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# Stochastic Equations through the Eye of the Physicist

## Basic Concepts, Exact Results and Asymptotic Approximations

**By**- Valery Klyatskin, 1988 Research Professor of Theoretical and Mathematical Physics, Russian Academy of Science; 1977 D. Sc. in Physical and Mathematical Sciences, Acoustical Institute, Russian Academy of Science; 1968 Ph.D. in Physical and Mathematical Sciences, Institute of Atmospheric Physics Russian Academy of Science; 1964 M.Sc. in Theoretical Physics, Moscow Institute of Physics and Technology (FIZTEX)., Russian Academy of Science, Russia

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 examples.Part IV takes up issues for the coherent phenomena in stochastic dynamical systems, described by ordinary and partial differential equations, like wave propagation in randomly layered 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) that give many technical mathematical details needed in the book.

**Audience**

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