SOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA I
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 s...
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Yaroslavl State University
2015-06-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/255 |
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| author | I. D. Remizov |
| author_facet | I. D. Remizov |
| author_sort | I. D. Remizov |
| collection | DOAJ |
| description | 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.The article is published in the author’s wording. |
| format | Article |
| id | doaj-art-ed917d576b4849dfb7cadada8a2718b2 |
| institution | Matheson Library |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2015-06-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-ed917d576b4849dfb7cadada8a2718b22025-08-04T14:06:41ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172015-06-0122333735510.18255/1818-1015-2015-3-337-355244SOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA II. D. Remizov0Bauman Moscow State Technical University; Lobachevsky University of Nizhny NovgorodA 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.The article is published in the author’s wording.https://www.mais-journal.ru/jour/article/view/255hilbert spacefeynman formulachernoff theoremmultiple integralsgaussian measure |
| spellingShingle | I. D. Remizov SOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA I Моделирование и анализ информационных систем hilbert space feynman formula chernoff theorem multiple integrals gaussian measure |
| title | SOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA I |
| title_full | SOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA I |
| title_fullStr | SOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA I |
| title_full_unstemmed | SOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA I |
| title_short | SOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA I |
| title_sort | solution to a parabolic differential equation in hilbert space via feynman formula i |
| topic | hilbert space feynman formula chernoff theorem multiple integrals gaussian measure |
| url | https://www.mais-journal.ru/jour/article/view/255 |
| work_keys_str_mv | AT idremizov solutiontoaparabolicdifferentialequationinhilbertspaceviafeynmanformulai |