Conditional Foruier-Feynman transform and convolution product for a vector valued conditioning function
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Abstract
Let $C_0 [0,T]$ denote the Wiener space, the space of continuous functions $x(t)$ on $[0, T]$ such that $x(0)=0$. Define a random vector $Z_{{\vec e},k} : C_0 [0, T] \rightarrow {\mathbb R}^k$ by
$$ Z_{{\vec e},k} (x) =(\int_0^T e_1 (t) dx(t), \ldots,\int_0^T e_k (t) dx(t) )$$
where $e_j \in L_2 [0, T] $ with $e_j \ne 0$ a.e., $ j=1, \ldots , k$.
In this paper we study the conditional Fourier-Feynman transform and a conditional convolution product for a cylinder type functionals defined on $C_0 [0,T]$ with a general vector valued conditioning functions $Z_{{\vec e},k}$ above which need not depend upon the values of $x$ at only finitely many points in $(0, T]$ rather than a conditioning function $X(x) = (x(t_1 ) , \ldots, x(t_n ))$ where $0<t_1 < \ldots < t_n =T$. In particular we show that the conditional Fourier-Feynman transform of the conditional convolution product is the product of conditional Fourier-Feynman transforms.
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