Fox, Ronald Forrest. “Analysis of nonstationary, Gaussian and non-Gaussian, generalized Langevin equations using methods of multiplicative stochastic processes.” Journal of Statistical Physics 16, no. 3 (1977): 259-279.
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Using the methods of multiplicative stochastic processes, a thorough analysis of “non-Markovian,” generalized Langevin equations is presented. For the Gaussian case, these methods are used to show that the nonstationary Fokker-Planck equation already found by Adelman and others is also obtainable from van Kampen’s lemma for stochastic probability flows. Here, results applicable to an arbitraryn -component process are obtained and the specific two-component case of the Brownian harmonic oscillator is presented in detail in order to explicitly exhibit the matrix algebraic methods. The non-Gaussian case is presented at the end of the paper and shows that the methods already used in the Gaussian case lead directly to results for the non-Gaussian case. In order to use the methods of multiplicative stochastic processes analysis, it is necessary to transform the “non-Markovian,” generalized Langevin equation using a stochastic extension of a transformation discussed by Adelman. This transformation removes the “memory kernel” term in the usual generalized Langevin equation and in the Gaussian case leads to the result that the original process was in fact not “non-Markovian” but actually nonstationary,Markovian .