Evaluation formulas for an analogue of conditional analytic Feynman integrals over a function space
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Let C^r[0, t] be the function space of the vector-valued continuous paths x :[0, t]→ R^r and define X_t : C^r[0, t]→ R^{(n+1)r} and Y_t : C^r[0, t]→ R^(nr) by X_t(x) = (x(t_0), x(t_1), ....., x(t_{n-1}), x(t_n)) and Y_t(x) = (x(t_0), x(t_1), ..., x(t_{n-1})), respectively, where 0=t_0 < t_1 < .... < t_n =t. In the present paper, with the conditioning functions X_t and Y_t, we introduce two simple formulas for the conditional expectations over C^r[0, t], an analogue of the r-dimensional Wiener space. We establish evaluation formulas for the analogues of the analytic Wiener and Feynman integrals for the function [수식]where θ(s,ㆍ) and ψ are the Fourier-Stieltjes transforms of the complex Borel measures on R^r. Using the simple formulas, we evaluate the analogues of the conditional analytic Wiener and Feynman integrals of the functional G.
Abstract
Let C^r[0, t] be the function space of the vector-valued continuous paths x :[0, t]→ R^r and define X_t : C^r[0, t]→ R^{(n+1)r} and Y_t : C^r[0, t]→ R^(nr) by X_t(x) = (x(t_0), x(t_1), ....., x(t_{n-1}), x(t_n)) and Y_t(x) = (x(t_0), x(t_1), ..., x(t_{n-1})), respectively, where 0=t_0 < t_1 < .... < t_n =t. In the present paper, with the conditioning functions X_t and Y_t, we introduce two simple formulas for the conditional expectations over C^r[0, t], an analogue of the r-dimensional Wiener space. We establish evaluation formulas for the analogues of the analytic Wiener and Feynman integrals for the function [수식]where θ(s,ㆍ) and ψ are the Fourier-Stieltjes transforms of the complex Borel measures on R^r. Using the simple formulas, we evaluate the analogues of the conditional analytic Wiener and Feynman integrals of the functional G.
참고문헌 (16)
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