastic Integrals and Stochastic Differential Equations
• Stochastic Modeling with Stochastic Differential Equations
• FKP Equation for the Dissipative Hamiltonian Dynamical Systems
• Stationary Response of Dissipative Dynamical Systems, Existence and Uniqueness, Explicit Solution of an Invariant Measure
• Complements for the Normalization Condition, Characteristic Function and Moments of the Invariant Measure
• Application I. Multidimensional Linear Oscillators Subject to External and Parametric Random Excitations
• Application II. Multidimensional Nonlinear Oscillators with Inertial Nonlinearity Subject to External Random Excitations
• Application III. Multidimensional Nonlinear Oscillators Subject to External and Parametric Random Excitations
• Symplectic Change of Variables in the Multidimensional Unsteady FKP Equation
• References
• Index

"This is a timely volume summarizing and unifying 30 years of search for explicit solutions of (stationary) FPE's. New articles in this area, which continue to appear, have to explain in which way they extend Soize's presentation. As such, this book is a useful reference for the random vibrations community."