Contents
Preface
Introduction
I Dynamical description of stochastic systems
1 Examples, basic problems, peculiar features of solutions
1.1 Ordinary differential equations: initial value problems
1.1.1 Particle under the random velocity field
1.1.2 Particles under the random velocity field
1.1.3 Particles under random forces
1.1.4 Systems with the blow-up singularities
1.1.5 Oscillator with randomly varying frequency (stochastic parametric resonance)
1.2 Linear ordinary differential equations: boundary-value problems
1.2.1 Plane waves in layered media: a wave incident on a medium layer
1.2.2 Plane waves in layered media: the source inside the medium
1.2.3 Plane waves in layered media: the two-layer model
1.3 First-order partial differential equations
1.3.1 Linear first-order partial differential equations: passive tracer in random velocity field
1.3.2 Quasilinear equations
1.3.3 Boundary-value problems for nonlinear ordinary differential equations
1.3.4 Nonlinear first-order partial differential equations
1.4 Partial differential equations of higher orders
1.4.1 Stationary problems for Maxwell’s equations
1.4.2 The Helmholtz equation (boundary-value problem) and the parabolic equation of quasioptics (waves in randomly inhomogeneous media)
1.4.3 The Navier–Stokes equation: random forces in hydrodynamic theory of turbulence
1.4.4 Equations of geophysical hydrodynamics
1.5 Solution dependence on medium parameters and initial value
1.5.1 Principle of dynamic causality
1.5.2 Solution dependence on initial value
2 Indicator function and Liouville equation
2.1 Ordinary differential equations