Abstract
A nonstandard construction of stable type Euclidean random fields via hyperfinite flat integrals and stable white noise is given. Moreover, a brief account on an extension of Cutland’s flat integral formula for (centered) Gaussian measures on the Hilbert space l 2 to the case of Banach spaces l p , 1 ≤ p < ∞, is presented.
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References
Albeverio, S., Fenstad, J.E., Hoegh-Krohn R. and Lindstrom T. (1986). Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press.
Albeverio, S. and Wu, J.-L. (1995). Euclidean random fields obtained by convolution from generalized white noise. J. Math. Phys. 36, 5217–5245.
Albeverio, S. and Wu, J.-L. (1996). A mathematical flat integral realization and a large deviation result for the free Euclidean field. Acta Appl. Math. 45, 317–348.
Albeverio, S. and Wu, J.-L. (2000). On hyperfinite integral representation of Euclidean random field measures. Bonn SFB 256-preprint.
Cutland, N.J. (1983). Nonstandard measure theory and its applications. Bull. London Math. Soc. 15, 529–589.
Cutland, N.J. (1987). Infinitesimals in action. J. London Math. Soc. 35, 202–216.
Cutland, N.J. (1989). The Brownian bridge as a flat integral. Math. Proc. Cambridge Phil. Soc. 106, 343–354.
Cutland, N.J. (1990). An action functional for Lévy Brownian motion. Acta Appl. Math., 487–495.
Cutland, N.J. (1991). On large deviations in Hilbert space. Proc. Edinburgh Math. Soc. 34, 487–495.
Cutland, N.J. (1990). Nonstandard representation of Gaussian measures. White Noise–Mathematics and Application (eds. Hida, T., Kuo, H.-H., Pottholf, J. and Streit, L.). Singapore: World Scientific, pp 73–92.
Deuschel, J.-D. and Stroock, D.W. (1989). Large Deviations. San Diego: Academic Press.
Fritz, J., Lebowitz, J.L. and Szâsz, D. (Editors) (1981). Random Fields. Rigorous Results in Statistical Mechanics and Quantum Field Theory (Vol. I, II). Colloquia Mathematica Societatis Janos Bolyai 27,, Amsterdam: North-Holland Publishing Co..
Gelfand, I.M. and Vilenkin, N.Ya. (1964). Generalized Functions, IV. Some Applications of Harmonic Analysis. New York: Academic Press.
Keßler, C. (1984). Nonstandard methods in random fields. Bochum: Dissertation.
Keßler, C. (1988). On hyperfinite representation of distributions. Bull. London Math. Soc. 20, 139–144.
Kuo, H.-H. (1975). Gaussian Measures in Banach Spaces. Lect. Notes Math. 463. Berlin: Springer-Verlag.
Lindenstrauss, J. and Tzafriri, L. (1977). Classical Banach Spaces. I. Sequence Spaces. Berlin: Springer-Verlag.
Lindstrom, T. (1988). An invitation to nonstandard analysis. Nonstandard Analysis and Its Applications (ed. by Cutland, N.J.). Cambridge: Cambridge University Press, ppl-105.
Osswald, H. (2000). Infinitesimal in abstract Wiener spaces. To appear in Stochastic Processes, Physics and Geometry: New Interplays. A Volume in Honor of Sergio Albeverio. Proceedings of the Leipzig International Conference on Infinite Dimensional (Stochastic) Analysis and Quantum Physics, the Canadian Mathematical Society (CMS) Conference Proceeings Series.
Rosen, J. and Simon, B. (1975). Global support properties of stationary ergodic processes. Duke Math. J. 42, 51–55.
Vakhania, N.N., Tarieladze, V.I. and Chobanyan, S.A. (1987). Probability Distributions on Banach Spaces. Dordrecht: D. Reidel Publishing Company.
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Albeverio, S., Wu, JL. (2001). Nonstandard Construction of Stable Type Euclidean Random Field Measures. In: Schuster, P., Berger, U., Osswald, H. (eds) Reuniting the Antipodes — Constructive and Nonstandard Views of the Continuum. Synthese Library, vol 306. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9757-9_1
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DOI: https://doi.org/10.1007/978-94-015-9757-9_1
Publisher Name: Springer, Dordrecht
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