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Fractional random fields associated with stochastic fractional heat equations

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

M. Ya. Kelbert ,
N. N. Leonenko  and
M. D. Ruiz-Medina
M. Ya. Kelbert*
Affiliation:
University of Wales Swansea
N. N. Leonenko*
Affiliation:
Cardiff University
M. D. Ruiz-Medina*
Affiliation:
University of Granada
*
∗ Postal address: Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK.
∗∗ Postal address: Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK. Email address: leonenkon@cardiff.ac.uk
∗∗∗ Postal address: Department of Statistics and Operational Research, University of Granada, Campus Fuente Nueva s/n., E-18071 Granada, Spain.
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Abstract

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This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

Keywords

Fractional random fieldoperator spectral methodfractional heat equationspatiotemporal spectral densityMatérn classgeostatistics

MSC classification

Primary: 60G60: Random fields 60G20: Generalized stochastic processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 108 - 133
Copyright
Copyright © Applied Probability Trust 2005 

References

Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York.Google Scholar
Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, New York.Google Scholar
Angulo, J. M., Ruiz-Medina, M. D. and Anh, V. V. (2000a). Estimation and filtering of fractional generalized random fields. J. Austral. Math. Soc. A 69, 336–361.CrossRefGoogle Scholar
Angulo, J. M., Ruiz-Medina, M. D., Anh, V. V. and Grecksch, W. (2000b). Fractional diffusion and fractional heat equation. Adv. Appl. Prob. 32, 1077–1099.CrossRefGoogle Scholar
Anh, V. V. and Leonenko, N. N. (2000). Scaling laws for fractional diffusion-wave equations with singular data. Statist. Prob. Lett. 48, 239–252.CrossRefGoogle Scholar
Anh, V. V. and Leonenko, N. N. (2001). Spectral analysis of fractional kinetic equations with random data. J. Statist. Phys. 104, 1349–1487.Google Scholar
Anh, V. V. and Leonenko, N. N. (2002). Renormalization and homogenization of fractional diffusion equations with random data. Prob. Theory Relat. Fields 124, 381–408.Google Scholar
Anh, V. V., Angulo, J. M. and Ruiz-Medina, M. D. (1999). Possible long-range dependence in fractional random fields. J. Statist. Planning Infer. 80, 95–110.CrossRefGoogle Scholar
Chambers, M. J. (1996). The estimation of continuous parameter long-memory time series models. Econom. Theory 12, 373–390.Google Scholar
Chentsov, N. N. (1960). Limit theorems for certain classes of random functions. In Proc. All-Union Conf. Theory Prob. Math. Statist., Armenian Academy of Sciences, Yerevan, pp. 280–285 (in Russian).Google Scholar
Christakos, G. (1991). A theory of spatio-temporal random fields and its application to space-time data processing. IEEE Trans. Systems Man. Cybernet. 21, 861–875.Google Scholar
Christakos, G. (2000). Modern Spatiotemporal Geostatistics. Oxford University Press.Google Scholar
Dautray, R. and Lions, J. L. (1985a). Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Functional and Variational Methods. Springer, New York.Google Scholar
Dautray, R. and Lions, J. L. (1985b). Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Spectral Theory and Applications. Springer, New York.Google Scholar
Dobrushin, R. L. (1979). Gaussian and their subordinated self-similar random generalized fields. Ann. Prob. 7, 1–28.Google Scholar
Dobrushin, R. L. and Major, P. (1979). Non-central limit theorem for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitsth. 50, 1–28.CrossRefGoogle Scholar
Donoghue, W. J. (1969). Distributions and Fourier Transforms. Academic Press, New York.Google Scholar
Dunford, N. and Schwartz, J. T. (1971). Linear Operators. John Wiley, New York.Google Scholar
Gay, R. and Heyde, C. C. (1990). On a class of random field model which allows long dependence. Biometrika 77, 401–403.Google Scholar
Geĺfand, I. M. and Vilenkin, N. Ya. (1964). Generalized Functions, Vol. 4. Academic Press, New York.Google Scholar
Gı¯hman, Ī. Ī. and Skorokhod, A. V. (1971). The Theory of Stochastic Processes, Vol. 1. Springer, Berlin.Google Scholar
Granger, C. W. J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Series Anal. 10, 233–257.Google Scholar
Heyde, C. C. (1997). Quasi-Likelihood and Its Applications: A General Approach to Optimal Parameter Estimation. Springer, New York.Google Scholar
Hilfer, R. (ed.) (2000). Applications of Fractional Calculus in Physics. World Scientific, Singapore.Google Scholar
Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68, 165–176.CrossRefGoogle Scholar
Ibragimov, I. A. (1983). On smoothness conditions for trajectories of random functions. Theory Prob. Appl. 28, 240–262 (in Russian).Google Scholar
Ibragimov, I. A. and Khasminskii, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Springer, New York.CrossRefGoogle Scholar
Ivanov, A. B. and Leonenko, N. N. (1989). Statistical Analysis of Random Fields. Kluwer, Dordrecht.CrossRefGoogle Scholar
Jones, R. H. and Zhang, Y. (1997). Models for continuous stationary space-time processes. In Modelling Longitudinal and Spatially Corrected Data, eds Gregoire, T. G. et al., Springer, New York, pp. 343–449.Google Scholar
Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Korolyuk, V. S., Portenko, N. I., Skorokhod, A. V. and Turbin, A. F. (1978). Handbook on Probability Theory and Mathematical Statistics. Naukova Dumka, Kiev (in Russian).Google Scholar
Mainardi, F., Luchko, Y. and Pagnini, G. (2001). The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153–192.Google Scholar
Major, P. (1981). Multiple Wiener–Itô Integrals (Lecture Notes Math. 849). Springer, Berlin.Google Scholar
Prakasa Rao, B. L. S. (1987). Asymptotic Theory of Statistical Inference. John Wiley, New York.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations (Appl. Math. (New York) 21). Springer, Berlin.Google Scholar
Ramm, A. G. (1990). Random Fields Estimation Theory. Longman Scientific and Technical, London.Google Scholar
Ruiz-Medina, M. D., Angulo, J. M. and Anh, V. V. (2003). Fractional generalized random fields on bounded domains. Stoch. Anal. Appl. 21, 465–492.Google Scholar
Ruiz-Medina, M. D., Angulo, J. M. and Anh, V. V. (2004). Fractional random fields on domains with fractal boundary. Infinit. Dimens. Anal. Quantum Prob. Relat. Top. 7, 395–417.Google Scholar
Ruiz-Medina, M. D., Anh, V. V. and Angulo, J. M. (2001). Stochastic fractional-order differential models with fractal boundary conditions. Statist. Prob. Lett. 54, 47–60.Google Scholar
Stein, E. M. (1970). Singular Integrals and Differential Properties of Functions. Princeton University Press.Google Scholar
Stein, M. L. (1999). Interpolation of Spatial Data. Springer, Berlin.CrossRefGoogle Scholar
Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitsth. 50, 53–83.Google Scholar
Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam.Google Scholar
Whittle, P. (1954). On stationary processes in the plane. Biometrika 41, 434–449.Google Scholar
Whittle, P. (1963). Stochastic processes in several dimensions. Bull. Inst. Internat. Statist. 40, 974–994.Google Scholar
Wong, E. and Hajek, B. (1985). Stochastic Processes in Engineering Systems. Springer, New York.Google Scholar
Woyczyński, W. A. (1998). Burgers-KPZ Turbulence (Lecture Notes Math. 1700). Springer, Berlin.Google Scholar
Yadrenko, M. Ī. (1983). Spectral Theory of Random Fields. Optimization Software, New York.Google Scholar