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Let $C[0,t]$ denote a generalized Wiener space, the space of real-valued continuous functions on the interval $[0,t]$ 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. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on $C[0,t]$ with the conditioning function $Z_n$ and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $F(x) = f(\int_0^t e(s)dx(s))$ for $x\in C[0,t]$, where $f\in L_p (\mathbb R)(1\le p\le\infty)$ and $e$ is a unit element in $L_2[0,t]$. Finally we express the generalized analytic conditional Feynman integral of $F$ as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of $e$ and the conditioning function $Z_n$ does not contain the present positions of the generalized Wiener paths.
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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 / Springer
조동현 / 2007 / Change of scale formulas for conditional Wiener integrals as integral transforms over Wiener paths in abstract Wiener space / 대한수학회논문집 22(1):91~109
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 / Springer
조동현 / 2007 / Change of scale formulas for conditional Wiener integrals as integral transforms over Wiener paths in abstract Wiener space / 대한수학회논문집 22(1):91~109
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