Fractional Elliptic, Hyperbolic and Parabolic Random Fields
Maria D. Ruiz-Medina (University of Granada)
Murad S. Taqqu (Boston University)
Abstract
This paper introduces new classes of fractional and multifractional random fields arising from elliptic, parabolic and hyperbolic equations with random innovations derived from fractional Brownian motion. The case of stationary random initial conditions is also considered for parabolic and hyperbolic equations.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1134-1172
Publication Date: June 5, 2011
DOI: 10.1214/EJP.v16-891
References
- Adler, R.J. (1981). The Geometry of Random Fields. Wiley, Chichester. MR0611857
- Amann, H. (2009). Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function Spaces. Jindrich Necas Center for Mathematical Modeling Lecture Notes, Prague, Volume 6.
- Angulo, J.M.,Anh, V.V., McVinish, R. and Ruiz-Medina, M.D. (2005). Fractional kinetic equation driven by gaussian or infinitely divisible noise. Advances in Applied Probability 37, 366--392. MR2144558
- Angulo, J.M. Ruiz-Medina, M.D., Anh, V.V. and Grecksch, W. (2000). Fractional diffusion and fractional heat equation. Advances in Applied Probability 32, 1077-1099. MR1808915
- Anh, V.V., Angulo J.M. and Ruiz-Medina, M.D. (1999). Possible long-range dependence in fractional random fields. Journal of Statistical Planning and Inference 80, 95--110. MR1713795
- Anh, V.V. and Leonenko, N.N. (2001). Spectral analysis of fractional kinetic equations with random data. J. Statis. Phys. 104, 1349--1387. MR1859007
- Benassi, A. Jaffard, S and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13, 19--90. MR1462329
- Biagini, F. , Hu, Y., Oksendal, B. and Zhang, T. (2008). Stochastic calculus for fractional Brownian motion and applications. Springer-Verlag, London. MR2387368
- Chow, P.-L. (2007). Stochastic partial differential equations. Chapman \& Hall. MR2295103
- Dachkovski, S. (2003). Anisotropic function spaces and related semilinear hypoelliptic equations. Math. Nachr. 248-249, 40-61. MR1950714
- Dautray, R. and Lions, J.L. (1985a). Mathematical analysis and numerical methods for science and technology. Volume~2 of Functional and variational methods. Springer-Verlag, New York. MR0969367
- Dautray, R. and Lions, J.~L.} (1985b). Mathematical Analysis and Numerical Methods for Science and Technology, Volume~3 of Spectral Theory and Applications. Springer-Verlag, New York.
- Duncan, T.E., Maslowski, B. and Pasik-Duncan, B. (2002). Fractional Brownian motion and stochastic equations in Hilbert space. Stoch. Dyn. 2, 225-250. MR1912142
- Edwards, R.E. (1965). Functional Analysis. Holt, Rinehart and Winston, New York. MR0221256
- Gibinelle, M., Lejay, A. and Tindel, S. (2006). Young integrals and SPDE. Pot. Analysis 25, 307-326. MR2255351
- Guyon, X . (1987). Variations de champs gaussiens stationnaites: application a l'identification. Probability Theory and Related Fields 75, 179--193. MR0885461
- Heine, V. (1955). Models for two-dimensional stationary stochastic processes. Biometrika 42, 170--178. MR0071673
- Hu, Y. (2001). Heat equation with fractional white noise potentials. Appl. Math. Optim. 43, 221-243. MR1885698
- Hu, Y. and Nualart, D. (2009b). Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361, 2689-2718. MR2471936
- Hu, Y., Oksendal, B. and Zhang, T. (2004). General fractional multiparameter noise theory and stochastic partial differential equations. Commun. Partial Differemt. Equations 29, 1-23. MR2038141
- Igloi, E. and Terdik, G. (1999). Bilinear stochastic systems with fractional Brownian motion input. Ann. Appl. Probab. 9, 46--77. MR1682600
- Jacob, N. (2005). Pseudodifferential operators: Markov processes III. Markov processes and applications, Imperial College Press. MR2158336
- Kelbert, M. Leonenko, N. and Ruiz-Medina, M.D. (2005). Fractional Random fields associated with stochastic fractional heat equations. Advances in Applied Probability 37, 108-133. MR2135156
- Kikuchi, K. and Negoro, A. (1997). On Markov processes generated by pseudodifferentail operator of variable order. Osaka Journal of Mathematics 34, 319--335. MR1483853
- Kozachenko, Yu. V. and Slivka, G. I. (2007). Modelling a solution of a hyperbolic equation with random initial conditions. Theor. Probability and Math. Statist. 74, 59-75. MR2336779
- Leonenko, N. N. and Ruiz-Medina, M. D. (2006) Scaling laws for the multidimensional Burgers equation with quadratic external potential. J. Stat. Phys. 124, 191--205. MR2256621
- Leonenko, N. N. and Ruiz-Medina, M. D. (2008) Gaussian scenario for the heat equation with quadratic potential and weakly dependent data with applications. Methodol. Comput. Appl. Probab., 10, 595—620. MR2443082
- Leopold, H.-G. (1991). On function spaces of variable order of differentiation. Forum Mathematicum 3, 1--21. MR1085592
- Lions, T. and Qian, Z. (2002). System control and rough paths. Oxford Mathematical Monographs, Oxford University Press, Oxford MR2036784
- Malowski, B. and Nualart, D. (2005). Evolution equations driven by a fractional Brownian motion. J. Func. Anal. 209, 277-305. MR1994773
- Mishura, Y.S. (2008). Stochastic calculus for fractional Brownian motion and related processes. Lecture Notes in Mathematics, 1929, Springer-Verlag, Berlín. MR2378138
- Mohapl, J. (1999). On estimation in random fields generated by linear stochastic partial differential equations. Mathematica Slovaca 49, 95--115. MR1804478
- Mueller, C. and Tribe, R. (2004). A singular parabolic Anderson model. Electron J. Probab. 9, 98-144. MR2041830
- Nualart, D. and Sanz-Solé, M. (1979). A Markov property for two-parameter Gaussian processes. Stochastica 3, 1--16. MR0562437
- Ramm, A.G. (2005). Random fields estimation. World Scientific, Singapore.
- Ratanov, N.E., Shuhov, A.G. and Suhov, Yu M. (1991). Stabilization of the statistical solution of the parabolic equation. Acta Applicandae Mathematicae 22, 103-115. MR1100768
- Robeva, R.S. and Pitt, L.D.} (2007). On the equality of sharp and germ \sigma -fields for Gaussian processes and fields. Pliska Stud. Math. Bulgar. 16, 183-205. MR2070315
- Ruiz-Medina, M.D., Angulo, J.M. and Anh, V.V. (2002). Stochastic fractional-order differential models on fractals. Theory of Probability and Mathematical Statistics 67, 130--146. MR1956627
- Ruiz-Medina, M.D., Angulo, J.M. and Anh, V.V. (2003). Fractional Generalized Random Fields on Bounded Domains. Stochastic Analysis and Applications 21, 465--492. MR1967723
- Ruiz-Medina, M.D., Angulo, J.M. and Anh, V.V. (2006). Spatial and spatiotemporal Karhunen-Loéve-type representations on fractal domains. Stochastic Analysis and Applications 24, 195--219. MR2198541
- Ruiz-Medina, M.D., Angulo, J.M. and Anh, V.V. (2008). Multifractality in space-time statistical models. Stochastic Environmental Research and Risk Assessment 22, 81--86. MR2418414
- Ruiz-Medina, M.D., Anh, V.V. and Angulo, J.M. (2004a). Fractional generalized random fields of variable order. Stochastic Analysis and Applications 22, 775--799. MR2047278
- Ruiz-Medina, M.D., Anh, V.V. and Angulo, J.M. (2004b). Fractal random fields on domains with fractal boundary. Infinite Dimensional Analysis, Quantum Probability and Related Topics 7, 395--417. MR2085640
- Ruiz-Medina, M.D., Anh, V.V. and Angulo, J.M.} (2010). Multifractional Markov processes in heterogeneous domains. Stochastic Analysis and Applications 29, 15--47.
- Sanz-Solé, M. and Torrecilla, I. (2009). A fractional Posson equation: existence, regularity and approximation of the solutions. Stochastics and Dynamics 9, 519-548. MR2589036
- Sanz-Solé, M. and Vuilermot, P.A. (2010). Mild solutions for a class of fractional SPDs and their sample paths. Preprint.
- Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. verw. Gebiete 50, 53-83. MR0550123
- Taqqu, M. S. (2003) Fractional Brownian motion and long-range dependence. Theory and applications of long-range dependence,5--38, Birkhäuser Boston, Boston, MA. MR1956042
- Tindel, S., Tudor, C. and Viens, F. (2003). Stochastic evolution equations with the drift in the first fractional chaos. Stoch. Anal. Appl. 22, 1209-1233.
- Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Co. Amsterdam. MR0503903
- Vecchia, A. V. (1985). A general class of models for stationary two-dimensional random processes. Biometrika 72, 281-291. MR0801769
- Wu, D. and Xiao, Y. (2006). Fractal properties of the random string processes. IMS Lecture Notes-Monograph Series. High Dimensional Probability 51, 128-147. MR2387765
- Yaglom. A.M. (1957). Some classes of random fields in n-dimensional space related to stationary random processes. Theor. Prob. Appl. 3, 273-320.
- Yaglom. A.M. (1986). Correlation Theory of Stationary and Related Random Functions I. Basic Results. Springer-Verlag, New-York. MR0893393

This work is licensed under a Creative Commons Attribution 3.0 License.