Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform
We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo&g...
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MDPI AG
2020-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/10/1666 |
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| author | Young Sik Kim |
| author_facet | Young Sik Kim |
| author_sort | Young Sik Kim |
| collection | DOAJ |
| description | We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo form="prefix">exp</mo><mo>{</mo><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mi>T</mi></msubsup><mi>θ</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>d</mi><mi>t</mi><mo>}</mo></mrow></semantics></math></inline-formula> successfully exist under the certain condition, where <inline-formula><math display="inline"><semantics><mrow><mi>θ</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∫</mo><mi>R</mi></msub><mo form="prefix">exp</mo><mrow><mo>{</mo><mi>i</mi><mi>u</mi><mi>v</mi><mo>}</mo></mrow><mi>d</mi><msub><mi>σ</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is a Fourier–Stieltjes transform of a complex Borel measure <inline-formula><math display="inline"><semantics><mrow><msub><mi>σ</mi><mi>t</mi></msub><mo>∈</mo><mi mathvariant="bold">M</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="bold">M</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> sucessfully holds on the Wiener space. |
| first_indexed | 2024-03-10T15:59:47Z |
| format | Article |
| id | doaj.art-71f5ac158dd84644b8e1856d49663394 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T15:59:47Z |
| publishDate | 2020-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-71f5ac158dd84644b8e1856d496633942023-11-20T15:21:05ZengMDPI AGMathematics2227-73902020-09-01810166610.3390/math8101666Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes TransformYoung Sik Kim0Department of Mathematics, College of Natural Sciences, Industry-University Cooperation Foundation, Hanyang University, 222 Wangshmni-ro, Seongdong-gu, Seoul 04763, KoreaWe prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo form="prefix">exp</mo><mo>{</mo><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mi>T</mi></msubsup><mi>θ</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>d</mi><mi>t</mi><mo>}</mo></mrow></semantics></math></inline-formula> successfully exist under the certain condition, where <inline-formula><math display="inline"><semantics><mrow><mi>θ</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∫</mo><mi>R</mi></msub><mo form="prefix">exp</mo><mrow><mo>{</mo><mi>i</mi><mi>u</mi><mi>v</mi><mo>}</mo></mrow><mi>d</mi><msub><mi>σ</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is a Fourier–Stieltjes transform of a complex Borel measure <inline-formula><math display="inline"><semantics><mrow><msub><mi>σ</mi><mi>t</mi></msub><mo>∈</mo><mi mathvariant="bold">M</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="bold">M</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> sucessfully holds on the Wiener space.https://www.mdpi.com/2227-7390/8/10/1666Wiener spaceWiener integralFeynman integralFourier–Stieltjes transformfirst variation |
| spellingShingle | Young Sik Kim Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform Mathematics Wiener space Wiener integral Feynman integral Fourier–Stieltjes transform first variation |
| title | Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform |
| title_full | Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform |
| title_fullStr | Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform |
| title_full_unstemmed | Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform |
| title_short | Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform |
| title_sort | feynman integral and a change of scale formula about the first variation and a fourier stieltjes transform |
| topic | Wiener space Wiener integral Feynman integral Fourier–Stieltjes transform first variation |
| url | https://www.mdpi.com/2227-7390/8/10/1666 |
| work_keys_str_mv | AT youngsikkim feynmanintegralandachangeofscaleformulaaboutthefirstvariationandafourierstieltjestransform |