maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2015-3-337-355mais-255Research ArticleОригинальные статьиArticlesРЕШЕНИЕ ПАРАБОЛИЧЕСКОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ В ГИЛЬБЕРТОВОМ ПРОСТРАНСТВЕ С ПОМОЩЬЮ ФОРМУЛЫ ФЕЙНМАНА – ISOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA IРeмизoвИ. Д.RemizovI. D.

Рeмизoв Иван Дмитриевич, Нижегородский государственный университет им. Н.И. Лобачевского, младший научный сотрудник; Московский Государственный Технический Университет им. Н.Э. Баумана, ассистент 

ivan.remizov@gmail.comМосковский Государственный Технический Университет им. Н.Э. Баумана; Нижегородский государственный университет им. Н.И. ЛобачевскогоРоссияBauman Moscow State Technical University; Lobachevsky University of Nizhny NovgorodRussian Federation201520062015223337355Copyright © Рeмизoв И.Д., 20152015Рeмизoв И.Д.Remizov I.D.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/255

В работе рассматривается параболическое дифференциальное уравнение u′t (t, x) = Lu(t, x) в частных производных, где L – это линейный дифференциальный оператор второго порядка с коэффициентами, не зависящими от времени, но зависящими от x. Предполагается, что пространственная переменная x принадлежит конечномерному или бесконечномерному вещественному сепарабельному гильбертову пространству H.

Из существования сильно непрерывной полугруппы, разрешающей рассматриваемое уравнение, в статье выводится представление этой полугруппы в виде формулы Фейнмана, т.е. полугруппа записывается в форме предела кратного интеграла по H при стремящейся к бесконечности кратности. Это представление дает единственное решение задачи Коши для рассматриваемого уравнения в классе функций, являющихся равномерными пределами гладких цилиндрических функций на H. Более того, это решение непрерывно зависит от начального условия. Для случая, когда в операторе L коэффициент при первой производной равен нулю, в настоящей работе доказано, что а) сильно непрерывная разрешающая полугруппа существует (это влечет за собой существование единственного решения для задачи Коши в упомянутом выше классе функций) и б) это решение непрерывно зависит от коэффициентов уравнения.

Статья публикуется в авторской редакции. 

A parabolic partial differential equation u′t (t, x) = Lu(t, x) is considered, where L is a linear second-order differential operator with time-independent coefficients, which may depend on x. We assume that the spatial coordinate x belongs to a finiteor infinite-dimensional real separable Hilbert space H.

Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup by a Feynman formula, i.e. we write it in the form of the limit of a multiple integral over H as the multiplicity of the integral tends to infinity. This representation gives a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on H. Moreover, this solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in L vanishes we prove that the strongly continuous resolving semigroup exists (this implies the existence of the unique solution to the Cauchy problem in the class mentioned above) and that the solution to the Cauchy problem depends continuously on the coefficients of the equation.

Гильбертово пространствоформула Фейнманатеорема Черновакратные интегралыгауссовская мераHilbert spaceFeynman formulaChernoff theoremmultiple integralsGaussian measureReferencesBogachev V. I., “Gaussian Measures”, Amer. Math. Soc., 1998.Bogachev V. I., “Gaussian Measures”, Amer. Math. Soc., 1998.Bogachev V. I., Smolyanov O. G., Real and functional analysis: university course (in Russian), RCD, M. Izevsk, 2009.Bogachev V. I., Smolyanov O. G., Real and functional analysis: university course (in Russian), RCD, M. Izevsk, 2009.Butko Ya. A., “The Feynman-Kac-Ito formula for an infinite-dimensional Schr¨odinger equation with scalar and vector potentials”.Butko Ya. A., “The Feynman-Kac-Ito formula for an infinite-dimensional Schr¨odinger equation with scalar and vector potentials”.Butko Ya. A., “Feynman formula for semigroups with multiplicatevely perturbed generators”, Science and Education (ISSN 1994-0408), 10 (October 2011), 77305691/239563.Butko Ya. A., “Feynman formula for semigroups with multiplicatevely perturbed generators”, Science and Education (ISSN 1994-0408), 10 (October 2011), 77305691/239563.Daletsky Yu. L., Fomin S. V., Measures and differential equations in infinite-dimensional space, Kluwer, 1991.Daletsky Yu. L., Fomin S. V., Measures and differential equations in infinite-dimensional space, Kluwer, 1991.Egorov A. D., Zidkov E. P., Lobanov Yu. Yu, Introduction to the theory and applications of the functional integration (in Russian), M. Fizmatlit, 2006.Egorov A. D., Zidkov E. P., Lobanov Yu. Yu, Introduction to the theory and applications of the functional integration (in Russian), M. Fizmatlit, 2006.Krylov N. V., “Lectures on Elliptic and Parabolic Equations in Holder Spaces”, AMS, Graduate Texts in Mathematics, 12 (1996).Krylov N. V., “Lectures on Elliptic and Parabolic Equations in Holder Spaces”, AMS, Graduate Texts in Mathematics, 12 (1996).Lobanov Yu. Yu, Methods of the approximate functional integrating for the numerical research in the quantum physics, Dr. Sci. dissertation thesis (in Russian), M., 2009. [9] Smolyanov O. G., Analysis on the topological linear spaces and its applications (in Russian), MSU, M., 1979.Lobanov Yu. Yu, Methods of the approximate functional integrating for the numerical research in the quantum physics, Dr. Sci. dissertation thesis (in Russian), M., 2009. [9] Smolyanov O. G., Analysis on the topological linear spaces and its applications (in Russian), MSU, M., 1979.Smolyanov O. G., Shavgulidze E. T., Continual integrals (in Russian), MSU, M., 1990.Smolyanov O. G., Shavgulidze E. T., Continual integrals (in Russian), MSU, M., 1990.Smolyanov O. G., Shamarov N. N., Kpekpassi M., “Feynman-Kac and Feynman Formulas for Infinite-Dimensional Equations with Vladimirov Operator”, Doklady Mathematics, 83:3 (2011), 389–393.Smolyanov O. G., Shamarov N. N., Kpekpassi M., “Feynman-Kac and Feynman Formulas for Infinite-Dimensional Equations with Vladimirov Operator”, Doklady Mathematics, 83:3 (2011), 389–393.Smolyanov O. G., Shamarov N. N., “Hamiltonian Feynman formulas for equations containing the Vladimirov operator with variable coefficients”, Doklady Mathematics, 84:2, 689–694.Smolyanov O. G., Shamarov N. N., “Hamiltonian Feynman formulas for equations containing the Vladimirov operator with variable coefficients”, Doklady Mathematics, 84:2, 689–694.Schwartz L., Analyse mathematique, I. Hermann, 1967.Schwartz L., Analyse mathematique, I. Hermann, 1967.Luiz C. L. Botelho, “Non-linear diffusion in RLuiz C. L. Botelho, “Non-linear diffusion in RButko Ya. A., Grothaus M., Smolyanov O. G., “Lagrangian Feynman formulas for secondorder parabolic equations in bounded and unbounded domains”, Infinite Dimansional Analyasis, Quantum Probability and Related Topics, 13:3 (2010), 377–392.Butko Ya. A., Grothaus M., Smolyanov O. G., “Lagrangian Feynman formulas for secondorder parabolic equations in bounded and unbounded domains”, Infinite Dimansional Analyasis, Quantum Probability and Related Topics, 13:3 (2010), 377–392.Yana A. Butko, Ren´e L. Schilling, Smolyanov Oleg G., “Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations”, arXiv: http://arxiv.org/abs/1203.1199v1.Yana A. Butko, Ren´e L. Schilling, Smolyanov Oleg G., “Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations”, arXiv: http://arxiv.org/abs/1203.1199v1.Cartan H., Differential Calculus, Kershaw Publishing Company, 1971.Cartan H., Differential Calculus, Kershaw Publishing Company, 1971.Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.Plyashechnik A. S., “Feynman formula for Schr¨odinger-Type equations with timeand space-dependent coefficients”, Russian Journal of Mathematical Physics, 19:3 (2012), 340–359.Plyashechnik A. S., “Feynman formula for Schr¨odinger-Type equations with timeand space-dependent coefficients”, Russian Journal of Mathematical Physics, 19:3 (2012), 340–359.Plyashechnik A. S., “Feynman formulas for second-order parabolic equations with variable coefficients”, Russian Journal of Mathematical Physics, 20:3 (2013), 377–379.Plyashechnik A. S., “Feynman formulas for second-order parabolic equations with variable coefficients”, Russian Journal of Mathematical Physics, 20:3 (2013), 377–379.G. Da Prato, Introduction to infinite-dimensional analysis, Springer, 2006.G. Da Prato, Introduction to infinite-dimensional analysis, Springer, 2006.G. Da Prato, Zabczyk J., “Second Order Partial Differential Equations in Hilbert Spaces”, London Mathematical Society Lecture Notes Series, 293 (2004).G. Da Prato, Zabczyk J., “Second Order Partial Differential Equations in Hilbert Spaces”, London Mathematical Society Lecture Notes Series, 293 (2004).Fernique X., “Int´egrabilit´e des vecteurs gaussiens”, S´er. A-B 270: A1698–A1699, C. R. Acad. Sci., Paris.Fernique X., “Int´egrabilit´e des vecteurs gaussiens”, S´er. A-B 270: A1698–A1699, C. R. Acad. Sci., Paris.Feynman R. P., “Space-time approach to nonrelativistic quantum mechanics”, Rev. Mod. Phys., 20 (1948), 367–387.Feynman R. P., “Space-time approach to nonrelativistic quantum mechanics”, Rev. Mod. Phys., 20 (1948), 367–387.Feynman R. P., “An operation calculus having applications in quantum electrodynamics”, Phys. Rev., 84 (1951), 108–128.Feynman R. P., “An operation calculus having applications in quantum electrodynamics”, Phys. Rev., 84 (1951), 108–128.Engel K.-J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.Engel K.-J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.Dieudonn´e J., Foundations of modern analysis, Academic Press, New York and London, 1969.Dieudonn´e J., Foundations of modern analysis, Academic Press, New York and London, 1969.Remizov I. D., “Solution of a Cauchy problem for a diffusion equation in a Hilbert space by a Feynman formula”, Russian Journal of Mathematical Physics, 19:3 (2012), 360–372.Remizov I. D., “Solution of a Cauchy problem for a diffusion equation in a Hilbert space by a Feynman formula”, Russian Journal of Mathematical Physics, 19:3 (2012), 360–372.Orlov Yu. N., Sakbaev V. Zh., Smolyanov O. G., “Feynman formulas as a method of averaging random Hamiltonians”, Proceedings of the Steklov Institute of Mathematics, 285:1 (August 2014), 222–232.Orlov Yu. N., Sakbaev V. Zh., Smolyanov O. G., “Feynman formulas as a method of averaging random Hamiltonians”, Proceedings of the Steklov Institute of Mathematics, 285:1 (August 2014), 222–232.Simon B., Functional Integration and Quantum Physics, Academic Press, 1979.Simon B., Functional Integration and Quantum Physics, Academic Press, 1979.Smolyanov O. G., Tokarev A. G., Truman A., “Hamiltonian Feynman path integrals via the Chernoff formula”, J. Math. Phys., 43:10 (2002), 5161–5171.Smolyanov O. G., Tokarev A. G., Truman A., “Hamiltonian Feynman path integrals via the Chernoff formula”, J. Math. Phys., 43:10 (2002), 5161–5171.Remizov I. D., The latest version of the preprint, arXiv: http://arxiv.org/abs/1409.8345.Remizov I. D., The latest version of the preprint, arXiv: http://arxiv.org/abs/1409.8345.Smolyanov O. G., “Feynman formulae for evolutionary equations”, London Mathematical Society Lecture Notes Series, 353, 2009.Smolyanov O. G., “Feynman formulae for evolutionary equations”, London Mathematical Society Lecture Notes Series, 353, 2009.Kuo H.-S., “Gaussian measures in Banach space”, Lecture notes in mathematics, 463 (1975).Kuo H.-S., “Gaussian measures in Banach space”, Lecture notes in mathematics, 463 (1975).Paul R. Chernoff, “Note on product formulas for operator semigroups”, J. Functional Analysis 2, 1968, 238–242.Paul R. Chernoff, “Note on product formulas for operator semigroups”, J. Functional Analysis 2, 1968, 238–242.Smolyanov O. G., Weizs¨acker H. V., Wittich O., “Chernoff’s Theorem and Discrete Time Approximations of Brownian Motion on Manifolds”, Potential Analysis, 26:1 (February 2007), 1–29.Smolyanov O. G., Weizs¨acker H. V., Wittich O., “Chernoff’s Theorem and Discrete Time Approximations of Brownian Motion on Manifolds”, Potential Analysis, 26:1 (February 2007), 1–29.Butko Ya. A., “Feynman formulae for evolution semigroups (in Russian)”, Electronic scientific and technical periodical ”Science and education”, 2014, № 3, 95–132.Butko Ya. A., “Feynman formulae for evolution semigroups (in Russian)”, Electronic scientific and technical periodical ”Science and education”, 2014, № 3, 95–132.

The authors declare that there are no conflicts of interest present.