The geometry of Gaussian random curves

Qingsong Wang; A. A. Dorogovtsev; K. V. Hlyniana; Naoufel Salhi; Qingsong Wang; A. A. Dorogovtsev; K. V. Hlyniana; Naoufel Salhi

doi:10.3934/math.20251049

AIMS Mathematics

2025, Volume 10, Issue 10: 23613-23638. doi: 10.3934/math.20251049

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Research article

The geometry of Gaussian random curves

1.
School of Mathematics, Jilin University, Changchun, 130012, China
2.
Institute of Mathematics, National Academy of Sciences of Ukraine, Ukraine
3.
Research laboratory of Stochastic Analysis, University of Tunis El Manar, Tunisia

Received: 12 August 2025 Revised: 30 September 2025 Accepted: 09 October 2025 Published: 16 October 2025
MSC : 60G15, 60J45, 60J65

In this paper, we investigated some geometric properties of non-smooth random curves within a stochastic flow. We considered a polygonal line $ \Gamma(\vec{u}_{1}, \cdots, \vec{u}_{n}) $, which connected the points $\vec{u}_{1}, \cdots, \vec{u}_{n}\in{\mathbb{R}^{d}}$ and was inscribed in a Brownian trajectory. Subsequently, we estimated the probability that a polygonal line was almost inscribed in a Brownian trajectory. Next, we turned to the study of the self-intersection local time of Brownian motion and demonstrated the asymptotic result of its conditional expectation as the size of the polygonal line increased. 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 proved that its visitation density exhibited an intermittency phenomenon.
- random curve,
- Brownian motion,
- self-intersection local time,
- SDE with interaction,
- stochastic flow,
- intermittency phenomenon
Citation: Qingsong Wang, A. A. Dorogovtsev, K. V. Hlyniana, Naoufel Salhi. The geometry of Gaussian random curves[J]. AIMS Mathematics, 2025, 10(10): 23613-23638. doi: 10.3934/math.20251049

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Abstract

In this paper, we investigated some geometric properties of non-smooth random curves within a stochastic flow. We considered a polygonal line $ \Gamma(\vec{u}_{1}, \cdots, \vec{u}_{n}) $, which connected the points $\vec{u}_{1}, \cdots, \vec{u}_{n}\in{\mathbb{R}^{d}}$ and was inscribed in a Brownian trajectory. Subsequently, we estimated the probability that a polygonal line was almost inscribed in a Brownian trajectory. Next, we turned to the study of the self-intersection local time of Brownian motion and demonstrated the asymptotic result of its conditional expectation as the size of the polygonal line increased. 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 proved that its visitation density exhibited an intermittency phenomenon.

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