First-passage properties of the jump process with a drift. Two exactly solvable cases
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913965517307904 |
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| author | Burenev, Ivan N. Majumdar, Satya N. |
| author_facet | Burenev, Ivan N. Majumdar, Satya N. |
| contents | We investigate the first-passage properties of a jump process with a constant drift, focusing on two key observables: the first-passage time $τ$ and the number of jumps $n$ before the first-passage event. By mapping the problem onto an effective discrete-time random walk, we derive an exact expression for the Laplace transform of the joint distribution of $τ$ and $n$ using the generalized Pollaczek-Spitzer formula. This result is then used to analyze the first-passage properties for two exactly solvable cases: (i) both the inter-jump intervals and jump amplitudes are exponentially distributed, and (ii) the inter-jump intervals are exponentially distributed while all jumps have the same fixed amplitude. We show the existence of two distinct regimes governed by the strength of the drift: (i) a survival regime, where the process remains positive indefinitely with finite probability; (ii) an absorption regime, where the first-passage eventually occurs; and (iii) a critical point at the boundary between these two phases. We characterize the asymptotic behavior of survival probabilities in each regime: they decay exponentially to a constant in the survival regime, vanish exponentially fast in the absorption regime, and exhibit power-law decay at the critical point. Furthermore, in the absorption regime, we derive large deviation forms for the marginal distributions of $τ$ and n. The analytical predictions are validated through extensive numerical simulations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_05409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First-passage properties of the jump process with a drift. Two exactly solvable cases Burenev, Ivan N. Majumdar, Satya N. Statistical Mechanics Mathematical Physics Probability We investigate the first-passage properties of a jump process with a constant drift, focusing on two key observables: the first-passage time $τ$ and the number of jumps $n$ before the first-passage event. By mapping the problem onto an effective discrete-time random walk, we derive an exact expression for the Laplace transform of the joint distribution of $τ$ and $n$ using the generalized Pollaczek-Spitzer formula. This result is then used to analyze the first-passage properties for two exactly solvable cases: (i) both the inter-jump intervals and jump amplitudes are exponentially distributed, and (ii) the inter-jump intervals are exponentially distributed while all jumps have the same fixed amplitude. We show the existence of two distinct regimes governed by the strength of the drift: (i) a survival regime, where the process remains positive indefinitely with finite probability; (ii) an absorption regime, where the first-passage eventually occurs; and (iii) a critical point at the boundary between these two phases. We characterize the asymptotic behavior of survival probabilities in each regime: they decay exponentially to a constant in the survival regime, vanish exponentially fast in the absorption regime, and exhibit power-law decay at the critical point. Furthermore, in the absorption regime, we derive large deviation forms for the marginal distributions of $τ$ and n. The analytical predictions are validated through extensive numerical simulations. |
| title | First-passage properties of the jump process with a drift. Two exactly solvable cases |
| topic | Statistical Mechanics Mathematical Physics Probability |
| url | https://arxiv.org/abs/2504.05409 |