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# Stochastic Equations through the Eye of the Physicist

## 1st Edition

**Basic Concepts, Exact Results and Asymptotic Approximations**

**Basic Concepts, Exact Results and Asymptotic Approximations**

Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come either as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. The well known example of Brownian particle suspended in fluid and subjected to random molecular bombardment laid the foundation for modern stochastic calculus and statistical physics. 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 naturally render to statistical description, where the input parameters and solutions are expressed by random processes and fields.
The fundamental problem of stochastic dynamics is to identify the essential characteristics of system (its state and evolution), and relate those to the input parameters of the system and initial data.
This raises a host of challenging mathematical issues. One could rarely solve such systems exactly (or approximately) in a closed analytic form, and their solutions depend in a complicated implicit manner on the initial-boundary data, forcing and system's (media) parameters . In mathematical terms such solution becomes a complicated "nonlinear functional" of random fields and processes.
Part I gives mathematical formulation for the basic physical models of transport, diffusion, propagation and develops some analytic tools.
Part II and III sets up and applies the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models, to produce exact or approximate solutions, or in worst case numeric procedures. The exposition is motivated and demonstrated with numerous e

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 or discussed in this book and also recent review papers with extensive bibliography on the subject.

Researches in physics (fluid dynamics, optics, acoustics, radiophysics), geosciences (ocean, atmosphere physics), applied mathematics (stochastic equations), applications (coherent phenomena), Senior and postgraduate students in different areas of physics, engineering and applied mathematics

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

- No. of pages:
- 556

- Language:
- English

- Copyright:
- © 2005

- Published:
- 20th May 2005

- Imprint:
- Elsevier Science

- Print ISBN:
- 9780444517975

- Electronic ISBN:
- 9780080457642

Born in 1940 in Moscow, USSR, Valery I. Klyatskin received his secondary education at school in Tbilisi, Georgia, finishing in 1957. Seven years later he graduated from Moscow Institute of Physics and Technology (FIZTEX), whereupon he took up postgraduate studies at the Institute of Atmospheric Physics USSR Academy of Sciences, Moscow gaining the degree of Candidate of Physical and Mathematical Sciences (Ph.D) in 1968. He then continued at the Institute as a researcher, until 1978, when he was appointed as Head of the Wave Process Department at the Pacific Oceanological Institute of the USSR Academy of Sciences, based in Vladivostok. In 1992 Valery I. Klyatskin returned to Institute of Atmospheric Physics Russian Academy of Sciences, Moscow when he was appointed to his present position as Chief Scientist. At the same time he is Chief Scientific Consultant of Pacific Oceanological Institute Russian Academy of Sciences, Vladivostok. In 1977 he obtained a doctorate in Physical and Mathematical Sciences and in 1988 became Research Professor of Theoretical and Mathematical Physics, Russian Academy of Science.

Russian Academy of Science, Russia