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Papers by Yana Belopolskaya

Research paper thumbnail of Probabilistic Interpretations of Nonlinear Second Order Parabolic Equations and Systems and Numerical Algorithms Based on Them

Stochastic Modelling & Computational Sciences

Research paper thumbnail of A Stochastic Approach to the Cauchy-Neumann Problem for Systems of Nonlinear Parabolic Equations

Journal of Mathematical Sciences

We derive stochastic equations to describe reflected diffusion processes associated with the Cauc... more We derive stochastic equations to describe reflected diffusion processes associated with the Cauchy-Neumann problem for systems of nonlinear parabolic equations in non-divergent form in the half-space. As a result, we obtain probabilistic representations of weak solutions to this problem. Keywords Reflected diffusion • Skorokhod's problem • Weak solutions of the Cauchy-Neuman • Systems of nonlinear parabolic equations * Belopolskaya Ya. I.

Research paper thumbnail of Probabilistic Algorithms for Numerical Construction of Classical Solutions to the Cauchy Problem for Nonlinear Parabolic Systems

Analytical and Computational Methods in Probability Theory, 2017

In this paper we have two main goals. One of them is to construct stochastic processes associated... more In this paper we have two main goals. One of them is to construct stochastic processes associated with a class of systems of semilinear parabolic equations which allow to obtain a probabilistic representations of classical solutions of the Cauchy problem for systems from this class. The second goal is to reduce the Cauchy problem solution of a PDE system to solution of a closed system of stochastic relations, prove the existence and uniqueness theorem for the correspondent stochastic system and apply it to develop algorithms to construct numerically the required solution of the PDE system.

Research paper thumbnail of A Probabilistic Representation of the Cauchy Problem Solution for the Multidimensional Schrödinger Equation

Journal of Mathematical Sciences, 2020

We construct a probabilistic representation of the Cauchy problem solution for the Schrödinger eq... more We construct a probabilistic representation of the Cauchy problem solution for the Schrödinger equation 2i∂ t u = −Δu. The result is an extension to the multidimensional case of the previous results by I. Ibragimov, N. Smorodina, and M. Faddeev. Bibliography: 8 titles.

Research paper thumbnail of Models of Markov processes with a random transition mechanism

arXiv: Probability, 2015

The paper deals with a certain class of random evolutions. We develop a construction that yields ... more The paper deals with a certain class of random evolutions. We develop a construction that yields an invariant measure for a continuous-time Markov process with random transitions. The approach is based on a particular way of constructing the combined process, where the generator is defined as a sum of two terms: one responsible for the evolution of the environment and the second representing generators of processes with a given state of environment. (The two operators are not assumed to commute.) The presentation includes fragments of a general theory and pays a particular attention to several types of examples: (1) a queueing system with a random change of parameters (including a Jackson network and, as a special case: a single-server queue with a diffusive behavior of arrival and service rates), (2) a simple-exclusion model in presence of a special `heavy` particle, (3) a diffusion with drift-switching, and (4) a diffusion with a randomly diffusion-type varying diffusion coefficie...

Research paper thumbnail of An approximation method for Navier–Stokes equations based on probabilistic approach

Statistics & Probability Letters, 2003

Research paper thumbnail of Вероятностная интерпретация метода исчезающей вязкости для систем законов сохранения и баланса

Matematicheskie Zametki, 2021

Рассматриваются системы нелинейных параболических уравнений с малым параметром при старшей произв... more Рассматриваются системы нелинейных параболических уравнений с малым параметром при старшей производной и ассоциированные с ними стохастические модели. Показано, что метод исчезающей вязкости, позволяющий выбирать физические решения задачи Коши для систем нелинейных законов сохранения, имеет естественное обоснование в терминах стохастических моделей. Аналогичный результат получен и для балансовых законов. Библиография: 10 названий.

Research paper thumbnail of Stochastic Models for Nonlinear Cross-Diffusion Systems

Statistics and Simulation, 2018

Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for... more Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for a system of quasilinear parabolic equations with crossdiffusion, we treat the PDE system as an analogue of systems of forward Kolmogorov equations for some unknown stochastic processes and derive expressions for their generators. This allows to construct a stochastic representation of the required solution. We prove that introducing stochastic test function we can check that the stochastic system gives rise to the required generalized solution of the original PDE system. Next, we derive a closed stochastic system which can be treated as a stochastic counterpart of the Cauchy problem for a parabolic system with cross-diffusion.

Research paper thumbnail of Stochastic models for fully coupled systems of nonlinear parabolic equations

We construct a probabilistic representation of a system of fully coupled parabolic equations aris... more We construct a probabilistic representation of a system of fully coupled parabolic equations arising as a model describing spatial segregation of interacting population species. We derive a closed system of stochastic equations such that its solution allows to obtain a probabilistic representation of a weak solution of the Cauchy problem for the PDE system. The corresponded stochastic system is presented in the form of a system of stochastic equations describing nonlinear Markov processes and their multiplicative functionals.

Research paper thumbnail of Vanishing Viscosity Method and Diffusion Processes

We construct diffusion processes associated with nonlinear parabolic equations and study their be... more We construct diffusion processes associated with nonlinear parabolic equations and study their behavior as the viscosity (diffusion) coefficients go to zero. It allows to construct regularizations for solutions to hyperbolic equations and systems and study their vnishing viscosity limits. 2000 MSC. 60H15, 35Q30.

Research paper thumbnail of Stochastic models for fully coupled systems of nonlinear parabolic equations

arXiv: Probability, 2017

We construct a probabilistic representation of a system of fully coupled parabolic equations aris... more We construct a probabilistic representation of a system of fully coupled parabolic equations arising as a model describing spatial segregation of interacting population species. We derive a closed system of stochastic equations such that its solution allows to obtain a probabilistic representation of a weak solution of the Cauchy problem for the PDE system. The corresponded stochastic system is presented in the form of a system of stochastic equations describing nonlinear Markov processes and their multiplicative functionals.

Research paper thumbnail of Probabilistic Representation of the Cauchy Problem Solutions for Systems of Nonlinear Parabolic Equations

In this paper we present some results concerning probabilistic approaches to construction of clas... more In this paper we present some results concerning probabilistic approaches to construction of classical and generalized solutions to the Cauchy problem for systems of parabolic equations from two different classes and show key points where there arises a crucial difference between them.

Research paper thumbnail of Stochastic Approach to the Vanishing Viscosity Method

Recent Developments in Stochastic Methods and Applications, 2021

We derive stochastic counterparts for solutions of the forward Cauchy problem for two classes of ... more We derive stochastic counterparts for solutions of the forward Cauchy problem for two classes of nonlinear parabolic equations. We refer to the first class parabolic systems such that coefficients of the higher order terms are the same for each equation in the system and to the second class parabolic systems with different higher order term coefficients. With a simple substitution we reduce a system of the first class to a system which may be interpreted as a system of backward Kolmogorov equations and construct a probabilistic representation of its solution. A different approach based on interpretation of a system under consideration as a system of forward Kolmogorov equations is developed to deal with stochastic counterparts of the second class systems. These approaches allow to reduce the investigation of the vanishing viscosity limiting procedure to a stochastic level which makes its justification to be much easier.

Research paper thumbnail of Vanishing Viscosity Method and Diffusion Processes

We construct diffusion processes associated with nonlinear parabolic equations and study their be... more We construct diffusion processes associated with nonlinear parabolic equations and study their behavior as the viscosity (diffusion) coefficients go to zero. It allows to construct regularizations for solutions to hyperbolic equations and systems and study their vnishing viscosity limits. 2000 MSC. 60H15, 35Q30.

Research paper thumbnail of Probabilistic Algorithms for Numerical Construction of Classical Solutions to the Cauchy Problem for Nonlinear Parabolic Systems

In this paper we have two main goals. One of them is to construct stochastic processes associated... more In this paper we have two main goals. One of them is to construct stochastic processes associated with a class of systems of semilinear parabolic equations which allow to obtain a probabilistic representations of classical solutions of the Cauchy problem for systems from this class. The second goal is to reduce the Cauchy problem solution of a PDE system to solution of a closed system of stochastic relations, prove the existence and uniqueness theorem for the correspondent stochastic system and apply it to develop algorithms to construct numerically the required solution of the PDE system.

Research paper thumbnail of Stochastic Models for Nonlinear Cross-Diffusion Systems

Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for... more Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for a system of quasilinear parabolic equations with cross-diffusion, we treat the PDE system as an analogue of systems of forward Kolmogorov equations for some unknown stochastic processes and derive expressions for their generators. This allows to construct a stochastic representation of the required solution. We prove that introducing stochastic test function we can check that the stochastic system gives rise to the required generalized solution of the original PDE system. Next, we derive a closed stochastic system which can be treated as a stochastic counterpart of the Cauchy problem for a parabolic system with cross-diffusion.

Research paper thumbnail of Yuri Daletskii and the Development of Infinite Dimensional Analysis

EMS Newsletter

ing the next year, Yuri passed all school exams and was subsequently allowed to complete his univ... more ing the next year, Yuri passed all school exams and was subsequently allowed to complete his university programme. He graduated in 1951, having written several research papers by that time. These works became the foundation of his PhD thesis completed under the supervision of Selim Krein, who played an important role in the development of the functional analysis school in Kiev. Soon after, Selim moved to Voronezh University. Yuri was very close to his teacher, always acknowledged his influence and remained in contact with him for his whole life. In 1951, Yuri took up a position at the Kiev Polytechnic Institute (The National Technical University of Ukraine at present), where he would remain for the entirety of his career, first as an assistant and eventually as a full professor and member of the Ukrainian Academy of Sciences. Yuri played a major role in forming the mathematical curriculum of the Institute. In the 70s and 80s he developed the mathematical programmes of the departments of Applied Mathematics, Mathematical Methods of System Analysis and the (new at that time) Faculty of Physics and Technology. Additionally, Yuri was one of the leaders of the successful independent postgraduate programme "Mathematics for Engineers", which was taught in Kiev for nearly two decades. Later, in the 90s, he also led mathematical programmes at the newly-founded Soros University. Very soon Yuri became a significant figure on the Kiev mathematical scene. At that time, mathematical life in Soviet research centres was concentrated around big inter-institutional seminars, famous examples being Gelfand's and Dobrushin's seminars in Moscow. Yuri supervised major Kiev seminars "Random processes and distributions in functional spaces" (together with A. Skorokhod) and "Algebraic Structures in Mathematical Physics". He was also an important contributor to the seminar "Group methods in solid-state physics". Due to his friendly and energetic personality and vast knowledge of a variety of mathematical fields, Yuri played a

Research paper thumbnail of Stochastic models for forward systems of nonlinear parabolic equations

Research paper thumbnail of Probabilistic Methods in Fluids: Proceedings of the Swansea 2002 Workshop: Wales, UK, 14-19 April 2002

Research paper thumbnail of Characterization of elliptic distributions

Journal of Mathematical Sciences, 2005

ABSTRACT

Research paper thumbnail of Probabilistic Interpretations of Nonlinear Second Order Parabolic Equations and Systems and Numerical Algorithms Based on Them

Stochastic Modelling & Computational Sciences

Research paper thumbnail of A Stochastic Approach to the Cauchy-Neumann Problem for Systems of Nonlinear Parabolic Equations

Journal of Mathematical Sciences

We derive stochastic equations to describe reflected diffusion processes associated with the Cauc... more We derive stochastic equations to describe reflected diffusion processes associated with the Cauchy-Neumann problem for systems of nonlinear parabolic equations in non-divergent form in the half-space. As a result, we obtain probabilistic representations of weak solutions to this problem. Keywords Reflected diffusion • Skorokhod's problem • Weak solutions of the Cauchy-Neuman • Systems of nonlinear parabolic equations * Belopolskaya Ya. I.

Research paper thumbnail of Probabilistic Algorithms for Numerical Construction of Classical Solutions to the Cauchy Problem for Nonlinear Parabolic Systems

Analytical and Computational Methods in Probability Theory, 2017

In this paper we have two main goals. One of them is to construct stochastic processes associated... more In this paper we have two main goals. One of them is to construct stochastic processes associated with a class of systems of semilinear parabolic equations which allow to obtain a probabilistic representations of classical solutions of the Cauchy problem for systems from this class. The second goal is to reduce the Cauchy problem solution of a PDE system to solution of a closed system of stochastic relations, prove the existence and uniqueness theorem for the correspondent stochastic system and apply it to develop algorithms to construct numerically the required solution of the PDE system.

Research paper thumbnail of A Probabilistic Representation of the Cauchy Problem Solution for the Multidimensional Schrödinger Equation

Journal of Mathematical Sciences, 2020

We construct a probabilistic representation of the Cauchy problem solution for the Schrödinger eq... more We construct a probabilistic representation of the Cauchy problem solution for the Schrödinger equation 2i∂ t u = −Δu. The result is an extension to the multidimensional case of the previous results by I. Ibragimov, N. Smorodina, and M. Faddeev. Bibliography: 8 titles.

Research paper thumbnail of Models of Markov processes with a random transition mechanism

arXiv: Probability, 2015

The paper deals with a certain class of random evolutions. We develop a construction that yields ... more The paper deals with a certain class of random evolutions. We develop a construction that yields an invariant measure for a continuous-time Markov process with random transitions. The approach is based on a particular way of constructing the combined process, where the generator is defined as a sum of two terms: one responsible for the evolution of the environment and the second representing generators of processes with a given state of environment. (The two operators are not assumed to commute.) The presentation includes fragments of a general theory and pays a particular attention to several types of examples: (1) a queueing system with a random change of parameters (including a Jackson network and, as a special case: a single-server queue with a diffusive behavior of arrival and service rates), (2) a simple-exclusion model in presence of a special `heavy` particle, (3) a diffusion with drift-switching, and (4) a diffusion with a randomly diffusion-type varying diffusion coefficie...

Research paper thumbnail of An approximation method for Navier–Stokes equations based on probabilistic approach

Statistics & Probability Letters, 2003

Research paper thumbnail of Вероятностная интерпретация метода исчезающей вязкости для систем законов сохранения и баланса

Matematicheskie Zametki, 2021

Рассматриваются системы нелинейных параболических уравнений с малым параметром при старшей произв... more Рассматриваются системы нелинейных параболических уравнений с малым параметром при старшей производной и ассоциированные с ними стохастические модели. Показано, что метод исчезающей вязкости, позволяющий выбирать физические решения задачи Коши для систем нелинейных законов сохранения, имеет естественное обоснование в терминах стохастических моделей. Аналогичный результат получен и для балансовых законов. Библиография: 10 названий.

Research paper thumbnail of Stochastic Models for Nonlinear Cross-Diffusion Systems

Statistics and Simulation, 2018

Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for... more Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for a system of quasilinear parabolic equations with crossdiffusion, we treat the PDE system as an analogue of systems of forward Kolmogorov equations for some unknown stochastic processes and derive expressions for their generators. This allows to construct a stochastic representation of the required solution. We prove that introducing stochastic test function we can check that the stochastic system gives rise to the required generalized solution of the original PDE system. Next, we derive a closed stochastic system which can be treated as a stochastic counterpart of the Cauchy problem for a parabolic system with cross-diffusion.

Research paper thumbnail of Stochastic models for fully coupled systems of nonlinear parabolic equations

We construct a probabilistic representation of a system of fully coupled parabolic equations aris... more We construct a probabilistic representation of a system of fully coupled parabolic equations arising as a model describing spatial segregation of interacting population species. We derive a closed system of stochastic equations such that its solution allows to obtain a probabilistic representation of a weak solution of the Cauchy problem for the PDE system. The corresponded stochastic system is presented in the form of a system of stochastic equations describing nonlinear Markov processes and their multiplicative functionals.

Research paper thumbnail of Vanishing Viscosity Method and Diffusion Processes

We construct diffusion processes associated with nonlinear parabolic equations and study their be... more We construct diffusion processes associated with nonlinear parabolic equations and study their behavior as the viscosity (diffusion) coefficients go to zero. It allows to construct regularizations for solutions to hyperbolic equations and systems and study their vnishing viscosity limits. 2000 MSC. 60H15, 35Q30.

Research paper thumbnail of Stochastic models for fully coupled systems of nonlinear parabolic equations

arXiv: Probability, 2017

We construct a probabilistic representation of a system of fully coupled parabolic equations aris... more We construct a probabilistic representation of a system of fully coupled parabolic equations arising as a model describing spatial segregation of interacting population species. We derive a closed system of stochastic equations such that its solution allows to obtain a probabilistic representation of a weak solution of the Cauchy problem for the PDE system. The corresponded stochastic system is presented in the form of a system of stochastic equations describing nonlinear Markov processes and their multiplicative functionals.

Research paper thumbnail of Probabilistic Representation of the Cauchy Problem Solutions for Systems of Nonlinear Parabolic Equations

In this paper we present some results concerning probabilistic approaches to construction of clas... more In this paper we present some results concerning probabilistic approaches to construction of classical and generalized solutions to the Cauchy problem for systems of parabolic equations from two different classes and show key points where there arises a crucial difference between them.

Research paper thumbnail of Stochastic Approach to the Vanishing Viscosity Method

Recent Developments in Stochastic Methods and Applications, 2021

We derive stochastic counterparts for solutions of the forward Cauchy problem for two classes of ... more We derive stochastic counterparts for solutions of the forward Cauchy problem for two classes of nonlinear parabolic equations. We refer to the first class parabolic systems such that coefficients of the higher order terms are the same for each equation in the system and to the second class parabolic systems with different higher order term coefficients. With a simple substitution we reduce a system of the first class to a system which may be interpreted as a system of backward Kolmogorov equations and construct a probabilistic representation of its solution. A different approach based on interpretation of a system under consideration as a system of forward Kolmogorov equations is developed to deal with stochastic counterparts of the second class systems. These approaches allow to reduce the investigation of the vanishing viscosity limiting procedure to a stochastic level which makes its justification to be much easier.

Research paper thumbnail of Vanishing Viscosity Method and Diffusion Processes

We construct diffusion processes associated with nonlinear parabolic equations and study their be... more We construct diffusion processes associated with nonlinear parabolic equations and study their behavior as the viscosity (diffusion) coefficients go to zero. It allows to construct regularizations for solutions to hyperbolic equations and systems and study their vnishing viscosity limits. 2000 MSC. 60H15, 35Q30.

Research paper thumbnail of Probabilistic Algorithms for Numerical Construction of Classical Solutions to the Cauchy Problem for Nonlinear Parabolic Systems

In this paper we have two main goals. One of them is to construct stochastic processes associated... more In this paper we have two main goals. One of them is to construct stochastic processes associated with a class of systems of semilinear parabolic equations which allow to obtain a probabilistic representations of classical solutions of the Cauchy problem for systems from this class. The second goal is to reduce the Cauchy problem solution of a PDE system to solution of a closed system of stochastic relations, prove the existence and uniqueness theorem for the correspondent stochastic system and apply it to develop algorithms to construct numerically the required solution of the PDE system.

Research paper thumbnail of Stochastic Models for Nonlinear Cross-Diffusion Systems

Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for... more Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for a system of quasilinear parabolic equations with cross-diffusion, we treat the PDE system as an analogue of systems of forward Kolmogorov equations for some unknown stochastic processes and derive expressions for their generators. This allows to construct a stochastic representation of the required solution. We prove that introducing stochastic test function we can check that the stochastic system gives rise to the required generalized solution of the original PDE system. Next, we derive a closed stochastic system which can be treated as a stochastic counterpart of the Cauchy problem for a parabolic system with cross-diffusion.

Research paper thumbnail of Yuri Daletskii and the Development of Infinite Dimensional Analysis

EMS Newsletter

ing the next year, Yuri passed all school exams and was subsequently allowed to complete his univ... more ing the next year, Yuri passed all school exams and was subsequently allowed to complete his university programme. He graduated in 1951, having written several research papers by that time. These works became the foundation of his PhD thesis completed under the supervision of Selim Krein, who played an important role in the development of the functional analysis school in Kiev. Soon after, Selim moved to Voronezh University. Yuri was very close to his teacher, always acknowledged his influence and remained in contact with him for his whole life. In 1951, Yuri took up a position at the Kiev Polytechnic Institute (The National Technical University of Ukraine at present), where he would remain for the entirety of his career, first as an assistant and eventually as a full professor and member of the Ukrainian Academy of Sciences. Yuri played a major role in forming the mathematical curriculum of the Institute. In the 70s and 80s he developed the mathematical programmes of the departments of Applied Mathematics, Mathematical Methods of System Analysis and the (new at that time) Faculty of Physics and Technology. Additionally, Yuri was one of the leaders of the successful independent postgraduate programme "Mathematics for Engineers", which was taught in Kiev for nearly two decades. Later, in the 90s, he also led mathematical programmes at the newly-founded Soros University. Very soon Yuri became a significant figure on the Kiev mathematical scene. At that time, mathematical life in Soviet research centres was concentrated around big inter-institutional seminars, famous examples being Gelfand's and Dobrushin's seminars in Moscow. Yuri supervised major Kiev seminars "Random processes and distributions in functional spaces" (together with A. Skorokhod) and "Algebraic Structures in Mathematical Physics". He was also an important contributor to the seminar "Group methods in solid-state physics". Due to his friendly and energetic personality and vast knowledge of a variety of mathematical fields, Yuri played a

Research paper thumbnail of Stochastic models for forward systems of nonlinear parabolic equations

Research paper thumbnail of Probabilistic Methods in Fluids: Proceedings of the Swansea 2002 Workshop: Wales, UK, 14-19 April 2002

Research paper thumbnail of Characterization of elliptic distributions

Journal of Mathematical Sciences, 2005

ABSTRACT