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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

Citations:

Zbl 1198.60005
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References:

[1] Bender, C., Marquardt, T.: Stochastic calculus for convoluted Lévy processes. Bernoulli 14(2), 499-518 (2008) · Zbl 1173.60017 · doi:10.3150/07-BEJ115
[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., Van der Hoek, J.: A general fractional white noise theory and applications to finance. Math. Finance 13, 301-330 (2003) · Zbl 1069.91047 · doi:10.1111/1467-9965.00018
[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 · doi:10.1016/j.jmaa.2006.02.020
[7] Huang, Z., Li, P.: Generalized fractional Lévy processes: a white noise approach. Stoch. Dyn. 6, 473-485 (2006) · Zbl 1109.60057 · doi:10.1142/S0219493706001839
[8] Huang, Z., Li, P.: Fractional generalized Lévy random fields as white noise functionals. Front. Math. China 2, 211-226 (2007) · Zbl 1135.60326 · doi:10.1007/s11464-007-0015-4
[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 · doi:10.1142/S0219493710002838
[10] Lokka, A., Oksendal, B., Proske, F.: Stochastic partial differential equations driven by Lévy space-time white noise. Ann. Appl. Probab. 14(3), 1506-1528 (2004) · Zbl 1053.60069 · doi:10.1214/105051604000000413
[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 · doi:10.1007/s11464-011-0130-0
[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 · doi:10.1007/s11464-008-0022-0
[14] Marquardt, T.: Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, 1099-1126 (2006) · Zbl 1126.60038 · doi:10.3150/bj/1165269152
[15] Nualart, D., Schoutens, W.: Chaotic and predictable representations for Lévy processes. Stoch. Process. Appl. 90(1), 109-122 (2000) · Zbl 1047.60088 · doi:10.1016/S0304-4149(00)00035-1
[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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