Evaluation formulas for an analogue of conditional analytic Feynman integrals over a function space
Bull. Korean Math. Soc. 2011 Vol. 48, No. 3, 655-672
https://doi.org/10.4134/BKMS.2011.48.3.655
Published online May 1, 2011
Dong Hyun Cho
Kyonggi University
Abstract : Let $C^r[0, t]$ be the function space of the vector-valued continuous paths $x:[0, t]\to\mathbb R^r$ and define $X_t: C^r[0, t]\to \mathbb R^{(n+1)r}$ and $Y_t: C^r[0, t]\to \mathbb R^{nr}$ by $X_t(x) = (x(t_0), x(t_1), \ldots, x(t_{n-1}),$ $ x(t_n))$ and $Y_t(x) = (x(t_0), x(t_1), \ldots,$ $ x(t_{n-1}))$, respectively, where $0=t_0 < t_1 < \cdots < t_n =t$. In the present paper, with the conditioning functions $X_t$ and $Y_t$, we introduce two simple formulas for the conditional expectations over $C^r[0, t]$, an analogue of the $r$-dimensional Wiener space. We establish evaluation formulas for the analogues of the analytic Wiener and Feynman integrals for the function \begin{eqnarray*} G (x)= \exp \biggl\{\int_0^t \theta(s, x(s))\,\, d\eta(s) \biggr\}\psi(x(t)), \end{eqnarray*} where $\theta(s, \cdot)$ and $\psi$ are the Fourier-Stieltjes transforms of the complex Borel measures on $\mathbb R^r$. Using the simple formulas, we evaluate the analogues of the conditional analytic Wiener and Feynman integrals of the functional $G$.
Keywords : analogue of Wiener measure, conditional Feynman integral, conditional Wiener integral, simple formula for conditional Wiener integral, Wiener space, Wiener integration formula
MSC numbers : Primary 28C20
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