A Translation Theorem for the Generalised Analytic Feynman Integral Associated With Gaussian Paths

Chang, Seung Jun; Choi, Jae Gil; Ko, Ae Young

doi:10.1017/s0004972715000878

Cited by 3 publications

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“…For more details, see [4,6]. For each real q ∈ R\{0}, (−iq) 1/2 denotes the principal square root of −iq; that is, (−iq) 1/2 is always chosen to have positive real part, so that Re(−iq) −1/2 = Re(1/(−iq) 1/2 ) > 0.…”

Section: Definition 22 Letmentioning

confidence: 99%

“…On the other hand, in [5][6][7]9,14,15], the authors defined the generalized analytic Feynman integral and the generalized analytic Fourier-Feynman transform on the function space C a,b [0, T ], and studied their properties and related topics. The function space C a,b [0, T ], induced by a GBMP, was introduced by Yeh in [30], and was used extensively in [4,8,[10][11][12][13]20].…”

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confidence: 99%

“…In this note, we set c = a(0) = b(0) = 0. Then the function space C a,b [0, T ] induced by the GBMP Y determined by the a(·) and b(·) can be considered as the space of continuous sample paths of Y , see [4][5][6][7][8][9][10][11][12][13][14][15][16]20], and one can see that for each t ∈ [0, T ],…”

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confidence: 99%

“…However, the stochastic process used in this paper, as well as in [4][5][6][7][8][9][10][11][12][13][14][15][16]18,20,21,25,30], is non-stationary in time and is subject to a drift because In this note, we present an example of a bounded functional which is not generalized analytic Feynman integrable on the function space C a,b [0, T ].…”

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Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

Chang¹,

Choi²

2016

Arch. Math.

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In the theory of the analytic Feynman integral, the integrand is a functional of the standard Brownian motion process. In this note, we present an example of a bounded functional which is not Feynman integrable. The bounded functionals discussed in this note are defined in sample paths of the generalized Brownian motion process.Mathematics Subject Classification. 28C20, 60J65, 46G12.1. Introduction. The purpose of this note is to illustrate an effect of drift of the generalized Brownian motion process (GBMP). To do this, we discuss the theory of analytic Feynman integrals. Frankly speaking, in order to emphasize an effect of drift of GBMPs, we present an example of a bounded functional which is not analytic Feynman integrable on the function space C a,b [0, T ]. The function space C a,b [0, T ] is a probability space induced by a GBMP. Let W ≡ C 0 [0, T ] denote one-parameter Wiener space; this is the space of all real-valued continuous functions x on [0, T ] with x(0) = 0. Let M denote the class of all Wiener measurable subsets of C 0 [0, T ] and let m be the Wiener measure. Then, as is well known, (C 0 [0, T ], M, m) is a complete measure space. The coordinate process W on C 0 [0, T ] × [0, T ] defined by (x, t)

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Section: Definition 22 Letmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

Chang¹,

Choi²

2016

Arch. Math.

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“…On the other hand, the translation theorem for the function space integral and the generalized analytic Fourier-Feynman transform (GFFT) have been developed for the functionals on the very general function space C a,b [0, T ] in [9], [10], and [13]. The function space C a,b [0, T ], induced by the generalized Brownian motion process (GBMP), was introduced by Yeh [24], [25] and used extensively in [8], [12]- [16], and [23] (for the precise definition of GBMP, see [24] and [25]).…”

Section: Introductionmentioning

confidence: 99%

Translation theorems for the Fourier–Feynman transform on the product function space $C_{a,b}^{2}[0,T]$

Chang¹,

Choi²,

Skoug³

2019

Banach J. Math. Anal.

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In this article, we establish the Cameron-Martin translation theorems for the analytic Fourier-Feynman transform of functionals on the productT ] is induced by the generalized Brownian motion process associated with continuous functions a(t) and b(t) on the time interval [0, T ]. The process used here is nonstationary in time and is subject to a drift a(t). To study our translation theorem, we introduce a Fresnel-type class F a,b A1,A2 of functionals on C 2 a,b [0, T ], which is a generalization of the Kallianpur and Bromley-Fresnel class F A1,A2 . We then proceed to establish the translation theorems for the functionals in F a,b A1,A2 .

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Translation Theorem for Function Space Integral Associated with Gaussian Paths and Applications

Chang

Choi

2019

Bull. Iran. Math. Soc.

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A Translation Theorem for the Generalised Analytic Feynman Integral Associated With Gaussian Paths

Abstract: In this paper, we establish a translation theorem for the generalised analytic Feynman integral of functionals that belong to the Banach algebra F (C a,b [0, T ]).2010 Mathematics subject classification: primary 60J65; secondary 28C20, 46G12.

Cited by 3 publications

References 11 publications

Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

Translation theorems for the Fourier–Feynman transform on the product function space $C_{a,b}^{2}[0,T]$

Translation Theorem for Function Space Integral Associated with Gaussian Paths and Applications

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