Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2022; 59(5): 1093-1103

Online first article August 22, 2022      Printed September 30, 2022

https://doi.org/10.4134/BKMS.b210542

Copyright © The Korean Mathematical Society.

Evaluation formula for Wiener integral of polynomials in terms of natural dual pairings on abstract Wiener spaces

Seung Jun Chang, Jae Gil Choi

Dankook University; Dankook University

Abstract

In this paper, we establish an evaluation formula to calculate the Wiener integral of polynomials in terms of natural dual pairings on abstract Wiener spaces $(H,B,\nu)$. To do this we first derive a translation theorem for the Wiener integral of functionals associated with operators in $\mathcal L(B)$, the Banach space of bounded linear operators from $B$ to itself. We then apply the translation theorem to establish an integration by parts formula for the Wiener integral of functionals combined with operators in $\mathcal L(B)$. We finally apply this parts formula to evaluate the Wiener integral of certain polynomials in terms of natural dual pairings.

Keywords: Abstract Wiener space, natural dual pairing, translation theorem, integration by parts formula

MSC numbers: Primary 28C20, 46G12; Secondary 46B09, 46B20

Supported by: The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1F1A1062770).

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