## Abstract

Hilfer [Physica A 329 (2003) 35] claims to give an example of a continuous time random walk (CTRW) model with long-tailed waiting time probability density that approaches a Gaussian behavior in the continuum limit. Rigorous limit theorems, derived previously, show however that in the limit of long-time such a CTRW converges to a non-Gaussian behavior. We discuss two types of continuum limits for the CTRW model: the fractional continuum limit and the one introduced by Hilfer. We show that the fractional limit yields the correct long-time behavior of the CTRW, while Hilfer's continuum limit does not. We discuss a general approach to find a continuum limit of the CTRW process.

Original language | English |
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Pages (from-to) | 231-236 |

Number of pages | 6 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 373 |

DOIs | |

State | Published - 1 Jan 2007 |

### Bibliographical note

Funding Information:EB thank the Israel Science Foundation and the center of complexity in Jerusalem, for financial support.

## Keywords

- Continuous time random walks
- Continuum limit
- Fractional diffusion equation