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DOI: 10.14529/mmp140412 MSC 60H30, 34K50, 34M99 Stochastic Leontiff–tupe equations with multiplicative effect in spase of complex–valued «noises» Alexander Leonidovich Shestakov, South Ural State University (national research University), Chelyabinsk, Rector, shal@susu.ac.ruMinzilya Almasovna Sagadeeva , South Ural State University (national research University), Chelyabinsk, Assistant Professor, sam79@74.ruAbstractWe consider a Leontieff–type stochastic equation, that is, a system of differential equations implicit with respect to the time derivative in the spaces of random processes. The concepts previously introduced for the spaces of differentiable "noise" using the Nelson-Gliklikh derivative carry over to the case of complex–valued "noise"; in addition, the right-hand side of the equation is subject to multiplicative effect of a special form. We construct a solution to the Showalter-Sidorov problem for Leontieff-type equations with multiplicative effect of a complex–valued process of special form. Aside from the introduction and references, the article consists of two parts. In the first part we carry over various concepts of the space of real-valued differentiable "noise" to the complex–valued case. In the second part we construct a Showalter-Sidorov solution to a Leontieff-type equation with multiplicative effect of a complex-valued process of special form. The list of references is not intended to be complete and reflects only the authors' personal preferences. KeywordsLeontieff–type equations, multiplicative effect, Wiener process, Nelson–Gliklikh derivative, space of complex–valued "noises", "white noise" References1. Arato M. Linear Stochastic Systems with Constant Coeffcients. A Statistical Approach. Berlin, Heidelberg, N.–Y., Springer, 1982. DOI: 10.1007/BFb0043631 2. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Dordrecht, Heidelberg, N.–Y., Springer, 2011. DOI: 10.1007/978–0–85729–163–9 3. Da Prato G., Zabczyk J. Stochastic Equations in In_nite Dimensions. Cambridge, Cambridge University Press, 1992. DOI: 10.1017/CBO9780511666223 4. Zamyshlyaeva A.A. Stochastic Mathematical Model of Ion–Acoustic Waves in Plasma. Estestvennye i Tekhnicheskie nauki [Natural and Technical Sciences], 2013, no. 4, pp. 284_292. (in Russian) 5. Zagrebina S.A., Soldatova E.A. The linear Sobolev–type Equations With Relatively pbounded Operators and Additive White Noise. Izvestija Irkutskogo gosudarstvennogo universiteta. Seriya "Matematika" [News of Irkutsk State University. Series "Mathematics"], 2013, vol. 6, no. 1, pp. 20_34. (in Russian) 6. Sviridyuk G.A., Manakova N.A. The Dynamical Models of Sobolev Type with Showalter –Sidorov Condition and Additive "Noise". Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2014, vol. 7, no. 1, pp. 90_103. (in Russian) DOI: 10.14529/mmp140108 7. 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. DOI: 10.1023/A:1024159908410 8. Shestakov A.L., Sviridyuk G.A. On a New Conception of White Noise. Obozrenie Prikladnoy i Promyshlennoy Matematiki, 2012, vol. 19, issue 2, pp. 287_288. (in Russian) 9. Shestakov A.L., Sviridyuk G.A., Hudyakov Yu.V. Dinamic Measurement in Spaces of "Noise". Bulletin of the South Ural State University. Series "Computer Technologies, Automatic Control, Radio Electronics" , 2013, vol. 13, no. 2, pp. 4_11. (in Russian) 10. 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. DOI: 10.1134/S0005117912010079 11. Shestakov A., Sviridyuk G., Sagadeeva M. Reconstruction of a Dynamically Distorted Signal with Respect to the Measuring Transducer Degradation. Applied Mathematical Sciences, 2014, vol. 8, no. 41–44, pp. 2125_2130. DOI: 10.12988/ams.2014.312718 12. Keller A.V., Sagadeeva M.A. The Numerical Solution of Optimal and Hard Control for Nonstationary System of Leontiev type. Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Seriya: Matematika. Fizika, 2013, vol. 32, no. 19, pp. 57_66. (in Russian) 13. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967. SourceBulletin of the South Ural State University. Series « Mathematical modeling and programming». 2014. Vol. 7. №4. С. 132-139. (The main) |