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A white noise approach to stochastic partial differential equations driven by the fractional Lévy noise. (English) Zbl 1448.60142
Summary: In this paper, based on the white noise theory for \(d\)-parameter Lévy random fields given by H. Holden et al. [Stochastic partial differential equations. A modeling, white noise functional approach. 2nd ed. New York, NY: Springer (2010; Zbl 1198.60005)], we develop a white noise frame for anisotropic fractional Lévy random fields to solve the stochastic Poisson equation and the stochastic Schrödinger equation driven by the \(d\)-parameter fractional Lévy noise. The solutions for the two kinds of equations are all strong solutions given explicitly in the Lévy-Hida stochastic distribution space.
MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H40 White noise theory
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
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[1] Bender, C.; Marquardt, T., Stochastic calculus for convoluted Lévy processes, Bernoulli, 14, 499-518, (2008) · Zbl 1173.60017
[2] Bers, L., John, F., Schechter, M.: Partial Differential Equations, Interscience (1964)
[3] Durrett, R.: Brownian Motion and Martingales in Analysis. Wadsworth, Belmont (1984) · Zbl 0554.60075
[4] Elliott, R. C.; Hoek, J., A general fractional white noise theory and applications to finance, Math. Finance, 13, 301-330, (2003) · Zbl 1069.91047
[5] Holden, H., Oksendal, B., Uboe, J., Zhang, T.: Stochastic Partial Differential Equations: a modeling, white noise functional approach, 2nd edn. Springer, (2010) · Zbl 1198.60005
[6] Huang, Z.; Li, C., On fractional stable processes and sheets: white noise approach, J. Math. Anal. Appl., 325, 624-635, (2007) · Zbl 1116.60018
[7] Huang, Z.; Li, P., Generalized fractional Lévy processes: a white noise approach, Stoch. Dyn., 6, 473-485, (2006) · Zbl 1109.60057
[8] Huang, Z.; Li, P., Fractional generalized Lévy random fields as white noise functionals, Front. Math. China, 2, 211-226, (2007) · Zbl 1135.60326
[9] Huang, Z.; Lü, X.; Wan, J., Fractional Lévy processes and noises on Gel’fand triple, Stoch. Dyn., 10, 37-51, (2010) · Zbl 1185.60053
[10] Lokka, A.; Oksendal, B.; Proske, F., Stochastic partial differential equations driven by Lévy space-time white noise, Ann. Appl. Probab., 14, 1506-1528, (2004) · Zbl 1053.60069
[11] Lü, X., Dai, W.: White noise analysis for fractional Lévy processes and its applications. (to appear)
[12] Lü, X.; Huang, Z.; Dai, W., Generalized fractional Lévy random fields on Gel’fand triple: a white noise approach, Front. Math. China, 6, 493-506, (2011) · Zbl 1288.60086
[13] Lü, X.; Huang, Z.; Wan, J., Fractional Lévy processes on Gel’fand triple and stochastic integration, Front. Math. China, 3, 287-303, (2008) · Zbl 1146.60309
[14] Marquardt, T., Fractional Lévy processes with an application to long memory moving average processes, Bernoulli, 12, 1099-1126, (2006) · Zbl 1126.60038
[15] Nualart, D.; Schoutens, W., Chaotic and predictable representations for Lévy processes, Stoch. Process. Appl., 90, 109-122, (2000) · Zbl 1047.60088
[16] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York (1987) · Zbl 0617.26004
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