Cameron and Storvick's function space integral for a Banach space of functionals generated by finite-dimensional functionals

Johnson, Glen O.; Skoug, David

doi:10.1007/bf02417011

Cited by 5 publications

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“…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].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 98%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

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Analytic operator-valued generalized Feynman integral on function space

Choi¹

2021

Preprint

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In this paper an analytic operator-valued generalized Feynman integral was studied on a general Wiener space C a,b [0, T ]. The function space C a,b [0, T ] is induced by the generalized Brownian motion process associated with continuous functions a and b. The existence of the analytic operatorvalued generalized Feynman integral is investigated as an operator from L 1 (R, ν δ,a ) to L ∞ (R) where ν δ,a is a measure on R given by dν δ,a = exp{δVar(a)u 2 }du, and where δ > 0 and Var(a) denotes the total variation of the mean function a of the generalized Brownian motion process. Keywords Generalized Brownian motion process • Analytic operator-valued generalized function space integra • Analytic operator-valued generalized Feynman integral.

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“…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].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analytic operator-valued generalized Feynman integral on function space

Choi¹

2021

Preprint

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The Cameron-Storvick function space integral: The L1 theory

Johnson

Skoug

1975

Journal of Mathematical Analysis and Applications

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The Cameron-Storvick function space integral: An L(L_p, L_p′) theory

Johnson¹,

Skoug²

1976

Nagoya Mathematical Journal

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Now the last expression above, except for absolute values on the 0's and ψ, is the same as the second expression in (2.1). Since we showed earlier that the second expression in (2.1) is an L r function of ξ and since that proof is unaffected by absolute values on the 0's and ψ, we see that the right hand side of (2.24) is an L p , function of ξ.Since we now know that both sides of (2.24) are L p , functions, their equality will be established if we show that for any φ in L p ,The formal part of the argument needed to establish (2.26) proceeds just as in (2.25) except for the absence of absolute values and the presence of the extra integral with respect to ξ. The fact that the appropriate integrals exist so that the Fubini Theorem and the fundamental Wiener integration formula may be applied follows from the fact that the last expression in (2.25) is an L p , function of ξ so that the product of it with \φ(ξ)\ is integrable.This finally completes the proof.Next we wish to establish a corresponding result for the case p = 2. When v is equal to 2 we can handle the endpoint r = 1 and, since L r ([a, &] for all values of the parameters λ and q.Proof. The proof given above for Theorem 2.1 also works when p = 2 (In fact the proof can be simplified considerably.) and establishes everything but the bounds (2.27). To establish (2.27) fix λ in C + and apply Lemma 1.4 to the second expression in (2.1) to obtain ), each integral over S(τ) may be interpreted either as a Bochner integral or as a Lebesgue integral.Proof. Let Re λ > 0, λ Φ 0. We consider a typical term ( of the sum (2.1). (One should keep in mind that when where we note in particular that the inner integral in the first expression in (2.29) must exist for almost all ξ. Since φ in L p was arbitrary the desired equalityfollows for almost all ξ.Remark. The finiteness of the left hand side of (2.28) was used to justify a use of the Fubini Theorem. When λ = -iq, it is again important to note that we can make the argument without taking absolute values all the way inside the integrals which define G iiT (s 19 ,s m )(£).We finish this section by dealing with the function F = 1. The necessary arguments here are easy because we already have Lemmas 1.1 and 1.2; hence we will merely state the proposition and indicate formally where the formula comes from. We do not need any dimension restriction here even when 1 < p < 2. J^n(F) is also given by the right hand side of (2.30) with λ = -iq. In addition we have Remark. Formula (2.30) comes from a fundamental Wiener integration formula as follows:3. The main existence theorem.In this section we shall establish the existence of If"(F) and Jl n (F) for series of the form (0.9) where each F k is in A = A (p,v,r). For each k,F k is either the function identically one or is given bywhere each θ kJ is in L rr . Given F k as in (3.1), we let &*(| >l| ) be the right side of (2.2) with m and g replaced by m k and g k respectively, where Hence the right hand side of (3.. Similarly the right hand side of (3.5) defines an element of L(L ...

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Cameron and Storvick's function space integral for a Banach space of functionals generated by finite-dimensional functionals

Cited by 5 publications

References 4 publications

Analytic operator-valued generalized Feynman integral on function space

Analytic operator-valued generalized Feynman integral on function space

The Cameron-Storvick function space integral: The L1 theory

The Cameron-Storvick function space integral: An L(L_p, L_p′) theory

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Cameron and Storvick's function space integral for a Banach space of functionals generated by finite-dimensional functionals

Cited by 5 publications

References 4 publications

Analytic operator-valued generalized Feynman integral on function space

Analytic operator-valued generalized Feynman integral on function space

The Cameron-Storvick function space integral: The L1 theory

The Cameron-Storvick function space integral: An L(Lp, Lp′) theory

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Product

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The Cameron-Storvick function space integral: An L(L_p, L_p′) theory