First-Passage Times for the Space-Fractional Spectral Fokker-Planck Equation
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914465509801984 |
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| author | Angstmann, Christopher N. Han, Daniel S. Henry, Bruce I. Huang, Boris Z. |
| author_facet | Angstmann, Christopher N. Han, Daniel S. Henry, Bruce I. Huang, Boris Z. |
| contents | We extend the random walk framework to include compounded steps, providing first-passage time (FPT) properties for a new class of superdiffusive processes, which are governed by the space-fractional spectral Fokker-Planck equation. This first-passage process leads to novel FPT properties, different from Lévy flights, that account for space dependent forces and hitting boundaries throughout the path of a jump. The FPT distribution can be derived for different types of barriers and potentials, for which we also provide specific examples. For the one-sided absorbing boundary with no potential on the semi-infinite line, we find that the FPT density scales asymptotically as $t^{-1/(2α)-1}$ for large times, where the parameter $α\in (0,1]$ relates to the power-law behavior for the distribution of the number of compounded steps. This is in agreement with the method of images but different to the Sparre-Andersen scaling $t^{-3/2}$ for corresponding Lévy flights of order $2α$. In this case, there exists an optimal space-fractional exponent $α$ to minimize the mean FPT. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First-Passage Times for the Space-Fractional Spectral Fokker-Planck Equation Angstmann, Christopher N. Han, Daniel S. Henry, Bruce I. Huang, Boris Z. Statistical Mechanics We extend the random walk framework to include compounded steps, providing first-passage time (FPT) properties for a new class of superdiffusive processes, which are governed by the space-fractional spectral Fokker-Planck equation. This first-passage process leads to novel FPT properties, different from Lévy flights, that account for space dependent forces and hitting boundaries throughout the path of a jump. The FPT distribution can be derived for different types of barriers and potentials, for which we also provide specific examples. For the one-sided absorbing boundary with no potential on the semi-infinite line, we find that the FPT density scales asymptotically as $t^{-1/(2α)-1}$ for large times, where the parameter $α\in (0,1]$ relates to the power-law behavior for the distribution of the number of compounded steps. This is in agreement with the method of images but different to the Sparre-Andersen scaling $t^{-3/2}$ for corresponding Lévy flights of order $2α$. In this case, there exists an optimal space-fractional exponent $α$ to minimize the mean FPT. |
| title | First-Passage Times for the Space-Fractional Spectral Fokker-Planck Equation |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2511.18256 |