Comments on “The Generalized Wiener Process for Colored Noise”

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Galleani and Cohen have developed a new approach to the study of random differential equations. Recently, they applied their method to the interesting case that they called the generalized Wiener process. We show that while the phase space equation they derived is correct, their solution to the equation is not the full solution but holds only under certain conditions. We obtain the general solution and discuss under what circumstances their solution is exact, and under what circumstances their solution is a good approximation to the exact solution. In addition, we pinpoint where in their method of solution they neglected a term, and give the correction. In many cases, their approach may be accurate enough, and hence preferable, as it is a simpler calculation than the exact solution. However, for processes with long correlation times (i.e., autocorrelation functions that do not decay rapidly to zero), the complete method of solution presented here may be required. </para>