×

Centre-of-mass like superposition of Ornstein-Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion. (English) Zbl 1436.60041

Summary: We consider an ensemble of Ornstein-Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with time-dependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively. Moreover, we show that this variable is equivalent in distribution to a randomly-scaled Gaussian process, i.e., a process built by the product of a Gaussian process times a non-negative independent random variable. This last result establishes a connection with the so-called generalized grey Brownian motion and suggests application to model fractional anomalous diffusion in biological systems.

MSC:

60G22 Fractional processes, including fractional Brownian motion
65C30 Numerical solutions to stochastic differential and integral equations
91B70 Stochastic models in economics
60J60 Diffusion processes
34A08 Fractional ordinary differential equations
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] O.E. Barndorff-Nielsen, N.N. Leonenko, Spectral properties of superpositions of Ornstein-Uhlenbeck type processes. Methodol. Comput. Appl. Probab. 7 (2005), 335-352.; Barndorff-Nielsen, O. E.; Leonenko, N. N., Spectral properties of superpositions of Ornstein-Uhlenbeck type processes, Methodol. Comput. Appl. Probab., 7, 335-352 (2005) · Zbl 1089.60014
[2] F.E. Benth, B. Rüdiger, A. Süss, Ornstein-Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility. Stoch. Process. Appl. 128 (2018), 461-486.; Benth, F. E.; Rüdiger, B.; Süss, A., Ornstein-Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility, Stoch. Process. Appl., 128, 461-486 (2018) · Zbl 1380.60010
[3] J.P.N. Bishwal, Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: Continuous and discrete sampling. Fract. Calc. Appl. Anal. 14, No 3 (2011), 375-410; ; .; Bishwal, J. P.N., Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: Continuous and discrete sampling, Fract. Calc. Appl. Anal., 14, 3, 375-410 (2011) · Zbl 1273.62056
[4] M. Csörgő, Z.Y. Lin, On moduli of continuity for Gaussian and \(l^2\)-norm squared processes generated by Ornstein-Uhlenbeck processes. Canad. J. Math. 42 (1990), 141-158.; Csörgő, M.; Lin, Z. Y., On moduli of continuity for Gaussian and \(l^2\)-norm squared processes generated by Ornstein-Uhlenbeck processes, Canad. J. Math., 42, 141-158 (1990) · Zbl 0723.60038
[5] O. Garet, Asymptotic behaviour of Gaussian processes with integral representation. Stoch. Process. Appl. 89 (2000), 287-303.; Garet, O., Asymptotic behaviour of Gaussian processes with integral representation, Stoch. Process. Appl., 89, 287-303 (2000) · Zbl 1045.60034
[6] S. Gheorghiu, M.-O. Coppens, Heterogeneity explains features of “anomalous” thermodynamics and statistics. Proc. Natl. Acad. Sci. USA101 (2004), 15852-15856.; Gheorghiu, S.; Coppens, M.-O., Heterogeneity explains features of “anomalous” thermodynamics and statistics, Proc. Natl. Acad. Sci. USA, 101, 15852-15856 (2004)
[7] D. Grahovac, N.N. Leonenko, A. Sikorskii, I. Tešnjak, Intermittency of superpositions of Ornstein-Uhlenbeck type processes. J. Stat. Phys. 165 (2016), 390-408.; Grahovac, D.; Leonenko, N. N.; Sikorskii, A.; Tešnjak, I., Intermittency of superpositions of Ornstein-Uhlenbeck type processes, J. Stat. Phys., 165, 390-408 (2016) · Zbl 1356.60121
[8] M. Grothaus, F. Jahnert, F. Riemann, J.L. da Silva, Mittag-Leffler analysis I: Construction and characterization. J. Funct. Anal. 268 (2015), 1876-1903.; Grothaus, M.; Jahnert, F.; Riemann, F.; da Silva, J. L., Mittag-Leffler analysis I: Construction and characterization, J. Funct. Anal., 268, 1876-1903 (2015) · Zbl 1322.46026
[9] M. Grothaus, F. Jahnert, Mittag-Leffler analysis II: Application to the fractional heat equation. J. Funct. Anal. 270 (2016), 2732-2768.; Grothaus, M.; Jahnert, F., Mittag-Leffler analysis II: Application to the fractional heat equation, J. Funct. Anal., 270, 2732-2768 (2016) · Zbl 1360.46034
[10] F. Höfling, T. Franosch, Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys. 76 (2013), 046602.; Höfling, F.; Franosch, T., Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys., 76, 046602 (2013)
[11] J. Klafter, S.-C. Lim, R. Metzler, Fractional Dynamics: Recent Advances. World Scientific, Singapore (2011).; Klafter, J.; Lim, S.-C.; Metzler, R., Fractional Dynamics: Recent Advances (2011) · Zbl 1238.93005
[12] R. Klages, G. Radons, I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications. Wiley-VCH, Weinheim (2008).; Klages, R.; Radons, G.; Sokolov, I. M., Anomalous Transport: Foundations and Applications (2008)
[13] N. Leonenko, E. Taufer, Convergence of integrated superpositions of Ornstein-Uhlenbeck processes to fractional Brownian motion. Stochastics77, No 6 (2005), 477-499.; Leonenko, N.; Taufer, E., Convergence of integrated superpositions of Ornstein-Uhlenbeck processes to fractional Brownian motion, Stochastics, 77, 6, 477-499 (2005) · Zbl 1082.60014
[14] Z.Y. Lin, On large increments of infinite series of Ornstein-Uhlenbeck processes. Stoch. Process. Appl. 60 (1995), 161-169.; Lin, Z. Y., On large increments of infinite series of Ornstein-Uhlenbeck processes, Stoch. Process. Appl., 60, 161-169 (1995) · Zbl 0854.60037
[15] F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, No 2 (2001), 153-192.; Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4, 2, 153-192 (2001) · Zbl 1054.35156
[16] F. Mainardi, A. Mura, G. Pagnini, The M-Wright function in time-fractional diffusion processes: A tutorial survey. Int. J. Differ. Equations2010 (2010), 104505.; Mainardi, F.; Mura, A.; Pagnini, G.; The, M-Wright function in time-fractional diffusion processes: A tutorial survey, Int. J. Differ. Equations, 2010, 104505 (2010) · Zbl 1222.60060
[17] R. Mayor, S. Etienne-Manneville, The front and rear of collective cell migration. Nat. Rev. Mol. Cell Biol. 17 (2016), 97-109.; Mayor, R.; Etienne-Manneville, S., The front and rear of collective cell migration, Nat. Rev. Mol. Cell Biol., 17, 97-109 (2016)
[18] Y. Meroz, I.M. Sokolov, A toolbox for determining subdiffusive mechanisms. Phys. Rep. 573 (2015), 1-29.; Meroz, Y.; Sokolov, I. M., A toolbox for determining subdiffusive mechanisms, Phys. Rep., 573, 1-29 (2015)
[19] R. Metzler, J. Klafter, The restaurant at the end of the random walk: Recent developments in fractional dynamics descriptions of anomalous dynamical processes. J. Phys. A: Math. Theor. 37, No 31 (2004), R161-R208.; Metzler, R.; Klafter, J., The restaurant at the end of the random walk: Recent developments in fractional dynamics descriptions of anomalous dynamical processes, J. Phys. A: Math. Theor., 37, 31, R161-R208 (2004) · Zbl 1075.82018
[20] D. Molina-García, T. Minh Pham, P. Paradisi, C. Manzo, G. Pagnini, Fractional kinetics emerging from ergodicity breaking in random media. Phys. Rev. E94 (2016), 052147.; Molina-García, D.; Minh Pham, T.; Paradisi, P.; Manzo, C.; Pagnini, G., Fractional kinetics emerging from ergodicity breaking in random media, Phys. Rev. E, 94, 052147 (2016)
[21] A. Mura, Non-Markovian Stochastic Processes and Their Applications: From Anomalous Diffusion to Time Series Analysis. Lambert Academic Publishing (2011). Ph.D. Thesis, Physics Department, University of Bologna, 2008.; Mura, A., Non-Markovian Stochastic Processes and Their Applications: From Anomalous Diffusion to Time Series Analysis (2008)
[22] A. Mura, F. Mainardi, A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics. Integr. Transf. Spec. Funct. 20, No 3-4 (2009), 185-198.; Mura, A.; Mainardi, F., A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics, Integr. Transf. Spec. Funct., 20, 3-4, 185-198 (2009) · Zbl 1173.26005
[23] A. Mura, G. Pagnini, Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. J. Phys. A: Math. Theor. 41 (2008), 285003.; Mura, A.; Pagnini, G., Characterizations and simulations of a class of stochastic processes to model anomalous diffusion, J. Phys. A: Math. Theor., 41, 285003 (2008) · Zbl 1143.82028
[24] G. Pagnini, Erdélyi-Kober fractional diffusion. Fract. Calc. Appl. Anal. 15, No 1 (2012), 117-127; ; .; Pagnini, G., Erdélyi-Kober fractional diffusion, Fract. Calc. Appl. Anal., 15, 1, 117-127 (2012) · Zbl 1276.26021
[25] G. Pagnini, The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes. Fract. Calc. Appl. Anal. 16, No 2 (2013), 436-453; ; .; Pagnini, G.; The, M-Wright function as a generalization of the Gaussian density for fractional diffusion processes, Fract. Calc. Appl. Anal., 16, 2, 436-453 (2013) · Zbl 1312.33061
[26] G. Pagnini, Short note on the emergence of fractional kinetics. Physica A409 (2014), 29-34.; Pagnini, G., Short note on the emergence of fractional kinetics, Physica A, 409, 29-34 (2014) · Zbl 1395.82216
[27] G. Pagnini, P. Paradisi, A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 19, No 2 (2016), 408-440; ; .; Pagnini, G.; Paradisi, P., A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 19, 2, 408-440 (2016) · Zbl 1341.60073
[28] B.M. Regner, D. Vučinić, C. Domnisoru, T.M. Bartol, M.W. Hetzer, D.M. Tartakovsky, T.J. Sejnowski, Anomalous diffusion of single particles in cytoplasm. Biophys. J. 104 (2013), 1652-1660.; Regner, B. M.; Vučinić, D.; Domnisoru, C.; Bartol, T. M.; Hetzer, M. W.; Tartakovsky, D. M.; Sejnowski, T. J., Anomalous diffusion of single particles in cytoplasm, Biophys. J., 104, 1652-1660 (2013)
[29] V. Sposini, A.V. Chechkin, F. Seno, G. Pagnini, R. Metzler, Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion. New J. Phys. 20 (2018), Art. # 043044.; Sposini, V.; Chechkin, A. V.; Seno, F.; Pagnini, G.; Metzler, R., Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion, New J. Phys., 20 (2018)
[30] S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi, G. Castellani, G. Pagnini, Langevin equation in complex media and anomalous diffusion. J. R. Soc. Interface15 (2018), 20180282.; Vitali, S.; Sposini, V.; Sliusarenko, O.; Paradisi, P.; Castellani, G.; Pagnini, G., Langevin equation in complex media and anomalous diffusion, J. R. Soc. Interface, 15, 20180282 (2018) · Zbl 1436.60041
[31] C. Zeng, Y.-Q. Chen, Q. Yang, The fBm-driven Ornstein-Uhlenbeck process: Probability density function and anomalous diffusion. Fract. Calc. Appl. Anal. 15, No 3 (2012), 479-492; ; .; Zeng, C.; Chen, Y.-Q.; Yang, Q., The fBm-driven Ornstein-Uhlenbeck process: Probability density function and anomalous diffusion, Fract. Calc. Appl. Anal., 15, 3, 479-492 (2012) · Zbl 1274.60127
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.