Fractional random fields on domains with fractal boundary. (English) Zbl 1055.60046
The authors study generalized random fields on fractional Sobolev spaces on \(R^n\) whose dual characterizes the restriction of its reproducing kernel Hilbert space (RKHS) to a compact fractal set. The orthogonal decomposition of the RKHS in terms of complementary domains is studied. A characterization of the weak-sense Markov property with respect to fractal boundaries is given. The fractional-order differential representation on domains with fractal boundary is derived. A strong version of the results is presented. As examples fractional Riesz-Bessel motion and Gaussian homogeneous fields with fractional-order rational spectra are considered.
Reviewer: Andrew Olenko (Kyïv)
MSC:
| 60G60 | Random fields |
| 60G20 | Generalized stochastic processes |
| 60J25 | Continuous-time Markov processes on general state spaces |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
fractal set; random fields; fractional generalized random field; Sobolev space; stochastic fractal drum; trace operator; Markov random fieldReferences:
| [1] | Adler R. J., The Geometry of Random Fields (1981) · Zbl 0478.60059 |
| [2] | DOI: 10.1017/S0001867800009617 · doi:10.1017/S0001867800009617 |
| [3] | Angulo J. M., J. Austral. Math. Soc. (Series A) 69 pp 1– |
| [4] | DOI: 10.1016/S0378-3758(98)00244-4 · Zbl 1039.62090 · doi:10.1016/S0378-3758(98)00244-4 |
| [5] | DOI: 10.1017/S0004972700018803 · Zbl 0968.60035 · doi:10.1017/S0004972700018803 |
| [6] | Anh V. V., J. Austral. Math. Soc. B 42 pp 1– |
| [7] | DOI: 10.1016/S1364-8152(98)00023-1 · doi:10.1016/S1364-8152(98)00023-1 |
| [8] | M. T. Barlow, Lectures on Probability Theory and Statistics (Ecole d’Eté de Probabilités de Saint-Flour XXV-1995), ed. P. Bernard (Springer, 1998) pp. 1–123, DOI: 10.1007/BFb0092537. · doi:10.1007/BFb0092537 |
| [9] | Barlow M. T., Ann. Inst. H. Poincaré 25 pp 225– |
| [10] | DOI: 10.1007/BF01192060 · Zbl 0739.60071 · doi:10.1007/BF01192060 |
| [11] | DOI: 10.1007/BF00318785 · Zbl 0635.60090 · doi:10.1007/BF00318785 |
| [12] | DOI: 10.1029/2000WR900409 · doi:10.1029/2000WR900409 |
| [13] | Benassi A., Revista Mat. Iber. 13 pp 19– |
| [14] | DOI: 10.1112/S0024611598000331 · Zbl 0891.35108 · doi:10.1112/S0024611598000331 |
| [15] | DOI: 10.1016/S0960-0779(00)00238-1 · Zbl 1030.74045 · doi:10.1016/S0960-0779(00)00238-1 |
| [16] | DOI: 10.1007/978-3-7091-2664-6 · doi:10.1007/978-3-7091-2664-6 |
| [17] | DOI: 10.1016/S0376-7388(00)00689-X · doi:10.1016/S0376-7388(00)00689-X |
| [18] | DOI: 10.1002/pen.11230 · doi:10.1002/pen.11230 |
| [19] | DOI: 10.1007/978-1-4757-1776-1 · doi:10.1007/978-1-4757-1776-1 |
| [20] | Conway J. B., A Course in Functional Analysis (1990) · Zbl 0706.46003 |
| [21] | Dautray R., Mathematical Analysis and Numerical Methods for Science and Technology. Functional and Variational Methods (1985) |
| [22] | DOI: 10.1142/S0218348X9900027X · Zbl 1032.35147 · doi:10.1142/S0218348X9900027X |
| [23] | DOI: 10.1017/CBO9780511662201 · doi:10.1017/CBO9780511662201 |
| [24] | Falconer K. J., Fractal Geometry (1990) |
| [25] | Falconer K. J., Techniques in Fractal Geometry (1997) · Zbl 0869.28003 |
| [26] | Fleckinger-Pellé J., Trans. AMS 337 pp 99– |
| [27] | Gel’fand I. M., Generalized Functions (1964) |
| [28] | DOI: 10.1007/978-1-4613-8734-3_8 · doi:10.1007/978-1-4613-8734-3_8 |
| [29] | He C. Q., Mem. Amer. Math. Soc. 127 |
| [30] | Iannaccone M., Fractal Geometry in Biological Systems (1996) |
| [31] | DOI: 10.1016/0304-4149(95)00074-7 · Zbl 0853.60058 · doi:10.1016/0304-4149(95)00074-7 |
| [32] | Kigami J., Trans. AMS 335 pp 721– |
| [33] | DOI: 10.2977/prims/1195173187 · Zbl 0694.60071 · doi:10.2977/prims/1195173187 |
| [34] | S. Kusuoka, Statistical Mechanics and Fractals, Lect. Notes in Math. 1567, eds. R. L. Dobrushin and S. Kusuoka (Springer-Verlag, 1993) pp. 38–98. |
| [35] | DOI: 10.1090/S0002-9947-1991-0994168-5 · doi:10.1090/S0002-9947-1991-0994168-5 |
| [36] | M. L. Lapidus, Ordinary and Partial Differential Equations, Pitman Research Notes in Math., Vol. IV Longman Sci. Tech. 289 (1993) pp. 126–209. |
| [37] | DOI: 10.1090/conm/208/02742 · doi:10.1090/conm/208/02742 |
| [38] | Levitin M., Proc. London Math. Soc. 72 pp 188– |
| [39] | Lindstrom T., Mem. AMS 420 |
| [40] | DOI: 10.1017/CBO9780511623813 · doi:10.1017/CBO9780511623813 |
| [41] | DOI: 10.1016/S0378-4371(98)00614-1 · doi:10.1016/S0378-4371(98)00614-1 |
| [42] | DOI: 10.1007/978-1-4757-2437-0 · doi:10.1007/978-1-4757-2437-0 |
| [43] | Park H. W., Energy Sources 22 pp 881– · doi:10.1080/00908310051128237 |
| [44] | Peetre J., Math. Scand. 8 pp 116– · Zbl 0097.10402 · doi:10.7146/math.scand.a-10598 |
| [45] | DOI: 10.1007/978-1-4757-4740-9 · doi:10.1007/978-1-4757-4740-9 |
| [46] | Pitt L. D., Arch. Rational Mech. Anal. 43 pp 367– |
| [47] | Ramm A. G., Random Fields Estimation Theory (1990) · Zbl 0712.47042 |
| [48] | DOI: 10.1016/S0378-4371(98)00550-0 · doi:10.1016/S0378-4371(98)00550-0 |
| [49] | Y. A. Rozanov, Developments in Statistics, ed. R. Krishnaiah (Academic Press, 1979) pp. 203–234. |
| [50] | DOI: 10.1007/978-1-4613-8190-7 · doi:10.1007/978-1-4613-8190-7 |
| [51] | Rozanov Y. A., Random Fields and Stochastic Partial Differential Equations (1996) · Zbl 0902.60049 |
| [52] | Ruiz-Medina M. D., Theor. Probab. Math. Statist. 67 pp 130– |
| [53] | DOI: 10.1081/SAP-120019295 · Zbl 1043.60042 · doi:10.1081/SAP-120019295 |
| [54] | DOI: 10.1016/S0167-7152(01)00043-8 · Zbl 0988.60052 · doi:10.1016/S0167-7152(01)00043-8 |
| [55] | DOI: 10.1016/S0960-0779(99)00049-1 · Zbl 0976.82025 · doi:10.1016/S0960-0779(99)00049-1 |
| [56] | Sato K., Lévy Processes and Infinitely Divisible Distributions (2000) · Zbl 0973.60001 |
| [57] | DOI: 10.1007/3-540-59222-9 · Zbl 0823.00016 · doi:10.1007/3-540-59222-9 |
| [58] | DOI: 10.1007/BF03167188 · Zbl 0715.60088 · doi:10.1007/BF03167188 |
| [59] | DOI: 10.1007/BF03167295 · Zbl 0861.58047 · doi:10.1007/BF03167295 |
| [60] | Strichartz R. S., Notices Amer. Math. Soc. 46 pp 1199– |
| [61] | Stroock D. W., Multidimensional Diffusion Processes (1979) |
| [62] | Stein E. M., Singular Integrals and Differential Properties of Functions (1970) · Zbl 0207.13501 |
| [63] | Triebel H., Interpolation Theory, Function Spaces, Differential Operators (1978) · Zbl 0387.46033 |
| [64] | DOI: 10.1007/978-3-0348-0034-1 · Zbl 1208.46036 · doi:10.1007/978-3-0348-0034-1 |
| [65] | Wang Z., Appl. Math. Mech. 21 pp 1145– |
| [66] | DOI: 10.1016/S0378-4371(99)00565-8 · doi:10.1016/S0378-4371(99)00565-8 |
| [67] | DOI: 10.1007/978-3-642-61859-8 · Zbl 0830.46001 · doi:10.1007/978-3-642-61859-8 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
