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Let $C[0,t]$ denote the space of real-valued continuous functions on $[0,t]$ and define a random vector $Z_n:C[0,t]\to\mathbb R^n$ by $Z_n(x)=(\int_0^{t_1}h(s) dx(s),\ldots,\int_0^{t_n}h(s) dx(s))$, where $0<t_1 < \cdots < t_n=t$ is a partition of $[0,t]$ and $h\in L_2[0,t]$ with $h\neq 0$ a.e. Using a simple formula for a conditional expectation on $C[0,t]$ with $Z_n$, we evaluate a generalized analytic conditional Wiener integral of the function $G_r(x) = F(x)\Psi(\int_0^t v_1 (s) dx(s),\ldots,\int_0^t v_r(s)dx(s))$ for $F$ in a Banach algebra and for $\Psi=f+\phi $ which need not be bounded or continuous, where $f\in L_p(\mathbb R^r)(1\le p\le\infty)$, $\{v_1,\ldots,v_r\}$ is an orthonormal subset of $L_2[0,t]$ and $\phi$ is the Fourier transform of a measure of bounded variation over $\mathbb R^r$. Finally we establish various change of scale transformations for the generalized analytic conditional Wiener integrals of $G_r$ with the conditioning function $Z_n$.
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참고문헌 (15)
참고문헌 신청
R. H. Cameron / 1954 / The translation pathology of Wiener space / Duke Math. J 21:623~627
R. H. Cameron / 1947 / The behavior of measure and measurability under change of scale in Wiener space / Bull. Amer. Math. Soc 53:130~137
R. H. Cameron / 1987 / Change of scale formulas for Wiener integral / Rend. Circ. Mat. Palermo (2) (Suppl. 17):105~115
R. H. Cameron / 1980 / Some Banach algebras of analytic Feynman integrable functionals / Analytic functions, Kozubnik 1979 (Proc. Seventh Conf.) :18~67
K. S. Chang / 1987 / Functions in the Fresnel class / Proc. Amer. Math. Soc 100(2):309~318
R. H. Cameron / 1947 / The behavior of measure and measurability under change of scale in Wiener space / Bull. Amer. Math. Soc 53:130~137
R. H. Cameron / 1987 / Change of scale formulas for Wiener integral / Rend. Circ. Mat. Palermo (2) (Suppl. 17):105~115
R. H. Cameron / 1980 / Some Banach algebras of analytic Feynman integrable functionals / Analytic functions, Kozubnik 1979 (Proc. Seventh Conf.) :18~67
K. S. Chang / 1987 / Functions in the Fresnel class / Proc. Amer. Math. Soc 100(2):309~318
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