Spatial and Spatiotemporal Karhunen-Loève-Type Representations on Fractal Domains

REFERENCES

• Adler1 , R.J. 1981 . The Geometry of Random Fields . New York : John Wiley & Sons .
   Google Scholar
• Alieva , T. , and F. Agulló-López . 1998 . Diffraction analysis of random fractal fields . Journal of the Optical Society of America A 15 : 669 – 674 . [CSA]
   Google Scholar
• Angulo , J.M. , V.V. Anh , R. McVinish , and M.D. Ruiz-Medina . 2005 . Fractional kinetic equation driven by Gaussian or infinitely divisible noise . Adv. Appl. Prob. 37 : 366 – 392 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Angulo , J. M. , M.D. Ruiz-Medina , and V.V. Anh . 2000 . Estimation and filtering of fractional generalized random fields . J. Austral. Math. Soc., Series A 69 : 336 – 361 . [CSA]
   Google Scholar
• Angulo , J.M. , M.D. Ruiz-Medina , V. V. Anh , and W. Grecksch . 2000. Fractional diffusion and fractional heat equation. Adv. Appl. Prob. 32: 1077–1099. [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Anh , V.V. , J.M. Angulo , and M.D. Ruiz-Medina . 1999 . Possible long-range dependence in fractional random fields . J. Statist. Planning & Inference 80 : 95 – 110 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Anh , V.V. , M.D. Ruiz-Medina , and J.M. Angulo . 2001 . Differential representation and Markov property of generalized random fields . Stoch. Anal. Appl. 19 : 481 – 498 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Barlow , M.T. 1993 . Random walks, electrical resistance, and nested fractals . Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals . Pitman Research Notes in Mathematics Series 283 , In eds. K.D. Elworthy and N. Ikeda , 131 – 157 . Essex : Longman .
   Google Scholar
• Barlow , M.T. 1993b . Fractals and diffusion-limited aggregation . Bull. Sci. Math. 117 : 161 – 169 . [CSA]
   Web of Science ®Google Scholar
• Barlow , M.T. , and R.F. Bass . 1992 . Transition densities for Browninan motion on the Sierpinski carpet . Prob. Theor. Rel. Fields 91 : 307 – 330 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Barabasi , A.L. , and H.E. Stanley . 1995 . Fractal Concepts in Surface Growth . Cambrige : Cambridge University Press .
   Google Scholar
• Becker-Kern , F. , M.M. Meerschaert , and H.P. Scheffler . 2004 . Limit theorem for continuous-time random walks with two time scales . J. Appl. Prob. 41 : 455 – 466 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Benassi , A. , S. Jaffard , and D. Roux . 1997 . Elliptic Gaussian random processes . Revista Matemática Iberoamericana 13 : 19 – 90 . [CSA]
   Web of Science ®Google Scholar
• Bouchaud , J.-P. , and A. Georges . 1990 . Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications . Physics Reports (Review Section of Physics Letters) 195 : 127 – 293 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Carpinteri , R. , and F. Mainardi (Eds.) 1997 . Fractals and Fractional Calculus in Continuum Mechanics . New York : Springer-Verlag .
   Google Scholar
• Chen , Z.-Q. , and T. Kumagai . 2003 . Heat kernel estimates for stable-like processes on d-sets . Stoch. Proc. Appl. 108 : 27 – 62 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Dautray , R. , and J.L. Lions . 1985 . Mathematical Analysis and Numerical Methods for Sciences and Technology , Vol. 3 . New York : Springer-Verlag .
   Google Scholar
• Dobrushin , R.L. , and S. Kusuoka . 1993 . Statistical Mechanics and Fractals . Lecture Notes in Mathematics 1567. New York : Springer-Verlag .
   Google Scholar
• Goldstein , S. 1987 . Random walks and diffusion on fractals . Percolation Theory and Ergodic Theory of Infinite Particle Systems . (IMA) Mathematics and Applications 8 . In ed. H. Kesten , 121 – 129 . New York : Springer-Verlag .
   Google Scholar
• Hambly , B. , and O.D. Jones . 2000 . Modelling transport in disordered media via diffusion on fractals . Mathematical and Computer Modelling 31 : 129 – 142 . [CSA] [CROSSREF]
   Google Scholar
• Hilfer , R. (Ed.) 2000 . Applications of Fractional Calculus in Physics . Singapore : World Scientific .
   Google Scholar
• Hutson , V. , and J.S. Pym . 1980 . Applications of Functional Analysis and Operator Theory . New York : Academic Press .
   Google Scholar
• Jacob , N. , and R.L. Schilling . 2001 . Lévy-type processes and pseudodifferential operators . Lévy Processes: Theory and Applications . In eds. O. Barndorff-Nielsen , T. Mikosch , and S. Resnick , 139 – 167 . Boston : Birkhauser .
   Google Scholar
• Jonsson , A. 2000. Dirichlet forms and Brownian motion penetrating fractals. Potential Analysis 13:69–80. [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Kigami , J. 1989 . A harmonic calculus on the Sierpinski spaces . Japan J. Appl. Math. 6 : 259 – 290 . [CSA]
   Google Scholar
• Kigami , J. 1993 . Harmonic metric and Dirichlet form on the Sierpinski gasket . Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals . Pitman Research Notes in Mathematics Series 283. In eds. K.D. Elworthy and N. Ikeda , 201 – 218 . Essex : Longman .
   Google Scholar
• Kumagai , T. 1993 . Construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket . Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals . Pitman Research Notes in Mathematics Series 283. In eds. K.D. Elworthy and N. Ikeda , 219 – 247 . Essex : Longman .
   Google Scholar
• Kumagai , T. 1995 . Rotation invariance and characterization of a class of self-similar diffusion processes on the Sierpinski gasket . Algorithms, Fractals, and Dynamics . In eds. K. Takahashi , 131 – 142 , New York : Plenum .
   Google Scholar
• Kusuoka , S.A. 1987 . A diffusion process on a fractal . Probabilistic Methods in Mathematical Physics, Proceedings of the Taniguchi International Symposium . In eds. K. Itô and N. Ikeda , 251 – 274 . Tokyo : Kinokuniya .
   Google Scholar
• Kusuoka , S.A. 1989 . Dirichlet forms on fractals and products of random matrices . Publ. Res. Inst. Mathematical Sci. Kyoto University 25 : 659 – 680 . [CSA]
   Web of Science ®Google Scholar
• Kusuoka , S.A. 1993 . Diffusion Processes on Nested Fractals. Statistical Mechanics and Fractals . Lecture Notes in Mathematics 1567 . New York : Springer .
   Google Scholar
• Lindstrom , T. 1993 . Brownian motion penetrating the Sierpinski gasket . Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals . Pitman Research Notes in Mathematics Series 283. In eds. K.D. Elworthy and N. Ikeda , 248 – 278 . Essex : Longman .
   Google Scholar
• Mainardi , F. , M. Raberto , R. Gorenflo , and E. Scalas . 2000 . Fractional calculus and continuous-time finance II: the waiting-time distribution . Physica A 287 : 468 – 481 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Metzler , R. , and J. Klafter . 2000 . The random walk's guide to anomalous diffusion: A fractional dynamic approach . Physical Reports 339 : 1 – 72 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Mortola , S. , and R. Peirone . 1996 . Homogenization of quadratic forms on fractals . Rendiconti del Seminario Matemàtico e Fisico di Milano 64 : 117 – 128 . [CSA]
   Google Scholar
• M.D. Ruiz-Medina , J.M. Angulo , and V.V. Anh . 2003 . Fractional generalized random fields on bounded domains . Stoch. Anal. Appl. 21 : 465 – 492 . [CSA]
   Web of Science ®Google Scholar
• M.D. Ruiz-Medina , V. V. Anh , and J.M. Angulo . 2001 . Stochastic fractional-order differential models with fractal boundary conditions . Stat. & Prob. Lett. 54 : 47 – 60 . [INFOTRIEVE] [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• M.D. Ruiz-Medina , J.M. Angulo , and V.V. Anh . 2002 . Stochastic fractional-order differential models on fractals . Theor. Prob. Math. Stat. 67 : 130 – 146 . [CSA]
   Google Scholar
• M.D. Ruiz-Medina , V.V. Anh , and J.M. Angulo . 2004 . Fractal random fields on domains with fractal boundary . Inf. Dim. Anal. Quant. Prob. Rel. Top. 7 : 395 – 417 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Ruzicka , M. 2000 . Electrorheological Fluids: Modeling and Mathematical Theory , Lecture Notes in Mathematics 1748. New York : Springer .
   Google Scholar
• Samorodnitsky , G. , and M.S. Taqqu . 1994. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance . New York : Chapman and Hall.
   Google Scholar
• Scalas , E. , R. Gorenflo , and F. Mainardi . 2000 . Fractional calculus and continuous-time finance . Physica A 284 : 376 – 384 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Schilling , R.L. 1998 . Growth and Hölder conditions for the sample paths of Feller processes . Probab. Theor. Rel. Fields 112 : 565 – 611 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Shlesinger , M.F. , G.M. Zaslavsky , and U. Frisch . 1995 . Lévy Flights and Related Topics in Physics . Lecture Notes in Physics, Vol. 450. New York : Springer .
   Google Scholar
• Schlichter , J. , J. Friedrich , L. Herenyi , and J. Fidy . 2000 . Protein dynamics at low temperatures , J Chem. Phys. 112 : 3045 – 3050 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar
• Triebel , H. 1978 . Interpolation Theory, Function Spaces, Differential Operators . Amsterdam : North-Holland Publishing .
   Google Scholar
• Triebel , H. 1997 . Fractals and Spectra . Basel : Birkhäuser .
   Google Scholar
• Zähle , M. 2004 . Riesz potentials and Liouville operators on fractals . Potential Analysis 21 : 193 – 208 . [CSA] [CROSSREF]
   Web of Science ®Google Scholar