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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.00895 |
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| _version_ | 1866917662383144960 |
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| author | Sarmiento, Yonathan Das, Debraj Roldán, Édgar |
| author_facet | Sarmiento, Yonathan Das, Debraj Roldán, Édgar |
| contents | Using martingale theory, we compute, in very few lines, exact analytical expressions for various first-exit-time statistics associated with one-dimensional biased diffusion. Examples include the distribution for the first-exit time from an interval, moments for the first-exit site, and functionals of the position, which involve memory and time integration. As a key example, we compute analytically the mean area swept by a biased diffusion until it escapes an interval that may be asymmetric and have arbitrary length. The mean area allows us to derive the hitherto unexplored cross-correlation function between the first-exit time and the first-exit site, which vanishes only for exit problems from symmetric intervals. As a colophon, we explore connections of our results with gambling, showing that betting on the time-integrated value of a losing game it is possible to design a strategy that leads to a net average win. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00895 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities Sarmiento, Yonathan Das, Debraj Roldán, Édgar Statistical Mechanics Using martingale theory, we compute, in very few lines, exact analytical expressions for various first-exit-time statistics associated with one-dimensional biased diffusion. Examples include the distribution for the first-exit time from an interval, moments for the first-exit site, and functionals of the position, which involve memory and time integration. As a key example, we compute analytically the mean area swept by a biased diffusion until it escapes an interval that may be asymmetric and have arbitrary length. The mean area allows us to derive the hitherto unexplored cross-correlation function between the first-exit time and the first-exit site, which vanishes only for exit problems from symmetric intervals. As a colophon, we explore connections of our results with gambling, showing that betting on the time-integrated value of a losing game it is possible to design a strategy that leads to a net average win. |
| title | On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2401.00895 |