2011-№3(32) Article 15

А.M. Lavrov

Pawula’s kinetic equation for non-stationary gaussian random processes. p. 180-190

UDC 517.958: 519.218

The paper deals with transitive density, i.e. the fundamental solution of Pawula’s kinetic equation for one-dimensional probability density function of non-stationary Gaussian random processes. The obtained results are applicable to statistical radiophysics, theory of casual processes and partial differential equations.

Markov and non-markov random processes, Fokker-Planck-Kolmogorov equation, kinetic coefficients, transition density function, Gaussian (normal) random processes.

References

1. I.M. Gel’fand, G.E. Shilov. Obobshhennye funkcii i dejstvija nad nimi [Generalized functions and operations on them]. Moscow: ​​GIFML, 1958. 439 p.

2. Kazakov, V.A. Kineticheskie kojefficienty v prjamom uravnenii dlja differenciruemyh processov s posledejstviem [Kinetic coefficients in direct equation for differentiable processes with aftereffect]. The famous universities’. Radiophysics, 1986. vol. 29, no. 11, pp. 1344-1354.

3. V.A. Kazakov, A.M. Lavrov. Kineticheskie uravnenija dlja nemarkovskih processov s pirsonovskimi raspredelenijami [Kinetic equations for non-Markov processes with Peirce distributions]. The famous universities’. Radiophysics, 1995. vol. 38, no. 7, pp. 695-704.

4. V.A. Kazakov, A.M. Lavrov. O svjazi mezhdu kineticheskimi uravnenijami razlichnyh vidov [About the connection between different types of kinetic equations]. The famous universities’. Radiophysics, 1993. vol. 36, no. 10, pp. 944-948.

5. P.I. Kuznecov, R.L. Stratonovich, V.I. Tihonov. Korreljacionnye funkcii v teorii brounovskogo dvizhenija. Obobshhenie uravnenija Fokkera – Planka [Correlation functions in the theory of Brownian motion. Generalization of the Fokker— Plank equation]. ZhJeTF, 1954. vol. 26, no. 2. pp. 189-207.

6. Lavrov, A.M. Vychislenie kineticheskih kojefficientov, vhodjashhih v obobshhennoe uravnenie FPK dlja nestacionarnogo gaussovskogo sluchajnogo processa [Calculation of kinetic coefficients which comprise generalized FPK equation for nonstationary Gaussian random process]. Mathematical methods in Scientific Research, RGRTA, Ryazan, 2002. pp. 23-29.

7. Lavrov, A.M. Issledovanie kineticheskogo uravnenija Pavuly v sluchae nemarkovskih processov s pirsonovskimi raspredelenijami [Study of Pavula’s kinetic equation in the case of non-Markov processes with Peirce distributions]. Bulletin of the Ryazan State Radio Engineering Academy. Ryazan, 1996. no. 1, pp. 58-64.

8. Lavrov, A.M. Issledovanie kineticheskogo uravnenija Pavuly v sluchae releevskogo sluchajnogo processa [Study of Pavula’s kinetic equation in the case of Rayleigh random process]. Bulletin of the Ryazan State Radio Engineering Academy. Ryazan, 1998. no. 5, pp. 45-49.

9. Stratonovich, R.L. Izbrannye voprosy teorii fljuktuacij v radiotehnike [Selected issues of the theory of fluctuations in the radio]. Moscow: Soviet radio, 1961. 558 p.

10. Feller, V. Vvedenie v teoriju verojatnostej i ee prilozhenija [Introduction to probability theory and its applications]. Moscow: Peace, 1984. vol. 1, 528 p. Vol. 2, 738 p.

11. Pawula, R.F. Generalizations and Extensions of the Fokker-Planck-Kolmogorov Equations. Trans. IEEE. 1967. Vol. 1. vol. 3, no. 1, pp. 33–41.