Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)
Ehrenfest‒Brillouin-type correlated continuous time random walk and fractional Jacobi diffusion
N. N. Leonenko, I. Papić, A. Sikorskii, N. Šuvak
Abstract: Continuous time random walks (CTRWs) have random waiting times between particle jumps. Based on Ehrenfest–Brillouin-type model motivated by economics, we define the correlated CTRW that converge to the fractional Jacobi diffusion Y(E(t)), t≥0, defined as a time change of Jacobi diffusion process Y (t) to the inverse E(t) of the standard stable subordinator. In the CTRW considered in this paper, the jumps are correlated so that in the limit the outer process Y(t) is not a Lévy process but a diffusion process with non-independent increments. The waiting times between jumps are selected from the domain of attraction of a stable law, so that the correlated CTRWs with these waiting times converge to Y(E(t)).
Keywords: Correlated continuous time random walk, Ehrenfest–Brillouin Markov chain, fractional diffusion, Jacobi diffusion, Pearson diffusion.
1. A. V. Chechkin, M. Hofmann, I. M. Sokolov, Continuous-time random walk with correlated waiting times, Physical Review E, 80 (2009), no. 3, 031112.
2. S. N. Ethier, T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, 2009.
3. J. L. Forman, M. Sørensen, The Pearson diffusions: A class of statistically tractable diffusion processes, Scand. J. Statist., 35 (2008), no. 3, 438–465.
4. U. Garibaldi, E. Scalas, Finitary probabilistic methods in econophysics, Cambridge University Press, 2010.
5. G. Germano, M. Politi, E. Scalas, R. L. Schilling, Stochastic calculus for uncoupled continuous-time random walks, Phys. Rev. E, 79 (2009), 066102.
6. O. Kallenberg, Foundations of modern probability, 2nd ed., Springer Series in Statistics. Probability and Its Applications, Springer, 2002.
7. S. Karlin, H. Taylor, A first course in stochastic processes, Academic Press, 1981.
8. S. Karlin, H. Taylor, A second course in stochastic processes, Academic Press, 1981.
9. J. Klafter, R. Silbey, Derivation of the continuous-time random-walk equation, Physical Review Letters, 44 (1980), no. 2, 55–58.
10. V. N. Kolokoltsov, Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics, Theory of Probability & Its Applications, 53 (2009), no. 4, 594–609.
11. V. N. Kolokoltsov, M. A. Veretennikova, A fractional hamilton Jacobi Bellman equation for scaled limits of controlled continuous time random walks, Communications in Applied and Industrial Mathematics, 6 (2014), no. 1, e–484.
12. N. Leonenko, I. Papić, A. Sikorskii, N. Šuvak, Correlated continuous time random walks and fractional Pearson diffusions, Bernoulli, 24 (2018), no. 4B, 3603–3627.
13. N. N. Leonenko, M. M. Meerschaert, A. Sikorskii, Fractional Pearson diffusions, Journal of Mathematical Analysis and Applications, 403 (2013), no. 2, 532–546.
14. L. Lv, F.-Y. Ren, J. Wang, J. Xiao, Correlated continuous time random walk with time averaged waiting time, Physica A: Statistical Mechanics and its Applications, 422 (2015), 101–106.
15. M. M. Meerschaert, E. Nane, Y. Xiao, Correlated continuous time random walks, Statistics & Probability Letters, 79 (2009), no. 9, 1194–1202.
16. M. M. Meerschaert, E. Scalas, Coupled continuous time random walks in finance, Physica A: Statistical Mechanics and its Applications, 370 (2006), no. 1, 114–118.
17. M. M. Meerschaert, H.-P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, Journal of Applied Probability (2004), 623–638.
18. M. M. Meerschaert, H.-P. Scheffler, Triangular array limits for continuous time random walks, Stochastic Processes and their Applications, 118 (2008), no. 9, 1606–1633.
19. M. M. Meerschaert, A. Sikorskii, Stochastic models for fractional calculus, De Gruyter, 2011.
20. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics reports, 339 (2000), no. 1, 1–77.
21. E. W. Montroll, G. H. Weiss, Random walks on lattices. II, Journal of Mathematical Physics, 6 (1965), no. 2, 167–181.
22. H. Scher, M. Lax, Stochastic transport in a disordered solid. I. theory, Physical Review B, 7 (1973), no. 10, 4491–4502.
23. H. Scher, E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B, 12 (1975), 2455–2477.
24. J. H. Schulz, A. V. Chechkin, R. Metzler, Correlated continuous time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics, Journal of Physics A: Mathematical and Theoretical, 46 (2013), no. 47, 475001.
25. P. Straka, B. Henry, Lagging and leading coupled continuous time random walks, renewal times and their joint limits, Stochastic Processes and their Applications, 121 (2011), no. 2, 324–336.
26. V. Tejedor, R. Metzler, Anomalous diffusion in correlated continuous time random walks, Journal of Physics A: Mathematical and Theoretical, 43 (2010), no. 8, 082002.