On a First-order Stochastic Differential Equation†

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Abstract
A first-order system with random parameters and random forcing is studied. The analysis is concentrated on the probability distributions. It is shown that considerable qualitative information can be obtained from Feller's classification of the singular points of the forward and backward Kolmogorov equations. It is found that there is a drastic difference between the cases of uncorrelated and strongly correlated disturbances. The existence of stationary distributions is shown and their structure is analysed; it is found that the steady-state distributions are of the Pearson type. Some examples exhibit in detail the differences between uncorrelated and strongly correlated disturbances, giving the rather surprising effect that by making the fluctuations of the parameters sufficiently large, the probability of finding the state of the system in tin interval around the origin can be made arbitrarily close to one ‘peaking’. The results of some numerical computations are presented. A case where the energy of the fluctuation in parameters is limited within a certain frequency band shows that this situation is different from the case of ‘ white noise ’. For example, ‘ peaking ’ of the distributions does not occur. It is found that, for the purpose of analysing probability distributions the system obtained can be approximated by a different system with white noise coefficients, These results are also illustrated by numerical computations.

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