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Banach algebra $S_{\mathbb{B}}^{\prime \prime }$; Banach space $S_{n, \mathbb{B}}^{\prime \prime }$; conditional Wiener integral; conditional Feynman integral; simple formula for conditional Wiener integrals
In this paper, we introduce a simple formula for conditional Wiener integrals over $C_0(\mathbb{B})$, the space of abstract Wiener space valued continuous functions. Using this formula, we establish various formulas for a conditional Wiener integral and a conditional Feynman integral of functionals on $C_0(\mathbb{B})$ in certain classes which correspond to the classes of functionals on the classical Wiener space introduced by Cameron and Storvick. We also evaluate the conditional Wiener integral and conditional Feynman integral for functionals of the form \[ \exp \biggl \lbrace \int _0^T \theta (s, x(s))\mathrm{d}\eta (s) \biggr \rbrace \] which are of interest in Feynman integration theories and quantum mechanics.
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