First-passage time of a Brownian motion: two unexpected journeys
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914930397020160 |
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| author | Mazzolo, Alain |
| author_facet | Mazzolo, Alain |
| contents | The distribution of the first-passage time (FPT)$T_a$ for a Brownian particle with drift $μ$ subject to hitting an absorber at a level $a>0$ is well-known and given by its density $γ(t) = \frac{a}{\sqrt{2 πt^3} } e^{-\frac{(a-μt)^2}{2 t}}, t>0$, which is normalized only if $μ\geq 0$. This article demonstrates the existence of two additional diffusion process categories (one with one parameter and the other with two) that have the same first passage-time distributions when $μ<0$. For both, we identify the transition densities and thoroughly investigate the processes. A substantial implication is that the first-passage time distribution does not indicate whether the process originates from a drifted Brownian motion or from one of the new processes presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17656 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | First-passage time of a Brownian motion: two unexpected journeys Mazzolo, Alain Statistical Mechanics Probability The distribution of the first-passage time (FPT)$T_a$ for a Brownian particle with drift $μ$ subject to hitting an absorber at a level $a>0$ is well-known and given by its density $γ(t) = \frac{a}{\sqrt{2 πt^3} } e^{-\frac{(a-μt)^2}{2 t}}, t>0$, which is normalized only if $μ\geq 0$. This article demonstrates the existence of two additional diffusion process categories (one with one parameter and the other with two) that have the same first passage-time distributions when $μ<0$. For both, we identify the transition densities and thoroughly investigate the processes. A substantial implication is that the first-passage time distribution does not indicate whether the process originates from a drifted Brownian motion or from one of the new processes presented. |
| title | First-passage time of a Brownian motion: two unexpected journeys |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2306.17656 |