Limit theorem for continuous-time random walks with two time scales

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. A physically realistic rescaling uses two different time scales for the mean waiting time and the deviation from the mean. This paper derives the scaling limits for such processes. These limit processes are governed by fractional partial differential equations that may be useful in physics. A transfer theorem for weak convergence of finite-dimensional distributions of stochastic processes is also obtained.

[1]  Sheldon M. Ross,et al.  Introduction to probability models , 1975 .

[2]  George M. Zaslavsky,et al.  Fractional kinetic equation for Hamiltonian chaos , 1994 .

[3]  M. Donsker,et al.  On the distribution of the supremum functional for processes with stationary independent increments , 1957 .

[4]  M. Caputo.  Linear models of dissipation whose Q is almost frequency independent , 1966 .

[5]  Charles M. Grinstead,et al.  Introduction to probability , 1986, Statistics for the Behavioural Sciences.

[6]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[7]  Sheldon M. Ross,et al.  Introduction to probability models , 1975 .

[8]  Hans-Peter Scheffler,et al.  Stochastic solution of space-time fractional diffusion equations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[10]  J. Mason,et al.  Operator-limit distributions in probability theory , 1993 .

[11]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[12]  B. Gnedenko,et al.  Limit Distributions for Sums of Independent Random Variables , 1955 .

[13]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[14]  Raisa E. Feldman,et al.  Limit Distributions for Sums of Independent Random Vectors , 2002 .

[15]  Mark M. Meerschaert,et al.  STOCHASTIC SOLUTIONS FOR FRACTIONAL CAUCHY PROBLEMS , 2003 .

[16]  D. Benson,et al.  Operator Lévy motion and multiscaling anomalous diffusion. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Sean Andrew McKenna,et al.  On the late‐time behavior of tracer test breakthrough curves , 2000 .

[18]  David A. Benson,et al.  Advection and dispersion in time and space , 2002 .

[19]  D. Benson,et al.  Multidimensional advection and fractional dispersion. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Ward Whitt,et al.  Some Useful Functions for Functional Limit Theorems , 1980, Math. Oper. Res..

[21]  Sheldon M. Ross,et al.  Introduction to Probability Models (4th ed.). , 1990 .