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Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel

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Original languageEnglish
Pages (from-to)1006-1038
Number of pages33
JournalFractional Calculus and Applied Analysis
Issue number4
Publication statusPublished - 1 Aug 2015
Publication typeA1 Journal article-refereed


We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck- Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.

ASJC Scopus subject areas


  • anomalous diffusion, continuous time random walk (CTRW), Fokker- Planck-Smoluchowski equation, Mittag-Leffler functions, multi-scaling

Publication forum classification

Field of science, Statistics Finland