Generalized fractional Lévy random fields on Gel’fand triple: A white noise approach

Abstract

In this paper, under the first-order moment condition of the infinitely divisible distribution on Gel’fand triple, we use Riesz potential to construct fractional Lévy random fields on Gel’fand triple by white noise approach. We investigate the distribution and sample properties of isotropic and anisotropic fractional Lévy random fields, respectively.

Keywords

Infinitely divisible distributions / Gel’fand triple / fractional Lévy noise / generalized Lévy random field / fractional generalized Lévy random field / anisotropic Lévy random fields

References

[1] Ahn V V, Angulo J M, Ruisz-Medina M D. Possible long range dependence in fractional random fields. J Statis Planning and Inference, 1999, 80: 95-110

[2] Dekking M, Lévy-Véhel J, Lutton E, Tricot C. Fractals: Theory and Applications in Engineering. London: Spring-Verlag, 1999

[3] Duncan T E, Jakubowski J, Pasik-Duncan B. Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stoch Proc Appl, 2005, 115: 1357-1387

[4] Duncan T E, Jakubowski J, Pasik-Duncan B. Stochastic integration for fractional Brownian motion in a Hilbert space. Stoch Dyn, 2006, 6: 53-75

[5] Gel’fand I M, Vilenkin N Ya. Generalized Functions, Vol 4. New York: Academic Press, 1964

[6] Huang Z Y, Li C J. On fractional stable processes and sheets: white noise approach. J Math Anal Appl, 2007, 325: 624-635

[7] Huang Z Y, Li C J, Wan J P, Wu Y. Fractional Brownian motion and sheet as white noise functionals. Acta Math Sinica, 2006, 22: 1183-1188

[8] Huang Z Y, Li P Y, Generalized fractional Lévy processes: a white noise approach. Stoch Dyn, 2006, 6: 473-485

[9] Huang Z Y, Li P Y. Fractional generalized Lévy random fields as white noise functionals. Front Math China, 2007, 2(2): 211-226

[10] Huang Z Y, Lü X B, Wan J P, Fractional Lévy processes and noises on Gel’fand triple. Stoch Dyn, 2010, 10: 37-51

[11] Itô K. Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol 47. Philadelphia: SIAM, 1984

[12] Lü X B, Huang Z Y, Wan J P. Fractional Lévy Processes on Gel’fand Triple and Stochastic Integration. Front Math China, 2008, 3(2): 287-303

[13] Mandelbrot B, Van Ness J. Fractional Brownian motion, fractional noises and application. SIAM Rev, 1968, 10: 427-437

[14] Marquardt T, Fractional Lévy processes with an application to long memory moving average processes. Bernoulli, 2006, 12(6): 1009-1126

[15] Maslowski B, Nualart D. Evolution equations driven by a fractional Brownian motion. J Funct Anal, 2003, 202: 277-305

[16] Ruiz-Medina M D, Angulo J M, Ahn V V. Fractional generalized random fields on bounded domains. Stoch Anal Appl, 2003, 21: 462-492

[17] Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives: Theory and Applications. London: Gordon and Breach, 1987

[18] Sato K. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999

[19] Tindel S, Tudor C A, Viens F. Stochastic evolution equations with fractional Brownian motion. Probab Th Rel Fields, 2003, 127: 186-204

[20] Treves F. Topological Vector Spaces, Distributions and Kernels. New York: Academic Press, 1967

[21] Wang C S, Qu MS, Chen J S. A white noise approach to infinitely divisible distributions on Gel’fand triple. J Math Anal Appl, 2006, 315: 425-435

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg