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## The probabilistic point of view on the generalized fractional partial differential equations

Fractional Calculus and Applied Analysis. 2019. Vol. 22. No. 3. P. 543-600.

Kolokoltsov V.

This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Diﬀerential Equations of Caputo and Riemann-Liouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (Monte-Carlo simulation). In many cases these solutions can be compactly presented via the wide class of operatorvalued analytic functions of the Mittag-Leﬄer type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.

Borodin A., Olshanski G., Journal of Functional Analysis 2012 Vol. 263 P. 248-303

We construct a four-parameter family of Markov processes on infinite Gelfand–Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces a Feller Markov process on the infinite-dimensional boundary of the Gelfand–Tsetlin graph or, equivalently, the space of extreme characters of the infinite-dimensional unitary group U(∞). The process has a ...

Added: February 25, 2013

Kolokoltsov V., Switzerland : Birkhauser/Springer, 2019

This is an advanced text on ordinary differential equations (ODES) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It yields the concise exposition of the fundamentals with the fast, but rigorous and systematic transition to the up-fronts of modern research in linear and nonlinear partial ...

Added: November 14, 2019

Kolokoltsov V., Троева М. С., Математические заметки 2022 Т. 112 № 4 С. 567-585

Мы развиваем идею аппроксимации немарковским случайным блужданием в непрерывном времени (CTRW) систем взаимодействующих частиц, которая приводит к общему классу дробных кинетических мерозначных эволюций с переменным порядком. Мы доказываем корректность полученных новых уравнений и приводим вероятностную формулу их решений. Хотя наш метод достаточно общий, для простоты мы подробно рассматриваем только дробные версии взаимодействующих диффузий. Данную статью можно рассматривать как ...

Added: October 31, 2022

Kozyrev O., European Journal of Scientific Research 2013 Vol. 95 No. 3 P. 436-443

Fractional differential equation of particle transfer in porous and tubular media was obtained in the paper. It differs from the generally accepted ones by the dependence of the effective diffusion coefficient on the concentration. Together with 1D case problem also 3D problem of diffusion in normal random field was analyzed. For scales, lager then correlation ...

Added: March 2, 2013

Olshanski G., Функциональный анализ и его приложения 2016 Т. 49 № 4 С. 61-75

Ранее А. М. Бородин и автор построили 4-параметрическое семейство марковских процессов на дуальном объекте к бесконечномерной унитарной группе. Основной новый результат состоит в том, что эти процессы являются пределами скачкообразных процессов на дуальных объектах к растущим компактным унитарным группам. ...

Added: December 29, 2016

Molchanov S., Bogachev L., Derfel G., , in : Dynamical Systems, Differential Equations and Applications. : AIMS, 2016. P. 132-141.

...

Added: June 22, 2016

A.Manita, , in : Analytical and computational methods in probability theory and its applications (ACMPT-2017). Proceedings of the International Scientific Conference. : M. : RUDN, 2017. P. 644-648.

We consider Markov models of multicomponent systems with synchronizing interaction. Under natural regularity assumptions about the message routing graph, they have nice longtime behavior. We are interested in limit probability laws related to the steady state viewed from the center-of-mass coordinate system. ...

Added: November 14, 2017

Olshanski G., Markov dynamics on the dual object to the infinite-dimensional unitary group / Cornell University. Series math "arxiv.org". 2013. No. 1310.6155.

These are notes for a mini-course of 3 lectures given at the St. Petersburg
School in Probability and Statistical Physics (June 2012). My aim was to
explain, on the example of a particular model, how ideas from the
representation theory of big groups can be applied in probabilistic problems.
The material is based on the joint paper arXiv:1009.2029 by ...

Added: November 16, 2013

Manita A., , in : Analytical and Computational Methods in Probability Theory. Vol. 10684: Analytical and Computational Methods in Probability Theory.: Berlin : Springer, 2017. P. 347-360.

We consider Markov models of multicomponent systems with synchronizing interaction. Under natural regularity assumptions about the message routing graph, they have nice longtime behavior. We are interested in limit probability laws related to the steady state viewed from the center-of-mass coordinate system. ...

Added: July 16, 2018

Tumanov M., Piunovskly A., Plakhov A., Optimal Control Applications and Methods 2019 P. 1-21

Using the tools of the Markov Decision Processes, we justify the dynamic programming approach to the optimal impulse control of deterministic dynamical systems. We prove the equivalence of the integral and diﬀerential forms of the optimality equation. The theory is illustrated by an example from mathematical epidemiology. The developed methods can be also useful for ...

Added: February 27, 2020

Kleptsyna M., Veretennikov A., Mathematical Methods of Statistics 2016 Vol. 25 No. 3 P. 207-218

A new result on stability of an optimal nonlinear filter for a Markov chain with respect to small perturbations on every step is established. An exponential recurrence of the signal is assumed. ...

Added: October 16, 2016

Piatnitski A., Pirogov S. A., Zhizhina E., Applicable Analysis 2019 Vol. 98 No. 1-2 P. 217-231

The goal of the paper is to describe the large time behaviour of a symmetric diffusion in a high-contrast periodic environment and to characterize the limit process under the diffusive scaling. We consider separately the C_0 and L^2 settings. ...

Added: December 5, 2020

Logvinova K. V., Kozyrev O., European Journal of Scientific Research 2010 Vol. 45 No. 3 P. 383-390

In this study we investigate the successive approximation procedure allowed us to derive new equation, valid for the transport of dissolved matter in porous media, or in random force fields, on the macroscopic scale. To do this the diagram technique was exploited. ...

Added: January 9, 2013

Decrouez G. G., Borovkov K., Gilson M., Journal of Applied Probability 2014 Vol. 51 No. 3 P. 837-857

The paper deals with nonlinear Poisson neuron network models with bounded memory dynamics, which can include both Hebbian learning mechanisms and refractory periods. The state of the network is described by the times elapsed since its neurons fired within the post-synaptic transfer kernel memory span, and the current strengths of synaptic connections, the state spaces ...

Added: September 29, 2014

Veretennikov A., Veretennikova M., On convergence rate for homogeneous Markov chains / Cornell University. Series "Working papers by Cornell University". 2019.

New convergence rate asymptotic bound for a class of homogeneous Markov chains is established. ...

Added: November 14, 2019

Anatoly Manita, Queueing Systems 2014 Vol. 76 No. 2 P. 149-180

We consider a stochastic model of clock synchronization in a wireless network of N sensors interacting with one dedicated accurate time server. For large N we find an estimate of the final time sychronization error for global and relative synchronization. The main results concern the behavior of the network on different timescales tN→∞ , N→∞ ...

Added: February 4, 2015

Molchanov S., Bogachev L., Derfel G., Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 2015 Vol. 471 P. 1-19

Added: June 22, 2016

You Z., Osipov Y., Bedrikovetsky P. et al., Chemical Engineering Journal 2014 Vol. 258 P. 374-385

Asymptotic solution for non-linear stage of suspension-colloidal flow in porous media is developed. The expansion is performed behind the concentration front; the zero order approximation coincides with the short-time solution of the linearised system. Using the first order approximation allows significantly enhancing the validity time period for the analytical model if compared with the linearised ...

Added: August 3, 2014

Semin V. G., Качество Инновации Образование, РФ.Москва.Европейский центр по качеству 2014 № 11 С. 61-66

The article presents the results of stochastic modeling of flow of power in a hierarchical structure of the political model of Government - civil society using the apparatus of the equations of Fokker-Planck-Kolmogorov equation. These results are a further development of the well-known mathematical implementation of the General scheme of the system "power-society"that describes the ...

Added: March 14, 2015

Kozyrev O., Logvinova K. V., European Journal of Scientific Research 2010 Vol. 45 No. 1 P. 64-78

In this study we investigate impurity dynamics in media with random characteristics for different physical assumptions. We first discuss possible models for diffusion in porous media on the small scale. Then we describe some random processes, which are able to represent the disorder in such a medium. After that, we recall the definition of a ...

Added: January 14, 2013

Furmanov K. K., Nikol'skii I. M., Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253

Added: December 22, 2016

Sirotin V., Arkhipova M., Dubrova T. A. et al., Bielsko-Biala : University of Bielsko-Biala Press, 2016

The main attributes of modern enterprises should be the flexibility and the ability of forecasting the future. Constant adaptation to the changing environment and the rapidity of undertaking certain actions which are conditioned by specific situations determine the rules for the future position of market competition. Effective and efficient adjustment of the company in line ...

Added: November 2, 2016

Litvin Y. V., Абрамов И. В., Технологии техносферной безопасности 2016 № 66

Advanced approach to the assessment of a random time of arrival fire fighting calculation on the object of protection, the time of their employment and the free combustion. There is some quantitative assessments with the review of analytical methods and simulation ...

Added: August 27, 2016

Maslov V., Теоретическая и математическая физика 2019 Т. 201 № 1 С. 65-83

We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint
using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes
a boson at the instant of separating from a fermionic nucleus. We study the further transformations of
boson and fermion states of separation in a ...

Added: November 1, 2019