# On the Measurement of the "White Noise"

A.L. Shestakov, G.A. SviridyukIn 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.Full text

- Keywords
- Leontieff type equations, weakened Showalter - Sidorov problem, symmetric mean derivative, Wiener process.
- References
- 1. Shestakov A.L. Dynamic Accuracy of the Transmitter with a Corrective Device in the Form of Sensor Model. Measurement Techniques, 1987, no. 2, pp. 26-34. (in Russian)

2. Sviridyuk G.A., Brychev S. V. Numerical Solution of Systems of Equations of Leontief Type. Russian Mathematics, 2003, vol. 47, no. 8, pp. 44-50.

3. Shestakov A.L. Dynamic Error Correction Transducer Linear Filter-based Sensor Model. Izvestiya VUZ. Priborostroenie, 1991, vol. 34, no. 4. pp. 8-13. (in Russian)

4. Shestakov A.L. Modal Synthesis of the Transmitter. J. of Computer and Systems Sciences International, 1995, no. 4, pp. 67-75. (in Russian)

5. Shestakov A.L., Sviridyuk G.A. A New Approach to Measuring Dynamically Distorted Signals. Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya "Matematicheskoe modelirovanie i programmirovanie" - Bulletin of South Ural State University. Seria "Mathematical Modelling, Programming & Computer Software", 2010, no. 16 (192), issue 5, pp. 116-120. (in Russian)

6. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya "Matematicheskoe modelirovanie i programmirovanie" - Bulletin of South Ural State University. Seria "Mathematical Modelling, Programming & Computer Software", 2011, no. 17 (234), issue. 8. - pp. 70-75.

7. Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal Measurement Problem. Automation and Remote Control, 2012, vol. 73, no. 1, pp. 97-104.

8. Gantmacher F.R. The Theory of Matrices. AMS Chelsea Publishing: Reprinted by American Mathematical Society, 2000. 660 p.

9. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003.

10. Keller A.V., Nazarova E.I. The Problem of Optimal Measurement: Numerical Solution, Algorithm for Programs. News of Irkutsk State University. Series "Mathematics", 2011, vol. 4, no. 3, pp. 74-82.

11. Showolter - Sidorov Problem (Shosid Problem): Certificate 2010616865 / Keller A.V.(RU); possessor of the right(s) - South Ural State University (Chelyabinsk, Russian Federation). - 210615137; application form 16.08.2010; registered 14.10.2010, Register of computer programs.

12. Ito K. Essentials of Stochastic Processes (Translations of Mathematical Monographs, V. 231. American Mathematical Society, 2006.

13. Stratonovich R.L. Conditional Markov Processes and Their Applications to the Theory of Optimal Control, N.-Y., Elsevier, 1968.

14. Skorokhod A.V. Markov Processes and Probabilistic Applications in Analysis. Probability theory - 1, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 43, VINITI, Moscow, 1989, pp. 147-188.

15. Kovacs M., Larsson S. Introduction to Stochastic Partial Differential Equations. Proceedings of "New Directions in the Mathematical and Computer Sciences", National Universities Commission, Abuja, Nigeria, October 8-12, 2007. Publications of the ICMCS, 2008, vol. 4, pp. 159-232.

16. Melnikova I.V., Filinkov A.I., Alshansky M.A. Abstract Stochastic Equations II. Solutions In Spaces Of Abstract Stochastic Distributions. J. of Mathematical Sciences, 2003, vol. 116, no. 5, pp. 3620-3656.

17. Nelson E. Dynamical Theories of Brownian Motion. Princeton: Princeton University Press, 1967.

18. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Dordrecht, Heidelberg, N.-Y., Springer, 2011.

19. Sviridyuk G.A., Zagrebina S.A. The Showalter - Sidorov Problem as Phenomena of the Sobolev-type Equations. News of Irkutsk State University. Series "Mathematics", 2010, vol. 3, no. 1, pp. 51-72.