About the Project
We are interested to study robustness of models based on SDEs from the point of asymptotic, Lyapunov stability, stable manifolds and from the view point of Kupka-Smale stability.
One of the most effective models of random processes is a stochastic differential equation, which arises in many problems in various fields of natural and social sciences such as the radio-physics, sound navigation and ranging, seismography, meteorology, evolution of biological populations, theory of signals and automatic control, filtration, econometrics, financial mathematics, and so on. It is natural to add a stochastic process to take into account the uncertainty about future system development.
Objectives
One of goals is to study the important types of ODEs, SDEs and PDEs and the analysis of the qualitative behavior of solutions of such equations, whose results will open new perspectives in stochastic analysis. Sufficient conditions of asymptotic equivalence of SDEs solutions and some nonrandom function (e.g., solutions of ODEs) will be obtained. Some classes of differential equations perturbed by the Winner and Poisson processes will be considered. Then I will study the asymptotic properties of multidimensional SDEs. The behavior of the polar angle of the solution will considered, as well as the question of when the absolute value of the solution of SDE is equivalent to the absolute value of the solution of an ODE as time tends to infinity. One of the main thrust of the research is to establish the existence of a finite Lyapunov spectrum and invariant (Sobolev) manifolds for multidimensional SDE. Novel techniques of the theory of dynamical systems of SDEs with singular coefficients and a stochastic version of the Kupka-Smale theorem will be used in the research.
Background
Several scientific and educational joint projects have been already initiated between Ukraine and Norway. The implementation of this project will be a natural continuation of the successful project "Norway-Ukrainian cooperation in mathematical education".
Financing
This project has received funding through the MSCA4Ukraine project, which is funded by the European Union
Cooperation
We will use international network of cooperation partners during the project period.
We aim at collaborating with A. Pilipenko (Kyiv), who is an internationally renowned expert in stochastic dynamical systems/stochastic analysis, O. Klesov (Kyiv), who is one of the authors theory of pseudo-regular veraing function and F. Benth (UiO), G. Di Nunno (UiO), who are internationally leading researchers in stochastic pde theory/stochastic analysis.
All collaborators will contribute with their complementary expertise to the successful realization of the project.