Conditional Integral Transforms on a Function Space
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Let Cr[0; t] be the function space of the vector-valued continuous paths x : [0; t] ! Rr and define Xt : Cr[0; t] ! R(n+1)r and Yt : Cr[0; t] ! Rnr by Xt(x) = (x(t0); x(t1); ; x(tn1); x(tn)) and Yt(x) = (x(t0); x(t1); ; x(tn1)), respectively, where 0 = t0 < t1 < < tn = t. In the present paper, using two simple formulas for the conditional expectations over Cr[0; t] with the conditioning functions Xt and Yt, we establish evaluation formulas for the analogue of the conditional analytic Fourier-Feynman transform for the function of the form
Abstract
Let Cr[0; t] be the function space of the vector-valued continuous paths x : [0; t] ! Rr and define Xt : Cr[0; t] ! R(n+1)r and Yt : Cr[0; t] ! Rnr by Xt(x) = (x(t0); x(t1); ; x(tn1); x(tn)) and Yt(x) = (x(t0); x(t1); ; x(tn1)), respectively, where 0 = t0 < t1 < < tn = t. In the present paper, using two simple formulas for the conditional expectations over Cr[0; t] with the conditioning functions Xt and Yt, we establish evaluation formulas for the analogue of the conditional analytic Fourier-Feynman transform for the function of the form
참고문헌 (12)
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