Change of Scale Formulas for Wiener Integrals and Fourier-Feynman Transforms

Keywords:Change of scale transformation Conditional analytic Feynman w ϕ -integral Conditional analytic Wiener w ϕ -integral Conditional w ϕ -integral Simple formula for conditional w ϕ -integralIn this paper, using simple formulas for conditional w ϕ -integrals on C [0, t] with the conditioning functions X n and X n+1 , we evaluate the conditional analytic Wiener w ϕ -integrals of the function G r having the formfor F ∈ S wϕ which is a Banach algebra, and for Ψ = f + φ which need not be bounded or continuous, where f ∈ L p (R r ) and φ ∈M(R r ). And also, we establish several change of scale transformations for the conditional analytic Wiener w ϕ -integrals of G r with the conditioning functions X n and X n+1 .

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“…, z r ) and T A, j is the jth component of T A ( z r ). Now we have by (16) and the change of variable theorem…”

Section: Conditional Analytic Wiener and Feynman W ϕ -Integrals Of Bomentioning

confidence: 99%

“…[15], the Wiener measure and Wiener measurability behave badly under change of scale and under translation [1,2]. Various kinds of the change of scale formulas for Wiener integrals were developed on the classical and abstract Wiener spaces [4,5,[13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Analogues of conditional Wiener integrals and their change of scale transformations on a function space

Cho

Kim

Yoo

2009

Journal of Mathematical Analysis and Applications

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“…As mentioned in [14] the Wiener measure and Wiener measurability behave badly under change of scale transformation and under translation [1,2]. Various kinds of the change of scale formulas for Wiener integrals of bounded functions were developed on the classical and abstract Wiener spaces [3,12,13,15]. Chang, Kim, Song and Yoo [14] established a change of scale formula for the Wiener integral of function on the abstract Wiener space B which have the form F 1 (x) = G(x)Ψ((e 1 , x) ∼ , .…”

Section: Introductionmentioning

confidence: 99%

Change of Scale Formulas for a Generalized Conditional Wiener Integral

Cho¹,

Yoo²

2016

Bulletin of the Korean Mathematical Society

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Abstract. Let C[0, t] denote the space of real-valued continuous functions on [0, t] and define a random vector Zn :with h = 0 a.e. Using a simple formula for a conditional expectation on C[0, t] with Zn, we evaluate a generalized analytic conditional Wiener integral of the functionvr(s)dx(s)) for F in a Banach algebra and for Ψ = f + φ which need not be bounded or continuous, whereand φ is the Fourier transform of a measure of bounded variation over R r . Finally we establish various change of scale transformations for the generalized analytic conditional Wiener integrals of Gr with the conditioning function Zn.

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“…Various kinds of those studies for Wiener integrals of bounded and unbounded functionals were developed on Yeh-Wiener space [16,21], abstract Wiener space [12], [17], [18], [19], [20] and space of abstract Wiener space valued continuous functions on compact interval in R [10,11].…”

Section: Introductionmentioning

confidence: 99%

“…The Wiener process used in [2], [3], [10], [11], [12], [16], [17], [18], [19], [20], [21] is free of drift and stationary in time while the stochastic process used in this paper, as well as in [6], [7], [8], [9], [14], is nonstationary in time and is subject to a drift a(t). It turns out, as noted in Remark 3.5 below, that including a drift term a(t) makes establishing the existence of generalized analytic Feynman integrals of functionals on C a,b [0, T ] very difficult.…”

Section: Introductionmentioning

confidence: 99%