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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.18398 |
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| _version_ | 1866913897121841152 |
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| author | Tian, Lulu Chen, Hanshuang Li, Guofeng |
| author_facet | Tian, Lulu Chen, Hanshuang Li, Guofeng |
| contents | We consider high-order stochastic processes $x(t)$ described by the Langevin equation $\frac{{{d^m}x\left( t \right)}}{d{t^m}}= \sqrt{2D} ξ(t)$, where $ξ(t)$ is a delta-correlated Gaussian noise with zero mean, and $D$ is the strength of noise. We focus on the short-time statistics of the first-passage functionals $A=\int_{0}^{T} \left[ x(t)\right] ^n dt$ along the trajectories starting from $x(0)=L$ and terminating whenever passing through the origin for the first-time at $t=T$. Using the optimal fluctuation method, we analytically obtain the most likely realizations of the first-passage processes for a given constraint $A$ with $n=0$ and 1, corresponding to the first-passage time itself and the area swept by the first-passage trajectory, respectively. The tail of the distribution of $A$ shows an essential singularity at $A \to 0$, $P_{m,n}(A |L) \sim \exp\left(-\frac{α_{m,n}L^{2mn-n+2}}{D A^{2m-1}} \right)$, where the explicit expressions for the exponents $α_{m,0}$ and $α_{m,1}$ for arbitrary $m$ are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18398 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Short-time large deviations of first-passage functionals for high-order stochastic processes Tian, Lulu Chen, Hanshuang Li, Guofeng Statistical Mechanics We consider high-order stochastic processes $x(t)$ described by the Langevin equation $\frac{{{d^m}x\left( t \right)}}{d{t^m}}= \sqrt{2D} ξ(t)$, where $ξ(t)$ is a delta-correlated Gaussian noise with zero mean, and $D$ is the strength of noise. We focus on the short-time statistics of the first-passage functionals $A=\int_{0}^{T} \left[ x(t)\right] ^n dt$ along the trajectories starting from $x(0)=L$ and terminating whenever passing through the origin for the first-time at $t=T$. Using the optimal fluctuation method, we analytically obtain the most likely realizations of the first-passage processes for a given constraint $A$ with $n=0$ and 1, corresponding to the first-passage time itself and the area swept by the first-passage trajectory, respectively. The tail of the distribution of $A$ shows an essential singularity at $A \to 0$, $P_{m,n}(A |L) \sim \exp\left(-\frac{α_{m,n}L^{2mn-n+2}}{D A^{2m-1}} \right)$, where the explicit expressions for the exponents $α_{m,0}$ and $α_{m,1}$ for arbitrary $m$ are obtained. |
| title | Short-time large deviations of first-passage functionals for high-order stochastic processes |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2409.18398 |