Abstract
Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w ϕ of Wiener measure and for a partition 0 = t 0 < t 1 < ... < t n < t n+1 = T of [0, T], let X n : C[0, T] → ℝn+1 and X n+1: C[0, T] → ℝn+2 be given by X n (x) = (x(t 0), x(t 1), ..., x(t n )) and X n+1(x) = (x(t 0), x(t 1), ..., x(t n+1)), respectively.
In this paper, using a simple formula for the conditional w ϕ-integral of functions on C[0, T] with the conditioning function X n+1, we derive a simple formula for the conditional w ϕ-integral of the functions with the conditioning function X n . As applications of the formula with the function X n , we evaluate the conditional w ϕ-integral of the functions of the form F m (x) = ∫ T0 (x(t))m for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X n , we evaluate the conditional w ϕ-integral of the functions in a Banach algebra
which is an analogue of the Cameron and Storvick’s Banach algebra
. Finally, we derive the conditional analytic Feynman w ϕ-integrals of the functions in
.
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Cho, D.H. A simple formula for an analogue of conditional wiener integrals and its applications II. Czech Math J 59, 431–452 (2009). https://doi.org/10.1007/s10587-009-0030-6
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DOI: https://doi.org/10.1007/s10587-009-0030-6
Keywords
- analogue of Wiener measure
- Cameron-Martin translation theorem
- conditional analytic Feynman w ϕ-integral
- conditional Wiener integral
- Kac-Feynman formula
- simple formula for conditional w ϕ-integral