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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.

MSC:

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
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