Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2004; 41(2): 265-294

Printed March 1, 2004

Copyright © The Korean Mathematical Society.

Conditional paths in abstract Wiener space : an ${\rm L}_p$ theory

Dong Hyun Cho

Yonsei University

Abstract

In this paper, using a simple formula, we evaluate the conditional Fourier-Feynman transforms and the conditional convolution products of cylinder type functions, and show that the conditional Fourier-Feynman transform of the conditional convolution product is expressed as a product of the conditional Fourier-Feynman transforms. Also, we evaluate the conditional Fourier-Feynman transforms of the functions of the forms \begin{eqnarray*} \exp\biggl \{ \int_0^T \theta (s, x(s)) ds \biggr \}, && \exp\biggl \{ \int_0^T \theta (s, x(s)) ds \biggr \} \phi (x(T)), \\ \exp\biggl \{ \int_0^T \theta (s, x(s)) d\zeta (s) \biggr \}, && \exp\biggl \{ \int_0^T \theta (s, x(s)) d\zeta( s) \biggr \} \phi (x(T)) \end{eqnarray*} which are of interest in Feynman integration theories and quantum mechanics.

Keywords: conditional analytic Feynman integral, conditional convolution product, conditional Fourier-Feynman transform, conditional Wiener integral, cylinder type function, simple formula for conditional Wiener integral

MSC numbers: 28C20