Should I stay or should I go? Zero-size jumps in random walks for Lévy flights

Abstract

We study Markovian continuous-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Lévy-like motion for wild animals.

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• Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators