Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths

Kim, Byoung Soo ; Cho, Dong Hyun

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 609-628.

Résumé

Let $C[0,t]$ denote a generalized Wiener space, the space of real-valued continuous functions on the interval $[0,t]$, and define a random vector $Z_n\colon C[0,t]\to \mathbb R^{n+1}$ by $$ Z_n(x)=\biggl (x(0)+a(0), \int _0^{t_1}h(s) {\rm d} x(s)+x(0)+a(t_1), \cdots ,\int _0^{t_n}h(s) {\rm d} x(s)+x(0)+a(t_n)\biggr ), $$ where $a\in C[0,t]$, $h\in L_2[0,t]$, and $0$ is a partition of $[0,t]$. Using simple formulas for generalized conditional Wiener integrals, given $Z_n$ we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions $F$ in a Banach algebra which corresponds to Cameron-Storvick's Banach algebra $\mathcal S$. Finally, we express the generalized analytic conditional Feynman integral of $F$ as a limit of the non-conditional generalized Wiener integral of a polygonal function using a change of scale transformation for which a normal density is the kernel. This result extends the existing change of scale formulas on the classical Wiener space, abstract Wiener space and the analogue of the Wiener space $C[0,t]$.

Détail

DOI : 10.21136/CMJ.2017.0248-15

Classification : 28C20, 60G05, 60G15, 60H05
Keywords: analogue of Wiener space; analytic conditional Feynman integral; change of scale formula; conditional Wiener integral; Wiener integral