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Ergodic Markov processes and Poisson equations (lecture notes)
P. 457-511.
Veretennikov A.
Ergodic properties of Markov chains are studied, with the emphasis to convergence rate bounds and to applications to Poisson equations
Publication based on the results of:
In book
Heidelberg : Springer, 2017
Blank M., , in : IX Moscow International Conference on Operations Research (ORM2018) Moscow, October 22–27, 2018. : [б.и.], 2018. P. 269-272.
The celebrated Nagel–Schreckenberg model allows to study a reasonably
large class of one-dimensional traffic flow models with parallel updates.
Unfortunately further generalizations of this construction turn out
to be not especially fruitful. Namely, numerical simulations of such
generalizations did not demonstrate stable behavior qualitatively
different from the original model, and more to the point their
mathematical treatment is still not available ...
Added: February 8, 2019
Blank M., Доклады Академии наук 2013 Т. 448 № 6 С. 629-632
We give conditions for unique ergodicity for a discrete time collective
random walk on a continuous circle. Individual particles in this collective
motion perform independent (and different) random walks conditioned
by the assumption that the particles cannot overrun each other.
Deterministic version of this system is studied as well. ...
Added: November 25, 2014
Decrouez G. G., Robinson A., Risk Analysis: An International Journal 2013 Vol. 33 No. 12 P. 2142-2153
We propose a new modeling approach for inspection data that provides a more useful interpretation of the patterns of detections of invasive pests, using cargo inspection as a motivating example. Methods that are currently in use generally classify shipments according to their likelihood of carrying biosecurity risk material, given available historical and contextual data. Ideally, ...
Added: September 29, 2014
Moltchanov D., Ometov A., Koucheryavy E., Computer Communications 2019 No. 146 P. 110-120
One of the recent major steps towards 5G cellular systems is standardization of 5G New Radio (NR) operatingin the millimeter wave (mmWave) frequency band. This radio access technology (RAT) will potentially provideextraordinary rates at the access interface enabling the set of new bandwidth-greedy applications. However,the blockage of the line-of-sight (LoS) path between 3GPP NR access ...
Added: October 31, 2019
Zlotnik A.A., Zlotnik I.A., Doklady Mathematics 2017 Vol. 95 No. 2 P. 129-135
A new fast direct algorithm for implementing a finite element method (FEM) of order on rectangles as applied to boundary value problems for Poisson-type equations is described that extends a well-known algorithm for the case of difference schemes or bilinear finite elements (n = 1). Its core consists of fast direct and inverse algorithms for ...
Added: February 28, 2017
Царев А. С., Хайров Э. М., Гайдамака Ю. В. et al., Системы и средства информатики 2021 Т. 31 № 2 С. 16-25
Industrial Internet of Things (IIoT) is one of the key features of Industry 4.0 that stands for a new approach to design industrial processes using the latest technological advances. However, before implementing new processes, it is necessary to understand the characteristics and limitations that arise due to technical complexity of the system. To solve this ...
Added: December 12, 2022
Konakov V., Markova A., Automation and Remote Control 2017 Vol. 78 No. 8 P. 1438-1448
We consider the diffusion process and its approximation by Markov chain with nonlinear unbounded trends. The usual parametrix method is not applicable because these models have unbounded trends. We describe a procedure that allows to exclude nonlinear unbounded trend and move to stochastic differential equation with bounded drift and diffusion coefficients. A similar procedure is ...
Added: August 28, 2017
Blank M., , in : IX Moscow International Conference on Operations Research (ORM2018) Moscow, October 22–27, 2018. : [б.и.], 2018. P. 325-328.
The aim of this work is to analyze a circle of questions related to the notion of
recurrence in general general Markov chains. Being well known in two very different
subfields of random systems: lattice random walks and ergodic theory of continuous
selfmaps, the recurrence property is next to being neglected in general theory of
Markov chains (perhaps except ...
Added: February 8, 2019
Veretennikov A., Veretennikova M., / 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
А. В. Ткаченко, Вестник Московского университета. Серия 1: Математика. Механика 2013 № 2 С. 12-17
This paper is devoted to $M|GI|1|\infty$ queueing system with unreliable server and customer service times depending on the system state. Condition of ergodicity and generating function are found in the stationary state. ...
Added: March 27, 2013
А.А. Злотник, И.А. Злотник, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2017 Т. 473 № 2 С. 131-137
Представлен новый быстрый прямой алгоритм реализации метода конечных элементов (МКЭ) порядка n>=2 на прямоугольниках для решения различных краевых задач для уравнений типа Пуассона. Он обобщает хорошо известный алгоритм для случая разностных схем или билинейных КЭ (n=1), использующий версии быстрого дискретного преобразования Фурье (БДПФ). Его ядром являются быстрые прямой и обратный алгоритмы разложения по собственным векторам ...
Added: October 25, 2016
Kashtanov V., Kondrashova E. V., Надежность 2012 № 1 (40) С. 52-68
The present paper analyses characteristics of an input flow controlled by Markov chain. At the base of construction of the given flow rests the model of semi-Markov process. The paper has stated and proved the theorem about stationarity of an input flow controlled by Markov chain. In addition to that the theorem proved that in ...
Added: February 25, 2013
Tkachenko A., Moscow University Mathematics Bulletin 2014 Vol. 69 No. 1 P. 37-40
This paper is focused on a multichannel queueing system with heterogeneous servers and regenerative input flow operating in a random environment. The environment can destroy the whole system and the system is reconstructed after that. The necessary and sufficient ergodicity condition is obtained for the system. ...
Added: August 20, 2014
Bezhaeva Z., Oseledets V. I., Обозрение прикладной и промышленной математики 2011 Т. 18 № 4 С. 622-623
Рассматривается стационарная марковская цепь с конечным алфавитом и неприводимой переходной матрицей. Скрытая марковская цепь возникает из марковской цепи при передаче через канал. Получена формула для энтропии скрытой марковской цепи. Эффективность формулы продемонстрирована на конкретных примерах. ...
Added: March 22, 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
Kashtanov V., Kondrashova E. V., Надежность 2012 No. 1 (40) P. 68-83
The present paper analyses characteristics of an input flow controlled by Marcov chain. At the base of construction of the given flow rests the model of semi-Markov process. The paper has stated and proved the teorem about stationarity of an input flow controlled by Markov chain. In addition to that the theorem proved that in ...
Added: April 8, 2013
Ткаченко А.В., Вестник Московского университета. Серия 1: Математика. Механика 2014 № 1 С. 53-57
This paper is focused on multichannel queueing system with heterogeneous servers and regenerative input flow in a random environment. The environment can destroy all the system and then system is reconstructed. Ergodicity condition of the system is obtained. ...
Added: May 11, 2013
Minsk : [б.и.], 2022
This collection of papers includes proceedings of the Thirteenth International Conference "Computer Data Analysis and Modeling: Stochastics and Data Science" organized by the Belarusian State University and held in September 2022 in Minsk. Papers are reviewed by qualied researchers from Belarus and Russia. ...
Added: October 27, 2022
Blank M., Moscow Mathematical Journal 2019 Vol. 19 No. 1 P. 37-50
Using ideas borrowed from topological dynamics and ergodic theory we introduce topological and metric versions of the recurrence property for general Markov chains. The main question of interest here is how large is the set of recurrent points. We show that under some mild technical assumptions the set of non-recurrent points is of zero reference measure. Necessary and ...
Added: October 8, 2019
Konakov V., Markova A., / Cornell University. Series math "arxiv.org". 2014. No. 1412.1607v1.
We consider a sequence of Markov chains weakly convergent to a diffusion. We suppose that a drift term contains a linearly increasing component. The usual parametrix method fails because of this unbounded drift term. We show how to modify the parametrix method to obtain local theorems for this case. ...
Added: January 21, 2015
Shchegolev A., Управление большими системами: сборник трудов 2021 № 90 С. 36-48
The paper studies an improved estimate for the rate of convergence for nonlinear homogeneous discrete-time Markov chains. These processes are nonlinear in terms of the distribution law. Hence, the transition kernels are dependent on the current probability distributions of the process apart from being dependent on the current state. Such processes often act as limits ...
Added: April 21, 2021
Malovichko M., Yavich N., Razorenova A. et al., / Cornell University Library. Series math.RT "arXiv:1808.06395 [math.RT]". 2022.
This paper introduces a novel numerical method for the inverse problem of electroencephalography(EEG). We pose the inverse EEG problem as an optimal control (OC) problem for Poisson’s equation. The optimality conditions lead to a variational system of differential equations. It is discretized directly in finite-element spaces leading to a system of linear equations with a ...
Added: October 31, 2022
Dynnikov I., Hubert P., Skripchenko A., International Mathematics Research Notices 2023 Vol. 2023 No. 8 P. 6461-6503
At the beginning of the 80s, H. Masur and W. Veech started the study of generic properties of interval exchange transformations (IETs) proving that almost every such transformation is uniquely ergodic. About the same time, S. Novikov’s school and French mathematicians independently discovered very intriguing phenomena for classes of measured foliations on surfaces and respective ...
Added: December 28, 2023
Kleptsyn V., Alvarez S., Malicet D. et al., / Cornell University. Series math "arxiv.org". 2015.
Added: June 22, 2016