@article{Wang2025, 
author = {Qingsong Wang and A. A. Dorogovtsev and K. V. Hlyniana and Naoufel Salhi},
title = {The geometry of Gaussian random curves},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {10},
pages = {23613-23638},
keywords = {random curve, Brownian motion, self-intersection local time, SDE with interaction, stochastic flow, intermittency phenomenon},
url = {https://www.sciopen.com/article/10.3934/math.20251049},
doi = {10.3934/math.20251049},
abstract = {In this paper, we investigated some geometric properties of non-smooth random curves within a stochastic flow. We considered a polygonal line    Γ  (                    u        →                    1        ,  ⋯  ,                    u        →                    n        ), which connected the points                      u        →                    1        ,  ⋯  ,                    u        →                    n        ∈                    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.}
}