Journal of the
Korean Mathematical Society
JKMS

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

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J. Korean Math. Soc. 2021; 58(3): 609-631

Published online May 1, 2021 https://doi.org/10.4134/JKMS.j200180

Copyright © The Korean Mathematical Society.

A generalized simple formula for evaluating Radon-Nikodym derivatives over paths

Dong Hyun Cho

Kyonggi University

Abstract

Let $C[0,T]$ denote a generalized analogue of Wiener space, the space of real-valued continuous functions on the interval $[0,T]$. Define $Z_{\vec e,n}:C[0,T]\to\mathbb R^{n+1}$ by \begin{align*} Z_{\vec e,n}(x)=\left(x(0),\int_0^Te_1(t)dx(t),\ldots,\int_0^Te_n(t)dx(t)\right), \end{align*} where $e_1, \ldots,e_n$ are of bounded variations on $[0,T]$. In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on $C[0,T]$ with the conditioning function $Z_{\vec e,n}$ which has an initial weight and a kind of drift. As applications of the formula, we evaluate the Radon-Nikodym derivatives of various functions on $C[0,T]$ which are of interested in Feynman integration theory and quantum mechanics. This work generalizes and simplifies the existing results, that is, the simple formulas with the conditioning functions related to the partitions of time interval $[0,T]$.

Keywords: Analogue of Wiener measure, Banach algebra, conditional Wiener integral, cylinder function, Feynman integral, Wiener integral, Wiener space

MSC numbers: Primary 28C20; Secondary 60G05, 60G15

Supported by: This work was supported by Kyonggi University Research Grant 2019