A change of scale formula for conditional Wiener integrals on classical Wiener space

摘要

Let $X_k(x) = (int_0^T lpha_1 (s) d x(s), ldots, int_0^T lpha_k (s)d x(s))$ and $X_au(x)$ $=$ $(x(t_1)$, $ldots$, $x(t_k))$ on the classical Wiener space, where ${lpha_1, ldots, lpha_k}$ is an orthonormal subset of $L_2[0, T]$ and $au : 0 < t_1< cdots < t_k =T$ 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 egin{eqnarray*} G_r(x) = F(x)Psiiggl(int_0^T v_1 (s) dx(s), ldots, int_0^T v_r (s) dx (s) iggr), end{eqnarray*} for $Fin mathcal S$ and $Psi =psi + phi,(psiin L_p (mathbb R^r),$ $phiinhat{M}(mathbb R^r))$, 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 $E[G_r| X_au]$ and $E[F| X_au]$. 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 $F$.