The geometry of Gaussian random curves

In this paper, we investigated some geometric properties of non-smooth random curves within a stochastic flow. We considered a polygonal line $\mathrm{\Gamma }({\overrightarrow{u}}_1,\cdots ,{\overrightarrow{u}}_n)$, which connected the points ${\overrightarrow{u}}_1,\cdots ,{\overrightarrow{u}}_n\in {\mathbb{R}}^d$ 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.