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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2508.16283 |
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| _version_ | 1866915456999227392 |
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| author | Wang, Qingsong Dorogovtsev, A. A. Hlyniana, K. V. Salhi, Naoufel |
| author_facet | Wang, Qingsong Dorogovtsev, A. A. Hlyniana, K. V. Salhi, Naoufel |
| contents | In this paper, we investigate some geometric properties of non-smooth random curves within a stochastic flow. We consider a polygonal line $Γ(\vec{u}_{1},\cdots,\vec{u}_{n})$, which connects the points \(\vec{u}_{1},\cdots,\vec{u}_{n}\in{\mathbb{R}^{d}}\) and is inscribed in a Brownian trajectory. Subsequently, we estimate the probability that a polygonal line is almost inscribed in a Brownian trajectory. Next, we turn to the study of the self-intersection local time of Brownian motion and demonstrate the asymptotic result of its conditional expectation as the size of the polygonal line increases. Finally, taking such a Brownian trajectory as the initial curve, we let it evolve according to the solution of the equation with interaction. Then, we prove that its visitation density exhibits an intermittency phenomenon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Geometry of Gaussian Random Curves Wang, Qingsong Dorogovtsev, A. A. Hlyniana, K. V. Salhi, Naoufel Probability In this paper, we investigate some geometric properties of non-smooth random curves within a stochastic flow. We consider a polygonal line $Γ(\vec{u}_{1},\cdots,\vec{u}_{n})$, which connects the points \(\vec{u}_{1},\cdots,\vec{u}_{n}\in{\mathbb{R}^{d}}\) and is inscribed in a Brownian trajectory. Subsequently, we estimate the probability that a polygonal line is almost inscribed in a Brownian trajectory. Next, we turn to the study of the self-intersection local time of Brownian motion and demonstrate the asymptotic result of its conditional expectation as the size of the polygonal line increases. Finally, taking such a Brownian trajectory as the initial curve, we let it evolve according to the solution of the equation with interaction. Then, we prove that its visitation density exhibits an intermittency phenomenon. |
| title | The Geometry of Gaussian Random Curves |
| topic | Probability |
| url | https://arxiv.org/abs/2508.16283 |