Cameron and Storvick's Function Space Integral for Certain Banach Spaces of Functionals

Analytic operator-valued generalized Feynman integral on function space

“…In this case, equation (3.1) is given by function space integrals I λ (F ) with λ > 0 defined in [1,2,3,4,5,15,16,17,18,19,20,21,22,23,24,25].…”

“…In [1] Cameron and Storvick introduced the analytic operator-valued function space "Feynman integral" J an q (F ), which mapped an L 2 (R) function ψ into an L 2 (R) function J an q (F )ψ. In [1] and several subsequent papers [2,3,16,17,18,19,20,21,22,23] the existence of this integral as an element of L(L 2 (R), L 2 (R)) was established for various functionals. Next the existence of the integral as an element of L(L 1 (R), L ∞ (R)) was established in [4,5,24].…”

“…The Wiener space C 0 [0, T ] can be considered as the space of sample paths of standard Brownian motion process (SBMP). Thus, in various Feynman integration theories, the integrand F of the Feynman integral (1.1) is a functional of the SBMP, see [1,2,3,4,5,16,17,18,19,20,21,22,23,24,25].…”

“…Note that when a(t) ≡ 0 and b(t) = t, the GBMP is an SBMP, and so the function space C a,b [0, T ] reduces to the classical Wiener space C 0 [0, T ]. But we are obliged to point out that an SBMP used in [1,2,3,4,5,16,17,18,19,20,21,22,23,24,25] is stationary in time and is free of drift. While, the GBMP used in this paper as well as in [6,7,8,9,10,11,12,13,14] is nonstationary in time and is subject to a drift a(t).…”