A GENERALIZED L_p-ANALYTIC CONDITIONAL FEYNMAN INTEGRAL OVER PATHS IN ABSTRACT WIENER SPACE

Let C₀[0, T] denote the Wiener space, the space of continuous real-valued functions x on [0, T] with x(0) = 0. Let C^𝔹[a, b] denote an analogue of Wiener space over paths in abstract Wiener space 𝔹, the space of 𝔹-valued continuous functions on [a, b]. Let W(x) = {x(t_n)}^∞_n=0 ∈ B^ℵ₀ for x ∈ C^𝔹[a, b], where {t_n}^∞_n=0 is a strictly increasing sequence in [a, b] with t₀ = a and lim_n→∞ t_n = b. In this paper, we introduce a positive finite measure with scale on C^𝔹[a, b] which is a generalized analogue of Wiener measure. Then we extend the time integral which is a Riemann integral of functions on C₀[0, T], to a Riemann integral of functions on C^𝔹[a, b] which is a more generalized time integral. Finally, using a simple formula for calculating a Radon-Nikodym derivative similar to the conditional Wiener integral of functions on C^𝔹[a, b] given W which has an initial weight, we evaluate a generalized L_p-analytic conditional Feynman integral of the extended time integral. The established L_p-analytic conditional Feynman integrals are of interest in quantum mechanics, especially in Feynman integration theory.