Bilateral Laplace–Feynman transforms on abstract Wiener spaces

ABSTRACT

In this paper, an extension of the bilateral Laplace transform on infinite-dimensional Banach spaces is introduced. In order to develop the concept of an analytic bilateral Laplace–Feynman transform (BLFT) of functionals on abstract Wiener spaces, we complete the following objectives: we first establish the existence of the analytic BLFT of certain bounded functionals on B. We next extend the operational properties (shifting properties) of the bilateral Laplace transform of functions on Euclidean spaces to the cases of the BLFT of functionals on abstract Wiener spaces. We also provide a convolution product (CP) corresponding to the BLFT and establish a relationship between the BLFT and the CP. We finally provide a representation of an inverse transform of the BLFT. Specifically, we establish a representation of the inverse transform for the BLFT having special parameter via the concept of the analytic Fourier–Feynman transform (FFT).

KEYWORDS:

• Abstract Wiener space
• analytic bilateral Laplace–Feynman transform
• shifting property
• convolution product
• analytic Fourier–Feynman transform

AMS SUBJECT CLASSIFICATIONS:

• Primary: 42B10
• 46G12
• Secondary: 28C20
• 44A15
• 46B09

Acknowledgments

The author would like to express his gratitude to the editor and the referees for their valuable comments and suggestions which have improved the original paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [grant number 2021R1F1A1062770].