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

Johnson, Glen O.; Kallianpur, G.

doi:10.1090/s0002-9947-1993-1124168-8

Cited by 17 publications

(9 citation statements)

References 20 publications

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“…The last term, apart from a well-determined constant factor, matches formula (1.3) in Johnson & Kallianpur (1993). The latter formula can thus be secured by the following recipe: Project the given x in L 2 on the subspace S ξp generated by the pth Wiener chaotic measure ξ p .…”

Section: Corollary (On Lifting Frommentioning

confidence: 77%

“…The equalities (1.24), (1.25), which depart from the existing vector measure theory, bear significantly on the recent efforts of Hu & Meyer (1980) to explicate mathematically the Feynman integral. In their paper (1980, eqn (5)), and in the later paper on this subject by Johnson & Kallianpur (1993) appears the so-called kth trace of a function f on R p , which is defined on R p−2k by…”

Section: (H ) Bearing On the Feynman Integralmentioning

confidence: 99%

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The homogeneous chaos from the standpoint of vector measures

Masani

1997

Philosophical Transactions of the Royal Society of London. Seri

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Contentspage 1. Introduction Part I. Chaotic measure theory 2. The Venn expansion and proof of Wiener's equality (75) 3. Wiener's p-homogeneous chaotic measure on the pre-ring P p of intervals 4. The diagonal skeletons and the canonical coefficients 5. The countable additivity of ξ p on the ring R p and its extendibility to the δ-ring D p 6. The permutation group and symmetric sets and functions 7. The Lebesgue negligibility of the diagonal skeletons and of the canonical coefficients 8. The subspaces S ξp spanned by the ξ p 9. Concordance of the orthogonal and Lebesgue decompositions η p + ζ p of ξ p Part II. Chaotic integration 10. Integrability and integration with respect to the measure η p 11. The projection theorem and the formula for the orthogonal decomposition of L (ξ) 2 12. The π, h sectioning of functions 13. Integrability and integration with respect to the measure ξ p 14. The Fubini theorem for tensor products of functions on R p and on R q 15. The inversion formulae 16. Symmetric intervals, symmetric functional tensor products, and the Hermite expansion Appendix A. Integrability and integration with respect to a vector measure ρ Appendix B. Integration with respect to measures given by Markovian kernels Appendix C. The ratio of finite sets of positive integers Index of notation References Norbert Wiener's 1938 theory of the homogeneous chaos involves measures ξ p over

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Section: Corollary (On Lifting Frommentioning

confidence: 77%

Section: (H ) Bearing On the Feynman Integralmentioning

confidence: 99%

The homogeneous chaos from the standpoint of vector measures

Masani

1997

Philosophical Transactions of the Royal Society of London. Seri

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“…To check (15), note that when t ∈ T , we have |P t | ≥ 2 by (18), and so we have in addition β jt ≤ 2γ jt < −1 in (22). Thus for some finite c 2 , c 3 > 0,…”

Section: (ω)-Definitenessmentioning

confidence: 99%

“…Hu and Meyer [13], however, considered integrals with diagonals and related them to the iterated Stratonovich integrals. Formal theories were later developed in Johnson and Kallianpur [15] and Budhiraja and Kallianpur [6]. We denote the k-tuple Wiener-Stratonovich integral asI k (·).…”

Section: Expressing the Nclt Limit As A Centered Multiple Wiener-stramentioning

confidence: 99%

Convergence of long-memory discrete kth order Volterra processes

Bai¹,

Taqqu²

2015

Stochastic Processes and their Applications

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We obtain limit theorems for a class of nonlinear discrete-time processes X(n) called the k-th order Volterra processes of order k. These are moving average k-th order polynomial forms:is a nonrandom coefficient, and where the diagonals are included in the summation. We specify conditions for X(n) to be well-defined in L 2 (Ω), and focus on central and non-central limit theorems. We show that normalized partial sums of centered X(n) obey the central limit theorem if a(·) decays fast enough so that X(n) has short memory. We prove a non-central limit theorem if, on the other hand, a(·) is asymptotically some slowly decaying homogeneous function so that X(n) has long memory. In the non-central case the limit is a linear combination of Hermite-type processes of different orders. This linear combination can be expressed as a centered multiple Wiener-Stratonovich integral.

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“…The refer the reader to e.g. [6,10,12,16,19]. The construction that we use below closely follows the functional setting presented in [13].…”

Section: Stratonovich Integrals and Duhamel Solutionsmentioning

confidence: 99%

Convergence to SPDEs in Stratonovich Form

Bal

2009

Commun. Math. Phys.

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We consider the perturbation of parabolic operators of the form ∂ t + P (x, D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator P (x, D), the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with a multiplicative term that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed.The behavior described above should be contrasted to the case of dimensions that are larger than or equal to the order of the elliptic pseudo-differential operator P (x, D). In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in the companion paper [2]. The stochastic model is therefore valid only for sufficiently small space dimensions in this class of parabolic problems. keywords: Partial differential equations with random coefficients, Stochastic partial differential equations, Gaussian potential, iterated Stratonovich integral, Wiener-Itô chaos expansion AMS: 35R60, 60H15, 35K15.

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

Cited by 17 publications

References 20 publications

The homogeneous chaos from the standpoint of vector measures

The homogeneous chaos from the standpoint of vector measures

Convergence of long-memory discrete kth order Volterra processes

Convergence to SPDEs in Stratonovich Form

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