Geodesic
Fourier series expansion of stochastic measures
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 2, pp. 389-401 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider processes of the form $\mu(t)=\mu((0,t])$, where $\mu$ is a $\sigma$-additive in probability stochastic set function. Convergence of a random Fourier series to $\mu(t)$ is proved, and the approximation of integrals with respect to $\mu$ using Fejèr sums is obtained. For this approximation, we prove the convergence of solutions of the heat equation driven by $\mu$.
Keywords: stochastic measure, random Fourier series, stochastic integral, stochastic heat equation, mild solution. 
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V. M. Radchenko. Fourier series expansion of stochastic measures. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 2, pp. 389-401. http://geodesic.mathdoc.fr/item/TVP_2018_63_2_a7/

[1] K. Dzhaparidze, H. van Zanten, “Krein's spectral theory and the Paley–Wiener expansion for fractional Brownian motion”, Ann. Probab., 33:2 (2005), 620–644 | DOI | MR | Zbl

[2] Y. Meyer, F. Sellan, M. S. Taqqu, “Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion”, J. Fourier Anal. Appl., 5:5 (1999), 465–494 | DOI | MR | Zbl

[3] V. Pipiras, “Wavelet-type expansion of the Rosenblatt process”, J. Fourier Anal. Appl., 10:6 (2004), 599–634 | DOI | MR | Zbl

[4] G. Didier, V. Pipiras, “Gaussian stationary processes: adaptive wavelet decompositions, discrete approximations, and their convergence”, J. Fourier Anal. Appl., 14:2 (2008), 203–234 | DOI | MR | Zbl

[5] Zh.-P. Kakhan, Sluchainye funktsionalnye ryady, Mir, M., 1973, 302 pp. ; J.-P. Kahane, Some random series of functions, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968, viii+184 pp. | MR | Zbl | MR | Zbl

[6] M. Talagrand, Upper and lower bounds for stochastic processes. Modern methods and classical problems, Ergeb. Math. Grenzgeb. (3), 60, Springer, Heidelberg, 2014, xvi+626 pp. | DOI | MR | Zbl

[7] M. B. Marcus, G. Pisier, Random Fourier series with applications to harmonic analysis, Ann. of Math. Stud., 101, Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo, 1981, v+151 pp. | DOI | MR | Zbl

[8] S. Ronsin, H. Biermé, L. Moisan, “The Billard theorem for multiple random Fourier series”, J. Fourier Anal. Appl., 23:1 (2017), 207–228 | DOI | MR | Zbl

[9] A. Zigmund, Trigonometricheskie ryady, v. 1, Mir, M., 1965, 615 pp. ; A. Zygmund, Trigonometric series, v. I, 2nd ed., Cambridge Univ. Press, New York, 1959, xii+383 pp. | MR | Zbl | MR | Zbl

[10] G. Samorodnitsky, M. S. Taqqu, Stable non-Gaussian random processes. Stochastic models with infinite variance, Stochastic Model., Chapman Hall, New York, 1994, xxii+632 pp. | MR | Zbl

[11] L. Drewnowski, “Topological rings of sets, continuous set functions, integration. {III}”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 20:6 (1972), 439–445 | MR | Zbl

[12] N. N. Vakhaniya, V. I. Tarieladze, S. A. Chobanyan, Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, Nauka, M., 1985, 368 pp. ; N. N. Vakhania, V. I. Tarieladze, S. A. Chobanyan, Probability distributions on Banach spaces, Math. Appl. (Soviet Ser.), 14, D. Reidel Publishing Co., Dordrecht, 1987, xxvi+482 pp. | MR | Zbl | DOI | MR | Zbl

[13] V. N. Radchenko, Integraly po obschim sluchainym meram, Tr. In-ta matem. NAN Ukrainy, 27, In-t matem. NAN Ukrainy, Kiev, 1999, 196 pp. | Zbl

[14] S. Kwapień, W. A. Woyczyński, Random series and stohastic integrals: single and multiple, Probab. Appl., Birkhäuser Boston, Inc., Boston, MA, 1992, xvi+360 pp. | MR | Zbl

[15] V. M. Radchenko, N. O. Stefanska, “Ryadi Fur'{\fontencoding{T2A}\selectfontє ta Fur'{\fontencoding{T2A}\selectfontє–Khaara dlya stokhastichnikh mir”, Teoriya imovirn. ta matem. statist., 96 (2017), 155–162 | MR

[16] V. N. Radchenko, “Vyborochnye funktsii sluchainykh mer i prostranstva Besova”, Teoriya veroyatn. i ee primen., 54:1 (2009), 161–169 ; V. N. Radchenko, “Sample functions of stochastic measures and Besov spaces”, Theory Probab. Appl., 54:1 (2010), 160–168 | DOI | MR | Zbl | DOI

[17] L. Carleson, “On convergence and growth of partial sums of Fourier series”, Acta Math., 116:1 (1966), 135–157 | DOI | MR | Zbl

[18] V. Radchenko, “Stratonovich-type integral with respect to a general stochastic measure”, Stochastics, 88:7 (2016), 1060–1072 | DOI | MR | Zbl

[19] L. Drewnowski, “Topological rings of sets, continuous set functions, integration. {I}”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 20:4 (1972), 269–276 | MR | Zbl

[20] I. M. Bodnarchuk, G. M. Shevchenko, “Heat equation in a multidimensional domain with a general stochastic measure”, Theory Probab. Math. Statist., 93 (2016), 1–17 | DOI | MR | Zbl

[21] V. Radchenko, “Mild solution of the heat equation with a general stochastic measure”, Studia Math., 194:3 (2009), 231–251 | DOI | MR | Zbl

[22] V. N. Radchenko, “Evolyutsionnye uravneniya s obschimi stokhasticheskimi merami v gilbertovom prostranstve”, Teoriya veroyatn. i ee prim., 59:2 (2014), 375–386 ; V. M. Radchenko, “Evolution equations with common stochastic measures in Hilbert space”, Theory Probab. Appl., 59:2 (2015), 328–339 | DOI | Zbl | DOI | MR

[23] V. M. Radchenko, “Approximation of integrals with respect to a random measure by integrals with respect to a real measure”, Theory Probab. Math. Statist., 55 (1997), 177–179 | MR | Zbl