Recurrence of the random process governed with the fractional Laplacian and the Caputo time derivative
Affili, Elisa; Kemppainen, Jukka T. (2023-07-06)
Affili, E., & Kemppainen, J. T. (2023). Recurrence of the random process governed with the fractional Laplacian and the Caputo time derivative. Bruno Pini Mathematical Analysis Seminar, 14(1), 1–14. https://doi.org/10.6092/issn.2240-2829/17264
© 2023 Elisa Affili, Jukka T. Kemppainen. This work is licensed under a Creative Commons Attribution 3.0 Unported License.
https://creativecommons.org/licenses/by/3.0/
https://urn.fi/URN:NBN:fi-fe20231018140536
Tiivistelmä
Abstract
We are addressing a parabolic equation with fractional derivatives in time and space that governs the scaling limit of continuous-time random walks with anomalous diffusion. For these equations, the fundamental solution represents the probability density of finding a particle released at the origin at time 0 at a given position and time. Using some estimates of the asymptotic behaviour of the fundamental solution, we evaluate the probability of the process returning infinite times to the origin in a heuristic way. Our calculations suggest that the process is always recurrent.
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