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Registration Number

0216U003583, 0111U001002 , R & D reports

Title

Stochastic analysis of complex systems

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Head

Dorogovtsev Andrey Anatolievich, Доктор фізико-математичних наук

Registration Date

01-02-2016

Organization

Institute of mathematics NAS of Ukraine

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The aim of the research is to study complex stochastic systems which cannot be analysed in the framework of existing classical theories and the description of which needs new approaches. Methods of research - stochastic analysis, stochastic differential equations, theory of Markov processes, functional analysis. The main directions of the research are connected with ordinary stochastic flows and stochastic flows with reflection, geometric structure of complex stochastic systems, stochastic differential equations with the Levy noise, Markov processes with discrete and continuous time, stochastic processes on Lie groups. Stochastic flows with coalescence and smooth stochastic flows were studied. Stochastic flows described by stochastic differential equations with non-regular coefficients and stochastic flows generated by stochastic differential equations with reflection were investigated. The asymptotic behaviour of all moments of the process of the interparticle distance and of mixed moments of the stochastic flow was established. Several conditions sufficient for a set to have finite unconditional entropy were given, and an example of a set in the Hilbert space with different geometric and unconditional entropies was constructed. The asymptotics of tending to zero of a sequence of functionals of discrete stochastic flows was obtained. The core and the form of the generator of the semigroup of the m-point motion of the Arratia flow were found. A full description of a continuous in square mean semigroup which consists of finite-dimensional projectors was presented. For an integrator a renormalization for the Fourier-Wiener transform of the self-intersection local time was constructed, and the rate of growth of the number of self-intersections with the growth of the distance between the initial and final points was established. A representation of the Vasilyev invariants for braids in the form of iterated Stratonovich integrals was obtained. It was shown that for the angle the Wiener process winds around the origin over small intervals of time the weak principle of large deviations is valid. For stochastic differential equations with the Levy noise with discontinuous drift a theorem of the existence and uniqueness of the solution, of the continuous dependence of the solution on the coefficients and initial conditions and of the differentiability with respect to the initial condition were proved, and the Lyapunov exponents were computed. For Markov processes sufficient conditions of the existence of an exponential coupling and of the hypocoercitivity were established, and for local perturbations of Markov chains the invariance principle was established. Several limit theorems for sequences of local perturbations of Markov processes, where the role of the limit processes is played by diffusions with singular drift and diffusions with non-local Wentzel-Feller limit conditions, were obtained.

Product Description

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Ізюмцева О.Л.
Глиняна К.В.
Дороговцев А.А.
Коренковська Я.А.
Кузнецов В.О.
Кулик О.М.
Пилипенко А.Ю.
Портенко М.І.
Руденко О.В.
Рябов Г.В.
Рябова Н.Ф.
Танцюра М.В.
Фомічьов В.В.

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2020-04-02