Recurrence of the random process governed with the fractional Laplacian and the Caputo time derivative
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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Bologna
2023-07-01
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| Series: | Bruno Pini Mathematical Analysis Seminar |
| Subjects: | |
| Online Access: | https://mathematicalanalysis.unibo.it/article/view/17264 |
| Summary: | 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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| ISSN: | 2240-2829 |