Fox, Ronald Forrest. “Gaussian stochastic processes in physics.” Physics Reports 48, no. 3 (1978): 179-283.
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1977 marked the two hundredth anniversary of the birth of Gauss. At the same time, I was engaged in reviewing the application of stochastic processes to physical problems. My survey began with Langevin’s equation for Brownian motion, a stationary, Markov, Gaussian process, and its generalization for near equilibrium irreversible thermodynamics proposed by Onsager and Machiup. Next to be studied was the generalized Langevin equation, a stationary, non-Markovian, Gaussian process, and its very close relationship with Mon’s exact dynamical theory of irreversible processes. Then came the study of macrovariable fluctuations for systems which are characterized by non-linear equations for the average values of the basic quantities used in the description. In this case, an analogue of the central limit theorem is obtained and in the appropriate variables a non-stationary, Markov Gaussian process is obtained for the fluctuations of the macrovariables around their averaged values. The macrovariable fluctuation theories based upon the approaches of van Kampen and of
Kubo originate in master equations, which, while not themselves on the same footing as exact microscopic dynamics, permit the rigorous analysis of the transition from master equation descriptions to macrovariable fluctuation descriptions. These three cases, with their varying degrees of rigor and firm foundation in exact dynamics are Gaussian although they may or may not be Markov or stationary