Stochastic Equations through the Eye of the Physicist

Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come from either the typical forces, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. The well-known example of Brownian particle in fluidity and subjected to molecular biomechanics. Other important examples include turbulent transport and diffusion of particle-tracers (pollutants), or continuous densities ('' oil slicks ''), wave propagation and scattering in randomly inhomogeneous media, for instance light or sound propagating in the turbulent atmosphere.

Such models can render to random description and fields.

The fundamental problem of stochastic dynamics is to identify the system and its data.

This raises a host of challenging mathematical issues. One could rarely solve such systems in a closed form, for example. In mathematical terms such a solution is complicated "nonlinear functional" of random fields and processes.

Part I gives a mathematical formula for the basic physical models of transport, diffusion, propagation and develops some analytical tools.

Part II and III sets up and uses the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models. The exposition is motivated and with numerous examples.

Part IV takes up issues for the coherent phenomena in dynamics systems, experiments in randomly directed media (localization), turbulent advection of passive tracers (clustering), wave propagation in disordered 2D and 3D media .

For the sake of reader I provide several appendixes (Part V).

For scientists dealing with stochastic dynamic systems in different areas, such as hydrodynamics, acoustics, radio wave physics, theoretical and mathematical physics, and applied mathematics

the theory of stochastic in terms of the functional analysis

Referencing those papers, which are used in the book and also the recent bibliography on the subject.

Contents / Preface / Introduction
I Dynamical description of stochastic systems
1 Examples, basic problems, peculiar features of solutions
2 Indicator function and Liouville equation
II Stochastic equations
3 Random quantities, processes and fields
4 Correlation splitting
5 General approaches to analyzing stochastic dynamic systems
6 Stochastic equations with the Markovian fluctuations of parameters
III. Asymptotic and approximate methods for analyzing stochastic equations
7 Gaussian random field delta-correlated in time (ordinary differential equations)
8 Methods for solving and analyzing the Fokker-Planck equation
9 Gaussian delta-correlated random field (causal integral equations)
10 Diffusion approximation
IV Coherent phenomena in stochastic dynamic systems
11 Passive tracer clustering and diffusion in random hydrodynamic flows
12 Wave localization in randomly layered media
13 Wave propagation in random inhomogeneous medium
14 Some problems of conventional hydrodynamics
A Variation (functional) derivatives
B Fundamental solutions of wave problems in empty and layered media
B.1 The case of empty space
B.2 The case of layered space
C Imbedding method in boundary-value wave problems
C.1 Boundary-value problems for ordinary differential equations
C.2 Stationary boundary-value wave problems
C.2.1. One-dimensional stationary boundary-value wave problems