Limit theorems for coupled continuous time random walks. (English) Zbl 1054.60052

The continuous time random walk is a jump process with independent identically distributed waiting times between the jumps. Mathematically this model process, for so-called anomalous diffusion, may be interpreted as a random walk subordinated to a suitable renewal process. Usually, CTRW is regarded to be uncoupled, meaning that the random walk is independent of the subordinating renewal process. The main focus of the paper are scaling limits (large time behavior) of the coupled process whose jump sizes and waiting times are not independent random variables. The pertinent infinite mean waiting time limit, in the coupled subordination case, allows to infer a simple formula for the limiting probability distribution. Links with pseudodifferential equations are conjectured. Explicit examples of coupled and uncoupled CTRWs are elaborated.


60G50 Sums of independent random variables; random walks
60F17 Functional limit theorems; invariance principles
60H30 Applications of stochastic analysis (to PDEs, etc.)
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
Full Text: DOI


[1] Baeumer, B. and Meerschaert, M. (2001). Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4 481–500. · Zbl 1057.35102
[2] Baeumer, B., Meerschaert, M. and Mortensen, J. (2003). Space–time fractional derivative operators. Available at http://unr.edu/homepage/mcubed/coupleop.pdf. · Zbl 1070.47043
[3] Barkai, E., Metzler, R. and Klafter, J. (2000). From continuous time random walks to the fractional Fokker–Planck equation. Phys. Rev. E 61 132–138.
[4] Berg, C., Reus Christensen, J. P. and Ressel, P. (1984). Harmonic Analysis on Semigroups . Springer, New York. · Zbl 0619.43001
[5] Bingham, N. H. (1971). Limit theorems for occupation times of Markov processes. Z. Wahrsch. Verw. Gebiete 17 1–22. · Zbl 0194.49503
[6] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2 , 2nd ed. Wiley, New York. · Zbl 0219.60003
[7] Jacob, N. (1996). Pseudo-Differential Operators and Markov Processes . Akademie Verlag, Berlin. · Zbl 0860.60002
[8] Jurek, Z. and Mason, J. D. (1993). Operator-Limit Distributions in Probability Theory . Wiley, New York. · Zbl 0850.60003
[9] Klafter, J., Blumen, A. and Shlesinger, M. F. (1987). Stochastic pathways to anomalous diffusion. Phys. Rev. A 35 3081–3085.
[10] Kotulski, M. (1995). Asymptotic distributions of the continuous time random walks: A probabilistic approach. J. Statist. Phys. 81 777–792. · Zbl 1107.60318
[11] Kotulski, M. (1995). Asymptotic behavior of generalized Lévy walks. Chaos—the interplay between stochastic and deterministic behaviour. Lecture Notes in Physics 457 471–477. Springer, Berlin. · Zbl 0835.60064
[12] Kotulski, M. and Weron, K. (1996). Random walk approach to relaxation in disordered systems. Athens Conference on Applied Probability and Time Series Analysis I . Lecture Notes in Statist. 114 379–388. Springer, New York. · Zbl 0858.60063
[13] Kozubowski, T. J., Meerschaert, M. M. and Scheffler, H. P. (2004). The operator \(\nu\)-stable laws. Publ. Math. Debrecen. · Zbl 1059.60019
[14] Meerschaert, M., Benson, D. and Baeumer, B. (1999). Multidimensional advection and fractional dispersion. Phys. Rev. E 59 5026–5028.
[15] Meerschaert, M., Benson, D. and Baeumer, B. (2001). Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63 1112–1117.
[16] Meerschaert, M. M., Benson, D. A., Scheffler, H. P. and Baeumer, B. (2002). Stochastic solution of space–time fractional diffusion equations. Phys. Rev. E 65 1103–1106. · Zbl 1244.60080
[17] Meerschaert, M. M., Benson, D. A., Scheffler, H. P. and Becker-Kern, P. (2002). Governing equations and solutions of anomalous random walk limits. Phys. Rev. E 66 102–105.
[18] Meerschaert, M. M. and Scheffler, H. P. (2001). Limit Distributions for Sums of Independent Random Vectors : Heavy Tails in Theory and Practice . Wiley, New York. · Zbl 0990.60003
[19] Meerschaert, M. M. and Scheffler, H. P. (2001). Limit theorems for continuous time random walks. Available at http://unr.edu/homepage/mcubed/LimitCTRW.pdf. · Zbl 0990.60003
[20] Metzler, R. and Klafter, J. (2000). The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 1–77. · Zbl 0984.82032
[21] Montroll, E. W. and Weiss, G. H. (1965). Random walks on lattices. II. J. Math. Phys. 6 167–181. · Zbl 1342.60067
[22] Ressel, P. (1991). Semigroups in probability theory. In Probability Measures on Groups X (H. Heyer, ed.) 337–363. Plenum, New York. · Zbl 0823.60011
[23] Saichev, A. I. and Zaslavsky, G. M. (1997). Fractional kinetic equations: Solutions and applications. Chaos 7 753–764. · Zbl 0933.37029
[24] Samko, S., Kilbas, A. and Marichev, O. (1993). Fractional Integrals and Derivatives : Theory and Applications . Gordon and Breach, London. · Zbl 0818.26003
[25] Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions . Cambridge Univ. Press. · Zbl 0973.60001
[26] Scher, H. and Lax, M. (1973). Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B 7 4491–4502.
[27] Seneta, E. (1976). Regularly Varying Functions . Lecture Notes in Math. 508 . Springer, Berlin. · Zbl 0324.26002
[28] Shlesinger, M., Klafter, J. and Wong, Y. M. (1982). Random walks with infinite spatial and temporal moments. J. Statist. Phys. 27 499–512. · Zbl 0521.60080
[29] Stone, C. (1963). Weak convergence of stochastic processes defined on semi-infinite time intervals. Proc. Amer. Math. Soc. 14 694–696. · Zbl 0116.35602
[30] Uchaikin, V. V. and Zolotarev, V. M. (1999). Chance and Stability. Stable Distributions and Their Applications. VSP, Utrecht. · Zbl 0944.60006
[31] Wheeden, R. L. and Zygmund, A. (1977). Measure and Integral . Dekker, New York. · Zbl 0362.26004
[32] Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res. 5 67–85. · Zbl 0428.60010
[33] Whitt, W. (2002). Stochastic Process Limits : An Introduction to Stochastic-Process Limits and Their Application to Queues . Springer, New York. · Zbl 0993.60001
[34] Zaslavsky, G. (1994). Fractional kinetic equation for Hamiltonian chaos. Chaotic advection, tracer dynamics and turbulent dispersion. Phys. D 76 110–122. · Zbl 1194.37163
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.