NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Introductions to parts are included in .pdf document.
Part 1. THE CONTINUOUS MARKOV PROCESS
Since the first treatments of Brownian Motion as an example of a continuous Markov Process, the applications of Markov Processes in physical situations have extended over a wide range which includes such extremes as barometric pressure distributions and structural responses to earthquakes.
In this part, the notion of a continuous Markov Process is presented and described in terms of a transition probability and a Fokker-Planck Equation. Two uniqueness theorems are presented here, as well as a heuristic discussion of the large time behavior of such a process.
Part 2. THE MARKOV PROCESS AS GENERATED BY DIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS
In Part 1, the Markov Process was treated from the point of view of the Fokker-Planck Equation alone, and no discussion was presented treating the process itself. In this part, it will be demonstrated that a system of differential equations can define a Markov Process, and the Fokker-Planck Equation for such a process will be derived.
Particular emphasis is given to a discussion of the differing results of various authors in the case of "parametric white noise." These differing results have led to a controversy concerning the coefficients A[subscript k] of the Fokker-Planck Equation, Eq. 1.14, even in some of the simplest examples.
Among the examples given will be the general linear differential equation with "parametric white noise," and an "equivalent" linear differential equation with no parametric white noise will be derived.
Part 3. SOME SUFFICIENT STABILITY CONDITIONS FOR LINEAR SYSTEMS WITH RANDOM (NON-WHITE) COEFFICIENTS
As pointed out in Part 2, the stability and instability of linear systems whose coefficients are sums of constants and Gaussian White Noise can be determined by Laplace transforms applied to appropriate moment equations. This will lead explicitly to stability boundaries for the various moments, and is most often useful in determining "mean square stability." Unfortunately, such a procedure cannot be extended to cover non-white parameters. At present, there is no general method that may be used to show instability when the coefficients are random but not white, and only conservative sufficient conditions for various forms of stability can be obtained.
Some attempts at determining stability boundaries have been p...