Randomly trapped random walks on \(\mathbb{Z}^d\). (English) Zbl 1322.60051
Summary: We give a complete classification of scaling limits of randomly trapped random walks and associated clock processes on \(\mathbb{Z}^d\), \(d \geq 2\). Namely, under the hypothesis that the discrete skeleton of the randomly trapped random walk has a slowly varying return probability, we show that the scaling limit of its clock process is either deterministic linearly growing or a stable subordinator. In the case when the discrete skeleton is a simple random walk on \(\mathbb{Z}^d\), this implies that the scaling limit of the randomly trapped random walk is either a Brownian motion or the fractional kinetics process, as conjectured in [G. Ben Arous et al., Ann. Probab. 43, No. 5, 2405–2457 (2015; Zbl 1329.60354)].
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 60F17 | Functional limit theorems; invariance principles |
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60J65 | Brownian motion |
Keywords:
random walk; trap model; clock process; scaling limit; fractional kinetics process; Brownian motionCitations:
Zbl 1329.60354References:
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