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Information × Registration Number 0221U102909, 0116U002066 , R & D reports Title Stochastic systems with singular interaction popup.stage_title Head Dorohovtsev Andrii A., Доктор фізико-математичних наук Registration Date 11-02-2021 Organization Institute of Mathematics of the National Academy of Sciences of Ukraine popup.description2 The aim of the work is to study the geometric properties of complicated stochastic processes and the development of new methodology of investigating the limitting behaviour of sequences of Markov processes and chains that are perturbed in the neighborhood of some singular point (or set) and are attracted to a known distribution outside an arbitrary neighborhood of this point. Methods of investigation include stochastic analysis, stochastic differential equations, theory of Markov processes, functional analysis. Main research directions: development of the new methodology of studying the limitting behaviour of sequences of Markov processes and chains that are perturbed in the neighborhood of a singular point (or set) and are attracted to a known distribution outside arbitrary neighborhood of this point. A new class of stochastic differential equations that are governed by stochastic flows with singular interaction is defined and studied. Solutions to initial-boundary value problems similar to Dirichlet problem, Neuman problem, third boundary problem, and also some conjugating problems for a class of pseudo-differential equations that correspond to multidimensional stable stochastic processes are constructed. Complex study of local properties of Levy processes is performed via the transition density investigation. Main attention in thtis project is devoted to asymptotic properties of transition densities in small time. Studied properties are necessary for further applications in problems of statistical estimation and stochastic approximation. Analogue of Le Gall's formula for the area of a small neighborhood of a two-dimensional Gaussian integrator's trajectory. Existence of local times for diffusion processes that are not uniformly elliptic is proved via uniform estimates on transition densities. Product Description popup.authors Glinyana Katerina V. Konarovskii Vitalii V. Pilipenko Andrii Yu. Portenko Мykola І. Rudenko Oleksii V. Riabov Georgii V. popup.nrat_date 2021-02-11 Close