Limit theorems for Lévy walks in d dimensions: rare and bulk fluctuations

摘要: We consider super-diffusive Levy walks in dimensions when the duration of a single step, i.e. ballistic motion performed by walker, is governed power-law tailed distribution infinite variance and finite mean. demonstrate that probability density function (PDF) coordinate random walker has two different scaling limits at large times. One limit describes bulk PDF. It d-dimensional generalization one-dimensional counterpart central theorem (CLT) for with dispersion. In contrast CLT this does not have universal shape. The PDF reflects anisotropy single-step statistics however time is. other limit, so-called 'infinite density', tail which determines second (dispersion) higher moments This repeats angular structure velocity one step. A typical realization walk consists anomalous diffusive (described anisotropic distribution) interspersed long flights density). are rare but due to them increases so much their contribution illustrate concept considering types walks, isotropic distributions velocities. Furthermore, we show otherwise arbitrary process can be reduced walk. briefly discuss consequences non-universality d > 1 dimensional fractional diffusion equation, particular non-uniqueness Laplacian.