Translation Theorem for Conditional Function Space Integrals and Applications

Sang Kil Shim and Jae Gil Choi ()
Additional contact information
Sang Kil Shim: Department of Mathematics, Dankook University, Cheonan 31116, Republic of Korea
Jae Gil Choi: Department of Mathematics, Basic Science and Mathematics Center, Dankook University, Cheonan 31116, Republic of Korea

Mathematics, 2025, vol. 13, issue 18, 1-22

Abstract: The conditional Feynman integral provides solutions to integral equations equivalent to heat and Schrödinger equations. The Cameron–Martin translation theorem illustrates how the Wiener measure changes under translation via Cameron–Martin space elements in abstract Wiener space. Translation theorems for analytic Feynman integrals have been established in many research articles. This study aims to present a translation theorem for the conditional function space integral of functionals on the generalized Wiener space C a , b [ 0 , T ] induced via a generalized Brownian motion process determined using continuous functions a ( t ) and b ( t ) . As an application, we establish a translation theorem for the conditional generalized analytic Feynman integral of functionals on C a , b [ 0 , T ] . We then provide explicit examples of functionals on C a , b [ 0 , T ] to which the conditional translation theorem on C a , b [ 0 , T ] can be applied. Our formulas and results are more complicated than the corresponding formulas and results in the previous research on the Wiener space C 0 [ 0 , T ] because the generalized Brownian motion process used in this study is neither stationary in time nor centered. In this study, the stochastic process used is subject to a drift function.

Keywords: translation theorem; generalized Wiener space; generalized Brownian motion process; conditional function space integral; conditional generalized analytic Feynman integral (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/18/3022/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/18/3022/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:18:p:3022-:d:1752615

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().