Theory of Probability and Mathematical Statistics
On asymptotic behavior of solutions to random fractional Riesz–Bessel equations with cyclic long memory initial conditions
Maha Mosaad A. Alghamdi and Andriy Olenko
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Abstract: This paper investigates fractional Riesz–Bessel equations with random initial conditions. The spectra of these random initial conditions exhibit singularities at both zero frequency and non-zero frequencies, corresponding to the cases of classical long-range dependence and cyclic long-range dependence, respectively. Using spectral methods and asymptotic theory, it is shown that the rescaled solutions of the equations converge to spatio-temporal Gaussian random fields. The limit fields are stationary in space and non-stationary in time. The covariance and spectral structures of the resulting asymptotic random fields are provided. The paper further establishes multiscaling limit theorems for the case of regularly varying asymptotics. A numerical example illustrating the theoretical results is also presented.
Keywords: Fractional Riesz–Bessel equations, random partial differential equations, seasonal long memory, spectral singularities, multiscaling limit theorems.
Bibliography: Sergio Albeverio, Stanislav A. Molchanov, and Donatas Surgailis, Stratified structure of the Universe and Burgers’ equation—a probabilistic approach, Probab. Theory Related Fields 100 (1994), no. 4, 457–484. MR 1305783, DOI 10.1007/BF01268990
M. M. A. Alghamdi, N. Leonenko, and A. Olenko, Multiscaling asymptotic behavior of solutions to random high-order heat equations, Submitted; arXiv:2510.14153 (2025), 28.
Maha Mosaad A. Alghamdi, Nikolai Leonenko, and Andriy Olenko, Multiscaling limit theorems for stochastic FPDE with cyclic long-range dependence, Braz. J. Probab. Stat. 39 (2025), no. 2, 226–247. MR 4961223, DOI 10.1214/25-BJPS633
J. M. Angulo, M. D. Ruiz-Medina, V. V. Anh, and W. Grecksch, Fractional diffusion and fractional heat equation, Adv. in Appl. Probab. 32 (2000), no. 4, 1077–1099. MR 1808915, DOI 10.1239/aap/1013540349
V. V. Anh, J. M. Angulo, and M. D. Ruiz-Medina, Possible long-range dependence in fractional random fields, J. Statist. Plann. Inference 80 (1999), no. 1-2, 95–110. MR 1713795, DOI 10.1016/S0378-3758(98)00244-4
V. V. Anh and N. N. Leonenko, Non-Gaussian scenarios for the heat equation with singular initial conditions, Stochastic Process. Appl. 84 (1999), no. 1, 91–114. MR 1720100, DOI 10.1016/S0304-4149(99)00053-8
V. V. Anh and N. N. Leonenko, Scaling laws for fractional diffusion-wave equations with singular data, Statist. Probab. Lett. 48 (2000), no. 3, 239–252. MR 1765748, DOI 10.1016/S0167-7152(00)00003-1
V. V. Anh and N. N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Statist. Phys. 104 (2001), no. 5-6, 1349–1387. MR 1859007, DOI 10.1023/A:1010474332598
V. V. Anh and N. N. Leonenko, Renormalization and homogenization of fractional diffusion equations with random data, Probab. Theory Related Fields 124 (2002), no. 3, 381–408. MR 1939652, DOI 10.1007/s004400200217
V. V. Anh and N. N. Leonenko, Harmonic analysis of random fractional diffusion-wave equations, Appl. Math. Comput. 141 (2003), no. 1, 77–85. Advanced special functions and related topics in differential equations (Melfi, 2001). MR 1984229, DOI 10.1016/S0096-3003(02)00322-3
Vo Anh, Nikolai Leonenko, Andriy Olenko, and Volodymyr Vaskovych, On rate of convergence in non-central limit theorems, Bernoulli 25 (2019), no. 4A, 2920–2948. MR 4003569, DOI 10.3150/18-BEJ1075
V. V. Anh, A. Olenko, and Y. G. Wang, Fractional stochastic partial differential equation for random tangent fields on the sphere, Theory Probab. Math. Statist. 104 (2021), 3–22. MR 4421350, DOI 10.1090/tpms
Luisa Beghin, Viktoria P. Knopova, Nikolai N. Leonenko, and Enzo Orsingher, Gaussian limiting behavior of the rescaled solution to the linear Korteweg-de Vries equation with random initial conditions, J. Statist. Phys. 99 (2000), no. 3-4, 769–781. MR 1766908, DOI 10.1023/A:1018687327580
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871, DOI 10.1017/CBO9780511721434
Philip Broadbridge, Illia Donhauzer, and Andriy Olenko, Stochastic diffusion within expanding space-time, Z. Angew. Math. Phys. 75 (2024), no. 2, Paper No. 42, 22. MR 4709563, DOI 10.1007/s00033-024-02191-1
Philip Broadbridge, Alexander D. Kolesnik, Nikolai Leonenko, Andriy Olenko, and Dareen Omari, Spherically restricted random hyperbolic diffusion, Entropy 22 (2020), no. 2, Paper No. 217, 31. MR 4144958, DOI 10.3390/e22020217
Phil Broadbridge, Alexander D. Kolesnik, Nikolai Leonenko, and Andriy Olenko, Random spherical hyperbolic diffusion, J. Stat. Phys. 177 (2019), no. 5, 889–916. MR 4031900, DOI 10.1007/s10955-019-02395-0
Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, and Josef G. Steinebach, Pseudo-regularly varying functions and generalized renewal processes, Probability Theory and Stochastic Modelling, vol. 91, Springer, Cham, 2018. MR 3839312, DOI 10.1007/978-3-319-99537-3
M. Caputo, Linear model of dissipation whose q is almost frequency independent, II, Geophysical Journal of the Royal Astronomical Society 13 (1967), no. 5, 529–539.
Roger Gay and C. C. Heyde, On a class of random field models which allows long range dependence, Biometrika 77 (1990), no. 2, 401–403. MR 1064814, DOI 10.1093/biomet/77.2.401
Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, and Sergei V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. MR 3244285, DOI 10.1007/978-3-662-43930-2
Nikolai Leonenko, Limit theorems for random fields with singular spectrum, Mathematics and its Applications, vol. 465, Kluwer Academic Publishers, Dordrecht, 1999. MR 1687092, DOI 10.1007/978-94-011-4607-4
Nikolai Leonenko and Andriy Olenko, Tauberian and Abelian theorems for long-range dependent random fields, Methodol. Comput. Appl. Probab. 15 (2013), no. 4, 715–742. MR 3117624, DOI 10.1007/s11009-012-9276-9
Nikolai Leonenko and Andriy Olenko, Sojourn measures of Student and Fisher-Snedecor random fields, Bernoulli 20 (2014), no. 3, 1454–1483. MR 3217450, DOI 10.3150/13-BEJ529
N. Leonenko, A. Olenko, and J. Vaz, On fractional spherically restricted hyperbolic diffusion random field, Commun. Nonlinear Sci. Numer. Simul. 131 (2024), Paper No. 107866, 15. MR 4693178, DOI 10.1016/j.cnsns.2024.107866
N. N. Leonenko and W. A. Woyczynski, Exact parabolic asymptotics for singular n-D Burgers’ random fields: Gaussian approximation, Stochastic Process. Appl. 76 (1998), no. 2, 141–165. MR 1642664, DOI 10.1016/S0304-4149(98)00031-3
Nikolai N. Leonenko and Wojbor A. Woyczynski, Scaling limits of solutions of the heat equation for singular non-Gaussian data, J. Statist. Phys. 91 (1998), no. 1-2, 423–438. MR 1632518, DOI 10.1023/A:1023060625577
Gi-Ren Liu and Narn-Rueih Shieh, Scaling limits for some P.D.E. systems with random initial conditions, Stoch. Anal. Appl. 28 (2010), no. 3, 505–522. MR 2739572, DOI 10.1080/07362991003704969
Gi-Ren Liu and Narn-Rueih Shieh, Homogenization of fractional kinetic equations with random initial data, Electron. J. Probab. 16 (2011), no. 32, 962–980. MR 2801457, DOI 10.1214/EJP.v16-896
Gi-Ren Liu and Narn-Rueih Shieh, Multi-scaling limits for relativistic diffusion equations with random initial data, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3423–3446. MR 3314812, DOI 10.1090/S0002-9947-2014-06498-2
G-R. Liu and N-R. Shieh, Multi-scaling limits for time-fractional relativistic diffusion equations with random initial data, Theory Probab. Math. Statist. 95 (2018), 109–130.
Francesco Mainardi, On some properties of the Mittag-Leffler function E
α(-t)
α, completely monotone for t>0 with 0<α<1, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 7, 2267–2278. MR 3253257, DOI 10.3934/dcdsb.2014.19.2267
Keith B. Oldham and Jerome Spanier, The fractional calculus, Mathematics in Science and Engineering, Vol. 111, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Theory and applications of differentiation and integration to arbitrary order; With an annotated chronological bibliography by Bertram Ross. MR 361633
A. Ya. Olenko, A Tauberian theorem for fields with the OR spectrum. I, Teor. Ĭmovīr. Mat. Stat. 73 (2005), 120–133 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 73 (2006), 135–149. MR 2213848, DOI 10.1090/S0094-9000-07-00688-6
A. Olenko, Tauberian theorem for fields with an OR spectrum. II, Theory Probab. Math. Statist. 74 (2007), 93–111.
Andriy Olenko, Limit theorems for weighted functionals of cyclical long-range dependent random fields, Stoch. Anal. Appl. 31 (2013), no. 2, 199–213. MR 3021486, DOI 10.1080/07362994.2013.741410
Igor Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR 1658022
Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 453936, DOI 10.1007/BFb0079658
Thomas Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab. 19 (2014), no. 16, 25. MR 3164769, DOI 10.1214/EJP.v19-3058
