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On the Ambrosio–Figalli–Trevisan Superposition Principle for Probability Solutions to Fokker–Planck–Kolmogorov Equations
Journal of Dynamics and Differential Equations. 2021. Vol. 33. No. 2. P. 715-739.
New results on the superposition principle and Fokker-Planck-Kolmogorov equations are obtained.
Bogachev V., Рёкнер М., Shaposhnikov S., 2023 Т. 68 № 3 С. 420-455
В статье дан обзор нескольких направлений исследований, связанных с работами А. Н. Колмогорова о параболических и эллиптических уравнениях Фоккера–Планка–Колмогорова для переходных и стационарных вероятностей диффузионных процессов. Приведены основные результаты о существовании решений, единственности, свойствах плотностей решений. Упомянуты открытые вопросы в этой области. ...
Added: December 14, 2023
Bogachev V., Shaposhnikov S., REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES 2021 Vol. 66 No. 1 P. 67-81
New results on elliptic equations degenerating at infinity and uniqueness of probability solutions to the Kolmogorov equation are obtained. ...
Added: October 29, 2021
Bogachev V., Shaposhnikov S., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2023 Т. 513 № 1 С. 21-26
Доказано существование измеримых по параметру решений уравнений Фоккера–Планка–Колмогорова с коэффициентами, измеримо зависящими от данного параметра. ...
Added: February 24, 2024
V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, Doklady Mathematics 2020 Vol. 102 No. 3 P. 464-467
We give a solution to the Kolmogorov problem on uniqueness of probability solutions to a parabolic Fokker–Planck–Kolmogorov equation. ...
Added: October 31, 2022
Bogachev V., Krasovitskiy T., Shaposhnikov S., Doklady Mathematics 2021 Vol. 103 No. 3 P. 108-112
New results on Fokker-Planck-Kolmogorov equations are obtained. ...
Added: October 29, 2021
Bogachev V., Krasovitskiy T., Shaposhnikov S., Sbornik Mathematics 2021 Vol. 212 No. 6 P. 745-781
New results on uniqueness of solutions to Fokker-Planck-Kolmogorov equations are obtained. ...
Added: October 28, 2021
Bogachev V., Roeckner M., Shaposhnikov S., Journal of Evolution Equations 2013 Vol. 13 No. 3 P. 577-593
We prove a new uniqueness result for solutions to the Cauchy problem for highly degenerate second order parabolic Fokker-Planck-Kolmogorov equations on the whole space. A novelty is also our class of solutions in which uniqueness holds. This result considerably improves a number of previously known uniqueness theorems in the theory of parabolic equations and is ...
Added: February 26, 2014
Bogachev V., Shaposhnikov A. V., Shaposhnikov S., Calculus of Variations and Partial Differential Equations 2019 Vol. 58 No. 5(176) P. 1-16
Log-Sobolev-type inequalities for solutions to stationary Fokker-Planck-Kolmogorov equations are obtained. ...
Added: November 16, 2019
Bogachev V., Roeckner M., Shaposhnikov S., Doklady Mathematics 2018 Vol. 98 No. 2 P. 452-457
New results are obtained on convergence to stationary measures in nonlinear Fokker-Planck-Kolmogorov equations. ...
Added: November 15, 2019
Bogachev V., Крылов Н. В., Рёкнер М. et al., М., Ижевск : НИЦ Регулярная и хаотическая динамика, 2013
Изложена современная теория уравнений Фоккера - Планка - Колмогорова. ...
Added: March 5, 2014
Bogachev V., Shaposhnikov S., Journal of Evolution Equations 2020 Vol. 20 No. 2 P. 355-374
We prove a formula representing solutions to parabolic Fokker-Planck-Kolmogorov equations with coefficients of low regularity. This formula is applied for proving the continuity of solution densities under broad assumptions and obtaining upper bounds for them. In the case of diffusion coefficients of class VMO, we show that the solution density is locally integrable to any power. ...
Added: October 23, 2020
Bogachev V., Da Prato G., Roeckner M., Communications in Partial Differential Equations 2011 Vol. 36 P. 925-939
We develop a general technique to prove uniqueness of solutions for Fokker–Planck equations on infinite dimensional spaces. We illustrate this method by implementing it for Fokker–Planck equations in Hilbert spaces with Kolmogorov operators with irregular coefficients and both non-degenerate or degenerate second order part. ...
Added: February 26, 2014
Bogachev V., Roeckner M., Shaposhnikov S., Journal of Mathematical Sciences 2015 Vol. 207 No. 2 P. 147-165
New uniquness results for degenerate Fokker-Planck-Kolmogorov equations are obtained ...
Added: November 15, 2017
Bogachev V., Shaposhnikov S., Doklady Mathematics 2017 Vol. 96 No. 3 P. 583-586
New results concerning the local integrability of any order and continuity of solution densities of Fokker–Planck–Kolmogorov equations with nondifferentiable coefficients are obtained. ...
Added: February 17, 2018
Bogachev V., Roeckner M., Shaposhnikov S., Journal of Functional Analysis 2019 Vol. 276 No. 12 P. 3681-3713
Convergence in variation of solutions of nonlinear Fokker-Planck-Kolmogorov equations to stationary measures has been studied. ...
Added: November 16, 2019
V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, Doklady Mathematics 2018 Vol. 98 No. 2 P. 475-479
New results are obtained on non-Uniqueness of probability solutions to the two-dimensional stationary Fokker-Planck-Kolmogorov equation. ...
Added: November 15, 2019
Bogachev V., Popova S., Shaposhnikov S., Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni 2019 Vol. 30 No. 1 P. 205-221
New results on the Sobolev regularity of solutions to Fokker-Planck-Kolmogorov equations with integrable drifts are obtained. ...
Added: November 16, 2019
Bogachev V., Shaposhnikov A. V., Shaposhnikov S., 2018 Vol. 98 No. 3 P. 559-563
New esatimates are obtained for solutions to Fokker-Planck-Kolmogorov equations with integrable drifts. ...
Added: November 15, 2019
Bogachev V., Салахов Д. И., Shaposhnikov S., Алгебра и анализ 2023 Т. 35 № 5 С. 11-38
Исследуются нелинейные уравнения Фоккера–Планка–Колмогорова. Получены достаточные условия существования и единственности неотрицательного решения с заданным значением интеграла. Обоснована сходимость решений задачи Коши к решению стационарного уравнения. Важным отличием от известных результатов является весьма общий вид нелинейности, позволяющий одновременно рассматривать локальную и нелокальную зависимость коэффициентов от решения. ...
Added: February 24, 2024
Bogachev V., Roeckner M., Shaposhnikov S., Journal of Mathematical Sciences 2019 Vol. 242 No. 1 P. 69-84
New results on convergence to stationary distributions for solutions to nonlinear Fokker-Planck-Kolmogorov equations are obtained. ...
Added: November 16, 2019
Bogachev V., Roeckner M., Shaposhnikov S., Doklady Mathematics 2019 Vol. 100 No. 1 P. 363-366
The superposition principle for Fokker-Planck-Kolmogorov equations is generalized. ...
Added: November 16, 2019
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
Pahomov F., Известия РАН. Серия математическая 2016 Т. 80 № 6 С. 173-216
Полимодальная логика доказуемости
GLP была введена Г. К. Джапаридзе в 1986 г. Она является логикой доказуемости для ряда цепочек предикатов доказуемости возрастающей силы. Всякой полимодальной логике соответствует многообразие полимодальных алгебр. Л. Д. Беклемишевым и А. Виссером был поставлен вопрос о разрешимости элементарной теории свободной GLP-алгебры, порожденной константами 0, 1 [1]. В этой статье для любого натурального n решается аналогичный вопрос для логик GLPn, являющихся ...
Added: December 4, 2017
Furmanov K. K., Nikol'skii I. M., Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253
Added: December 22, 2016