Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Wang, Qingsong, Dorogovtsev, A. A., Hlyniana, K. V., Salhi, Naoufel
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7077674425
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