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The Itô-Nisio theorem, quadratic Wiener functionals, and 1-solitons. (English) Zbl 1203.60059

Among Professor Kiyosi Itô’s achievements, there is the Itô-Nisio theorem, a completely general theorem relative to the Fourier series decomposition of Brownian motion. In this paper, some of its applications are reviewed, and new applications to 1-soliton solutions to the Korteweg-de Vries (KdV for short) equation and Eulerian polynomials are given.

MSC:

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
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[1] Ciesielski, Z., Hölder conditions for realizations of Gaussian processes, Trans. Amer. Math. Soc., 99, 403-413 (1961) · Zbl 0133.10502
[2] Cohen, A., Eulerian polynomials of spherical type, Münster J. of Math., 1, 1-8 (2008) · Zbl 1184.05137
[3] Doob, J. L., Stochastic Processes (1953), John Wiley · Zbl 0053.26802
[4] Euler, L., Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques, Académie des sciences de Berlin, Lu en 1749, Opera Omnia Serie I, 15, 70-90 (1749)
[5] Feller, W., An Introduction to Probability Theory and its Applications, vol. 1, 2 (1950), John Wiley, p. 1966 · Zbl 0039.13201
[6] Feynman, R. P.; Hibbs, A. R., Quantum Mechanics and Path Integrals (1965), McGraw-Hill · Zbl 0176.54902
[7] Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139, 95-153 (1977) · Zbl 0366.22010
[8] Hara, K.; Ikeda, N., Quadratic Wiener functionals and dynamics on Grassmannians, Bull. Sci. Math., 125, 481-528 (2001) · Zbl 0995.60051
[9] Hardy, G. H., Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc., 17, 301-325 (1916) · JFM 46.0401.03
[10] Hirzebruch, F., Eulerian polynomials, Münster, J. Math., 1, 9-14 (2008) · Zbl 1255.11004
[11] Ikeda, N.; Kusuoka, S.; Manabe, S., Lévy’s stochastic area formula and related problems, (Stochastic Analysis (Ithaca, NY, 1993). Stochastic Analysis (Ithaca, NY, 1993), Proc. Sympos. Pure Math., vol. 57 (1995), Amer. Math. Soc.), 281-305 · Zbl 0837.60052
[12] Ikeda, N.; Manabe, S., Integral of differential forms along the path of diffusion processes, Publ. Res. Inst. Math. Sci., 15, 827-852 (1979) · Zbl 0462.60056
[13] Ikeda, N.; Manabe, S., Van Vleck-Pauli formula for Wiener integrals and Jacobi fields, (Itô’s Stochastic Calculus and Probability Theory (1996), Springer: Springer Tokyo), 141-156 · Zbl 0865.60067
[14] Ikeda, N.; Taniguchi, S., Quadratic Wiener functionals, Kalman-Bucy filters, and the KdV equation, (Kunita, H.; Watanabe, S.; Takahashi, Y., Stochastic Analysis and Related Topics in Kyoto, In honor of Kiyosi Itô. Stochastic Analysis and Related Topics in Kyoto, In honor of Kiyosi Itô, Adv. Studies Pure Math., vol. 41 (2004), Math. Soc. Japan: Math. Soc. Japan Tokyo), 167-187 · Zbl 1057.60055
[15] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1989), North-Holland: North-Holland Kodansha · Zbl 0684.60040
[16] Itô, K., Differential equations determining a Markov process, J. Pan-Japan Math. Coll., 244, 1352-1400 (1942), (in Japanese) (English translation: [21] pp. 42-75)
[17] Itô, K., Foundation of Probability Theory (1943), Iwanami, (in Japanese) (The reference Books[a] in the Bibliography of Kiyosi Itô, Kiyosi Itô Selected Papers [21])
[18] Itô, K., A review on the Japanese translation of Kurt Gödel: The consistency continuum hypothesis, Sūgaku, 1, 47-48 (1947), (in Japanese)
[19] Itô, K., Multiple Wiener integral, J. Math. Soc. Japan, 3, 157-169 (1951) · Zbl 0044.12202
[20] Itô, K., Probability Theory (1953), Iwanami, (in Japanese) (The reference Books[b] in the Bibliography of Kiyosi Itô, Kiyosi Itô Selected Papers [21])
[21] Itô, K., (Stroock, D. W.; Varadhan, S. R.S., Kiyosi Itô Selected Papers (1986), Springer)
[22] Itô, K.; Nisio, M., On stationary solutions of a stochastic differential equation, Jour. Math. Kyoto Univ., 4, 1-75 (1964) · Zbl 0131.16402
[23] Itô, K.; Nisio, M., On the convergence of sums of independent Banach space valued random variables, Osaka Jour. Math., 5, 35-48 (1968) · Zbl 0177.45102
[24] M. Kac, Integration in function space and some of its applications, Lezioni Fermiane, Accademia Nazionale del Lincei Scuola Norm. Sup. Pisa, 1980.; M. Kac, Integration in function space and some of its applications, Lezioni Fermiane, Accademia Nazionale del Lincei Scuola Norm. Sup. Pisa, 1980.
[25] Kolmogorov, A. N., Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104, 415-458 (1931) · JFM 57.0613.03
[26] Lévy, P., Théorie de l’addition des variables aléatoires (1937), Gauthier-Villars · JFM 63.0490.04
[27] Lévy, P., Le mouvement brownien plan, Amer. J. Math., 62, 487-550 (1940) · JFM 66.0619.02
[28] Lévy, P., Wiener’s random function, and other Laplacian random functions, (Proc. 2nd Berkeley Symp. Math. Statistics Probab. 1950 (1951), University of California Press), 171-187
[29] Lyons, T.; Qian, Z., System Control and Rough Paths, Oxford Math. Monographs (2002), Oxford University Press · Zbl 1029.93001
[30] Maruyama, G., On the transition probability functions of the Markov process, Natur Sci. Rep. Ochanomizu Univ., 5, 10-20 (1954) · Zbl 0059.12301
[31] Matsumoto, H., Semiclassical asymptotics of eigenvalues for Schrödinger operators with magnetic fields, J. Funct. Anal., 129, 168-190 (1995) · Zbl 0859.35081
[32] Matsumoto, H.; Taniguchi, S., Wiener functionals of second order and their Lévy measures, Electron J. Probab., 7, 14, 1-30 (2002) · Zbl 1007.60084
[33] Miwa, T.; Jimbo, M.; Date, E., Solitons (2000), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0986.37068
[34] Paley, R. E.A. C.; Wiener, N., (Fourier Transforms in the Complex Domain. Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Publ., vol. 19 (1934), AMS: AMS Providence, R.I.) · Zbl 0011.01601
[35] Perrin, J., Les Atomes (1913), Félix Alcan · JFM 44.0913.12
[36] Pinsky, M., Introduction to Fourier Analysis and Wavelets (2002), Pacific Grove: Pacific Grove Brooks/Cole · Zbl 1065.42001
[37] Prohorov, Y., Convergence of random processes and limit theorems in probability theory, Theor. Prob. Appl., 1, 157-214 (1956)
[38] Taniguchi, S., Brownian sheet and reflectionless potentials, Stochastic Process Appl., 116, 293-309 (2006) · Zbl 1088.60077
[39] Van Vleck, J. H., The correspondence principle in the statistical interpretation of quantum mechanics, Proc. Natl Acad. Sci. USA, 14, 178-188 (1928) · JFM 54.0976.01
[40] Wiener, N., Differential space, Math. Phys., 2, 131-174 (1923)
[41] Wiener, N., Un problème de probabilités dénombrables, Bull. Soc. Math. France, 52, 569-578 (1924) · JFM 51.0382.06
[42] Wiener, N., (Masani, P., Collected Works with Commentaries/Norbert Wiener, vol. 1 (1976), MIT Press)
[43] Wiener, N., I am a Mathematician, The Later Life of a Prodigy (1956), Doubleday & Co
[44] Yor, M., Remarques sur une formule de Paul Lévy, Séminaire Prob. XIV. Séminaire Prob. XIV, Lect. Notes Math., 784, 343-346 (1980) · Zbl 0429.60045
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