Analytic and Sequential Feynman Integrals on Abstract Wiener and Hilbert Spaces, and A Cameron-Martin Formula. Revision.

Kallianpur, G.; Kannan, D.; Karandikar, Rajeeva L.

doi:10.21236/ada146969

Cited by 36 publications

(47 citation statements)

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“…Hu and Meyer [5] sought to extend Ip(fp) in such a way as to preserve polynomials in first order stochastic integrals. That this is a reasonable strategy in connection with the Feynman integral was quite believable to the present authors since it is consistent with their earlier work, for example [7 or 12], relating the Fresnel integral of Albeverio and Hoegh-Krohn [1] which deals with certain functions on a Hilbert space H to the Fresnel (or Feynman) integral of corresponding functions on the Wiener space [3,10] or abstract Wiener space [12] associated with H.…”

Section: The Case Of Finite Expansionssupporting

confidence: 87%

Homogeneous chaos, 𝑝-forms, scaling and the Feynman integral

Johnson¹,

Kallianpur²

1993

Trans. Amer. Math. Soc.

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Abstract. In a largely heuristic but fascinating recent paper, Hu and Meyer have given a "formula" for the Feynman integral of a random variable / on Wiener space in terms of the expansion of / in Wiener chaos. The surprising properties of scaling in Wiener space make the problem of rigorously connecting this formula with the usual definition of the analytic Feynman integral a subtle one. One of the main tools in carrying this out is our definition of the 'natural extension' of pth homogeneous chaos in terms of the 'scale-invariant lifting' of p-forms on "the white noise space L2(R+) connected with Wiener space. The key result in our development says that if fp is a symmetric function in L2(W+) and Vp(fp) is the associated p-form on L2(R+), then ifp(fp) has a scaled L2-lifting if and only if the ' kth limiting trace' of fp exists for k = 0, 1, ... , [p/2]. This necessary and sufficient condition for the lifting of a p-fonn on white noise space to a random variable on Wiener space is a worthwhile contribution to white noise theory apart from any connection with the Feynman integral since p-forms play a role in white noise calculus analogous to the role played by pth homogeneous chaos in Wiener calculus.Various ¿-traces arise naturally in this subject; we study some of their properties and relationships. The limiting A:-trace plays the most essential role for us.

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Section: The Case Of Finite Expansionssupporting

confidence: 87%

Homogeneous chaos, 𝑝-forms, scaling and the Feynman integral

Johnson¹,

Kallianpur²

1993

Trans. Amer. Math. Soc.

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“…Transform approaches to the Feynman integral have their limitations as pointed out in [21] but also many notable advantages. In most (not all) respects the Feynman integral is a simpler, more fully developed, and better unified subject under transform assumptions [12,24].…”

Section: 2) 7(u) = Jhexv{^\\hf}dp(h)mentioning

confidence: 99%

“…Furthermore, the recent work of Kallianpur and Bromley [23] and Kallianpur, Kannan, and Karandikar [24] has brought the Fresnel and Feynman integrals into close contact with the theory of Gaussian stochastic process [14,22,26]. We note that iii is the so-called 'reproducing kernel Hilbert space' (or RKHS) [22, 25; 26, pp.…”

Section: Jomentioning

confidence: 99%

“…Johnson and Kallianpur are hoping to investigate the ramifications of [23,24] and Theorem 1 above for that process.…”

Section: Jomentioning

confidence: 99%

“…4. Because of the close relationship between 7(Hi) and an analogous Banach algebra of functions on Wiener space [15] and, more generally, between 7(H) and the 'Fresnel class' 7(B) of functions on the 'abstract Wiener space' B associated with H [23,24,26], one would guess that results analogous to Theorem 1 ought to be true in these settings. We have substantial progress in this direction and conjecture that the full results hold.…”

Section: Jomentioning

confidence: 99%

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Functions in the Fresnel class

Chang¹,

Johnson²,

Skoug³

1987

Proc. Amer. Math. Soc.

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ABSTRACT. Let H be a separable infinite-dimensional Hubert space over R. The Fresnel class T(H) of H consists of all Fourier-Stieltjes transforms of bounded Borel measures on H. There are several results insuring that various functions of interest in connection with the Feynman integral and quantum mechanics are in 7{H). We give a theorem which has most of these results as corollaries as well as many further corollaries involving the reproducing kernel Hubert spaces of Gaussian stochastic processes.

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A Probabilistic Approach to Semi-classical Approximations

Arous

Castell

1996

Journal of Functional Analysis

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We study in this paper the semi-classical expansion of the Schro dinger equation, using a probabilistic approach based on the Wiener measure. Using almost-analytic extensions, we exhibit a probabilistic ansatz for the wave function. We show that this ansatz approximates very well the wave function in the semi-classical regime, and gives the semi-classical expansion under mild hypothesis on the potential at infinity, and no analyticity conditions. In this paper, the study takes place before the caustics.

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Analytic and Sequential Feynman Integrals on Abstract Wiener and Hilbert Spaces, and A Cameron-Martin Formula. Revision.

Cited by 36 publications

References 4 publications

Homogeneous chaos, 𝑝-forms, scaling and the Feynman integral

Homogeneous chaos, 𝑝-forms, scaling and the Feynman integral

Functions in the Fresnel class

A Probabilistic Approach to Semi-classical Approximations

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