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Diffusion processes on the Thoma cone
Functional Analysis and Its Applications. 2016. Vol. 50. No. 3. P. 237–240.
The Thoma cone is a certain infinite-dimensional space that arises in the representation theory of the infinite symmetric group. The present note is a continuation of a paper by A. M. Borodin and the author (Electr. J. Probab. 18 (2013), no. 75), where a 2-parameter family of continuous-time Markov processes on the Thoma cone was constructed. The purpose of the note is to show that these processes are diffusions.
Девяткова И. Е., Olshanski G., Алгебра и анализ 2026 Т. 38 № 1 С. 139–197
Denote by S(\infty) the infinite symmetric group formed by finitary permutations of the set of natural numbers; this is a countable group. We define its virtual group algebra; it is a completion of the standard group algebra С[S(\infty)]. The virtual group algebra is obtained from the finite-dimensional group algebras C[S(n)] as the result of a ...
Added: February 18, 2026
Smirnov E., Тутубалина А. А., М.: МЦНМО, 2026.
Книга написана по материалам семестрового курса «Симметрические функции», читавшегося авторами в Независимом московском университете и на факультете математики Высшей школы экономики. В ней излагаются как классические, так и недавние результаты о симметрических функциях и их обобщениях, причем основное внимание уделяется комбинаторным аспектам теории. Курс снабжен большим количеством задач и упражнений, ко многим из которых приводятся ...
Added: December 2, 2025
Smirnov E., Тутубалина А. А., Математический сборник 2021 Т. 212 № 10 С. 131–151
Subword complexes were defined by A.Knutson and E.Miller in 2004 for describing Gröbner degenerations of matrix Schubert varieties. The facets of such a complex are indexed by pipe dreams, or, equivalently, by the monomials in the corresponding Schubert polynomial. In 2017 S.Assaf and D.Searles defined a basis of slide polynomials, generalizing Stanley symmetric functions, and ...
Added: September 29, 2021
Olshanski G., Cuenca C., Moscow Mathematical Journal 2020 Vol. 20 No. 4 P. 645–694
The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. At the top of the hierarchy, there are two closely related families, the Askey–Wilson and q
-Racah polynomials. As it is well known, their construction admits a generalization leading to remarkable orthogonal symmetric polynomials in several variables.
We construct an analogue of the ...
Added: January 19, 2021
Olshanski G., Функциональный анализ и его приложения 2018 Т. 52 № 4 С. 86–88
The Thoma simplex Ω is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are probability measures on Ω depending on three continuous parameters. One of them is the parameter of the Jack symmetric functions, and in the limit as it goes to 0, the z-measures turn into the ...
Added: May 26, 2019
Olshanski G., Journal of Combinatorial Theory, Series A 2019 Vol. 162 P. 65–117
Let Sym denote the algebra of symmetric functions and P_μ( · ; q, t) and Q_μ( · ; q, t) be the Macdonald symmetric functions (recall that they differ by scalar factors only). The (q, t)-Cauchy identity
expresses the fact that the P_μ( · ; q, t)’s form an orthogonal basis in Sym with respect to ...
Added: May 25, 2019
Бородин А. Н., Switzerland: Birkhauser/Springer, 2017.
This book provides a rigorous yet accessible introduction to the theory of stochastic processes. A significant part of the book is devoted to the classic theory of stochastic processes. In turn, it also presents proofs of well-known results, sometimes together with new approaches. Moreover, the book explores topics not previously covered elsewhere, such as distributions of ...
Added: January 25, 2019
Veretennikov A., Теорiя Ймовiрностей та Математична Статистика 2016 Vol. 95 P. 178–188
Poisson equation in the whole space was studied earlier for so called ergodic generators L corresponding to homogeneous Markov diffusions. Solving this equation is one of the main tools for diffusion approximation in the theory of stochastic averaging and homogenisation. Here a similar equation with a potential is considered, firstly because it is natural for ...
Added: October 17, 2017
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
Konakov V., Menozzi S., Electronic Journal of Probability 2017 Vol. 22 P. 1–47
We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of Hölder continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices. ...
Added: May 12, 2017
Konakov V., Kozhina A., Menozzi S., ESAIM: Probability and Statistics 2017 Vol. 21 P. 88–112
We study the sensitivity of the densities of non degenerate diffusion processes and related Markov Chains with respect to a perturbation of the coefficients. Natural applications of these results appear in models with misspecified coefficients or for the investigation of the weak error of the Euler scheme with irregular coefficients. ...
Added: April 14, 2017
Olshanski G., Borodin A., Cambridge: Cambridge University Press, 2017.
Representation theory of big groups is an important and quickly developing part of modern mathematics, giving rise to a variety of important applications in probability and mathematical physics. This book provides the first concise and self-contained introduction to the theory on the simplest yet very nontrivial example of the infinite symmetric group, focusing on its ...
Added: March 21, 2017
Shaposhnikov S., Manita O. A., Journal of Dynamics and Differential Equations 2016 Vol. 28 No. 2 P. 493–518
We study the Cauchy problem for Fokker–Planck–Kolmogorov equations for finite measures with unbounded and degenerate coefficients. Sufficient conditions for the existence and uniqueness of solutions are given. ...
Added: February 15, 2017
Zakharov V., Varnovsky N. P., Shokurov A. V., Moscow University Computational Mathematics and Cybernetics 2017 Vol. 41 No. 1 P. 60–70
We study some simple models of confidential databases in cloud computing systems. In the framework of these models we introduce a concept of deductive security for queries to such databases, find necessary and sufficient conditions of deductive security, and describe some classes of queries which satisfy these requirements. ...
Added: October 13, 2016
Zakharov V., Варновский Н. П., Шокуров А. В., Вестник Московского университета. Серия 15: Вычислительная математика и кибернетика 2017 № 1 С. 38–44
We study some simple models of confidential databases in cloud computing systems. In the framework of these models we introduce a concept of deductive security for queries to such databases, find necessary and sufficient conditions of deductive security, and describe some classes of queries which satisfy these requirements. ...
Added: October 13, 2016
Ulyanov V. V., Goetze F., A. Naumov, Journal of Theoretical Probability 2017 Vol. 30 No. 3 P. 876–897
In this paper we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. Applications to classical and free probability theory are discussed. ...
Added: March 17, 2016
Podolskaya O., В кн.: Материалы X молодежной научной школы по дискретной математике и ее приложениям.: М.: Издательство ИПМ РАН, 2015. С. 56–58.
Доказано, что сложность реализации произвольной симметрической булевой функции f от n переменных для f не равной тождественно единице при реализации схемами в базисе антицепных функций, т.е. функций, принимающих значение 1 лишь на попарно несравнимых наборах, равна min(k(f),n-k(f)+2), где k(f) - количество слоев, на которых функция f равна 1. ...
Added: January 11, 2016
Bogachev V., Veretennikov A., Shaposhnikov S., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2015 Т. 460 № 5 С. 507–511
Методами уравнений в частных производных установлены достаточные условия дифференцируемости инвариантных мер диффузионных процессов по параметру ...
Added: October 11, 2015