zbMATH — the first resource for mathematics

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.
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H40 White noise theory
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.