Abstract
This chapter considers basic methods of determining statistical characteristics of solutions to the stochastic equations. Averaging of a linear stochastic equation over an ensemble of realizations of fluctuating parameters will not result generally in a closed equation for the corresponding average value. It is found that to obtain the closed equation, one must deal with an additional extended space whose dimension appears infinite in most cases. This approach makes it possible to derive the linear equation for an average quantity of interest, but this equation will contain variational derivatives. The backward Liouville equation for the indicator function, which describes the evolution of a dynamic system, is obtained. It is found that the backward equation is more convenient for studying the statistical characteristics that concern the residence of random process within certain region of space, such as residence duration within this region and time of arrival at region boundary. It is observed that stationary and homogeneous hydrodynamic turbulence can be described in terms of the Fourier transform of the velocity field.
Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
