On the Measurement of the «White Noise»

In the framework of the Leontieff type equations theory we consider the mathematical model of the measuring transducer, demonstrating the mechanical lag effect. In studying of the model with deterministic external signal the methods and results of the Sobolev type equations theory and degenerate groups of operators are very useful, because they helped to create an efficient computational algorithm. Now, the model assumes a presence of white noise along with the deterministic signal. Since the model is represented by a degenerate system of ordinary differential equations, it is difficult to apply existing nowadays approaches such as Ito–Stratonovich–Skorohod and Melnikova–Filinkov–Alshansky in which the white noise is understood as a generalized derivative of the Wiener process. Instead of it, we propose a new concept of the «white noise», which is equal to the symmetric mean derivative (in the paper — the derivative of the Nelson–Gliklikh) of the Wiener process, and in the framework of the Einstein–Smoluchowsky coincides with the «ordinary» derivative of Brownian motion. The first part of the paper contains the basic facts of the Nelson–Gliklikh derivative theory adapted to this situation. The second part deals with the weakened Showalter–Sidorov problem and gives exact formulas for its solution. As an example, we present a concrete model of a measuring transducer.