Nonlocal fractional equations from random walks

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2023-12
Authors
Kunz, Christopher
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Stinga, Pablo
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Herzog, David
Wu, Ruoyu
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Abstract
If a particle is undergoing a continuous random walk, it is of interest to model the probability of observing said particle at some position and observing it at some time. Under "normal'' circumstances, this probability function satisfies the heat equation. However, there are natural phenomena where particles get stuck in one spot before moving again or make jumps of arbitrary length. Such behaviors are examples of anomalous diffusion, and we are interested in modeling this same probability under these scenarios and how it impacts the heat equation. The fractional left derivative and fractional Laplacian are developed and utilized in our formulations. Finally, we compare the kernels that are obtained from experimental observations with kernels that result from the computations of the discrete left fractional derivative and discrete fractional Laplacian. We show that the difference between the restriction of the original fractional derivative to the mesh and the corresponding discrete derivative can be made arbitrarily small. This proves that the discrete fractional derivative converges to its corresponding continuous fractional derivative, and it allows us to compute the kernels without relying on experiments. Extra care is taken in preserving coefficients in computations to show the importance of the gamma function in these models.
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Attribution 3.0 United States, 2023