Journal of the
Korean Mathematical Society JKMS

Let $X_k(x) = (\int_0^T \alpha_1 (s) d x(s), \ldots, \int_0^T \alpha_k (s)d x(s))$ and $X_\tau(x)$ $=$ $(x(t_1)$, $\ldots$, $x(t_k))$ on the classical Wiener space, where $\{\alpha_1, \ldots, \alpha_k\}$ is an orthonormal subset of $L_2[0, T]$ and $\tau : 0 < t_1< \cdots < t_k =T$ is a partition of $[0, T]$. In this paper, we establish a change of scale formula for conditional Wiener integrals $E[G_r| X_k]$ of functions on classical Wiener space having the form \begin{eqnarray*} G_r(x) = F(x)\Psi\biggl(\int_0^T v_1 (s) dx(s), \ldots, \int_0^T v_r (s) dx (s) \biggr), \end{eqnarray*} for $F\in \mathcal S$ and $\Psi =\psi + \phi\,(\psi\in L_p (\mathbb R^r),$ $\phi\in\hat{M}(\mathbb R^r))$, which need not be bounded or continuous. Here $\mathcal S$ is a Banach algebra on classical Wiener space and $\hat{M}(\mathbb R^r)$ is the space of Fourier transforms of measures of bounded variation over $\mathbb R^r$. As results of the formula, we derive a change of scale formula for the conditional Wiener integrals $E[G_r| X_\tau]$ and $E[F| X_\tau]$. Finally, we show that the analytic Feynman integral of $F$ can be expressed as a limit of a change of scale transformation of the conditional Wiener integral of $F$ using an inversion formula which changes the conditional Wiener integral of $F$ to an ordinary Wiener integral of $F$, and then we obtain another type of change of scale formula for Wiener integrals of $F$.

Keywords: change of scale formula, conditional analytic Feynman integral, conditional analytic Wiener integral, conditional Wiener integral

MSC numbers: 28C20