{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:30:13Z","timestamp":1760239813880,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,23]],"date-time":"2020-12-23T00:00:00Z","timestamp":1608681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001321","name":"National Research Foundation","doi-asserted-by":"publisher","award":["NRF-2017R1A6A3A11030667"],"award-info":[{"award-number":["NRF-2017R1A6A3A11030667"]}],"id":[{"id":"10.13039\/501100001321","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We shall prove the existence of the Wiener integral and the analytic Wiener and Feynman integral and we obtain those relationships and later, we prove the change of scale formula for the Wiener integral about the first variation of a function defined on the product abstract Wiener space. Later, we obtain those relationships in the Fresnel class as it\u2019s corollaries.<\/jats:p>","DOI":"10.3390\/sym13010012","type":"journal-article","created":{"date-parts":[[2020,12,23]],"date-time":"2020-12-23T12:19:51Z","timestamp":1608725991000},"page":"12","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Change of Scale Formula for Wiener Integrals about the First Variation on the Product Abstract Wiener Space"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0752-2667","authenticated-orcid":false,"given":"Young Sik","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Natural Sciences, Industry-University Cooperation Foundation, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1090\/S0002-9904-1947-08762-0","article-title":"The behavior of measure and measurability under change of scale in Wiener space","volume":"53","author":"Cameron","year":"1947","journal-title":"Bull. 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Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1007\/BFb0097256","article-title":"Some Banach algebras of analytic Feynman integrable functionals, An analytic functions","volume":"798","author":"Cameron","year":"1980","journal-title":"Lect. Notes Math."},{"key":"ref_10","unstructured":"Kim, Y.S. (2019, January 23\u201325). A change of scale formula for Wiener integrals about the first variation on the product abstract Wiener space. In Proceeding of the International Conference, \u201cModern Problems in Mathematics and Mechanics\u201d, Baku, Azerbeijan."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kim, Y.S. (2020). Feynman integral and a change of scale formula about the first variation and the Fourier-Stieltjes transform. 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