Discrete-space time-fractional processes. (English) Zbl 1312.60040
Summary: A time-fractional diffusion process defined in a discrete probability setting is studied. Working in continuous time, the infinitesimal generators of random processes are discretized and the diffusion equation generalized by allowing the time derivative to be fractional, i.e. of non-integer order. The properties of the resulting distributions are studied in terms of the Mittag-Leffler function. We discuss the computation of these distribution functions by deriving new global rational approximations for the Mittag-Leffler function that account for both its initial Taylor series and asymptotic power-law tail behaviours. Furthermore, we derive integral representations for both the continuous and the discrete time-fractional distributions and use these to prove a convergence theorem.
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 33E12 | Mittag-Leffler functions and generalizations |
Keywords:
fractional calculus; time-fractional process; discrete-space process; Mittag-Leffler functionReferences:
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