Probability theory, Kolmogorov

The first complete statement of the theory was given by A. N. Kolmogorov, Foundation of Probability Theory, Chelsea Publishing Co., New York, I960. 

Finite-additive invariant measures on non-compact groups were studied by Birkhoff (1936) (see also the book of Hewitt and Ross, 1963, Chapter 4). The frequency-based Mises approach to probability theory foundations (von Mises, 1964), as well as logical foundations of probability by Carnap (1950) do not need cr-additivity. Non-Kolmogorov probability theories are discussed now in the context of quantum physics (Khrennikov, 2002), nonstandard analysis (Loeb, 1975) and many other problems (and we do not pretend provide here is a full review of related works). 

Ad b. This transformation is pure mathematics. It is the main subject of textbooks on probability theory. The subject has been freed from all contamination of reality by the axiomatics of Kolmogorov. ... 

In his treatise "The local structure of turbulence in an incompressible viscous liquid at very high Reynolds numbers , Kolmogorov [289] considered the elements of free turbulence as random variables, which are in general terms accessible to probability theory. This assumes local isotropic turbulence. Thus the probability distribution law is independent of time, since a temporally steady-state condition is present. For these conditions Kolmogorov postulated two similarity hypotheses ... 

Kolmogorov, A. N. (1968a). Logical basis for information theory and probability theory. IEEE Transactions on Information Theory, 14(5), 662-664. Available from http //ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm arnumber= 1054210, doi 10.1109/TIT.1968.1054210... 

Not only is the master equation more convenient for mathematical operations than the original Chapman-Kolmogorov equation, it also has a more direct physical interpretation. The quantities W(y y ) At or Wnn> At are the probabilities for a transition during a short time At. They can therefore be computed, for a given system, by means of any available approximation method that is valid for short times. The best known one is time-dependent perturbation theory, leading to Fermi s Golden Rule f)... 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

A. N. Kolmogorov, Foundations of the Theory of Probability (N. Morrison, transl.), Chelsea, New York, 1950. The original, Grundbegriffe der Wahrscheinlichkeitsrechnung, was published in Berlin in 1933. 

In classical Brownian motion theory, one usually assumes that the time evolution of the probability density p(F, t) is governed by the Chapman-Kolmogorov... 

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