Dynamics of Stochastic Systems

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

Part III takes on issues in randomly organized media (localization), turbulent advection of passive tracers (clustering).

Each chapter is solved by the problems of the reader, which will be a good training for independent investigations.

· This book is translated from Russian.
· The book develops the mathematical tools of stochastic analysis, and applies them to a wide range of physical models of particles, fluids, and waves.
· Accessible to a broad audience with general background in mathematical physics, but no special expertise in stochastic analysis, wave propagation or turbulence

Contents
Preface
Introduction
I Dynamical description of stochastic systems
1 Examples, basic problems, peculiar features of solutions
2 Indicator function and Liouville equation
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
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
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
V Appendix
A Variation (functional) derivatives
B Fundamental solutions of wave problems in empty and layered media
C Imbedding method in boundary-value wave problems 380
Bibliography
index