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Asymptotic Behavior of Randomly Perturbed Dynamical Systems

Kolomiyets, Yuriy V.
http://rave.ohiolink.edu/etdc/view?acc_num=kent1164124028

Abstract Details

2006, PHD, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
We consider systems of random differential equations. The coefficients of the equations depend on a small parameter. The first equation, “slow” component, Ordinary Differential Equation (ODE), has unbounded highly oscillating in space variable coefficients and random disturbances, which are described by the second equation, “fast” component, with periodic coefficients. Sufficient conditions for weak convergence, as the small parameter goes to zero of the solutions of the “slow” components to the certain random process, are proved. The Classical Diffusion Approximation Theorem (DAT) says that the drift coefficient, of the approximated Stochastic Differential Equation (SDE), includes a derivative with respect to a space variable of the unbounded coefficients (see, e.g. the monograph of A. V. Skorokhod [13], and the bibliography). So, we cannot apply the classical DAT because of the highly oscillating character of dependency on the small parameter of the unbounded coefficient of the “slow” component. On other hand, we cannot apply the Limit Theorem for SDE’s (in the sense of G. L. Kulinich [8], N. I. Portenko [11], M. Freidlin, A. D. Wentzell [4], S. Ya. Makhno [9]), because the “slow” component is an ODE, and consequently has no nonzero diffusion coefficient (the presence of strongly positive diffusion coefficient is a necessary conditions for such kind of the theorems).
Hassan Allouba (Advisor)
38 p.
Mathematics
Diffusion approximation; random equations; asymptotic behavior; hidh oscillation

Recommended Citations

Citations

  • Kolomiyets, Y. V. (2006). Asymptotic Behavior of Randomly Perturbed Dynamical Systems [Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1164124028

    APA Style (7th edition)

  • Kolomiyets, Yuriy. Asymptotic Behavior of Randomly Perturbed Dynamical Systems. 2006. Kent State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1164124028.

    MLA Style (8th edition)

  • Kolomiyets, Yuriy. "Asymptotic Behavior of Randomly Perturbed Dynamical Systems." Doctoral dissertation, Kent State University, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=kent1164124028

    Chicago Manual of Style (17th edition)

kent1164124028
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© 2006, all rights reserved.
This open access ETD is published by Kent State University and OhioLINK.