The Geometry of Gaussian Random Curves

TL;DR

This paper explores the geometric properties of non-smooth random curves in stochastic flows, focusing on polygonal lines inscribed in Brownian trajectories, their self-intersections, and the evolution of visitation densities showing intermittency.

Contribution

It introduces new estimates for polygonal lines in Brownian motion, analyzes self-intersection local times, and studies the evolution of visitation densities with intermittency phenomena.

Findings

Probability estimates for polygonal lines inscribed in Brownian trajectories

Asymptotic behavior of self-intersection local times

Intermittency in visitation densities during curve evolution

Abstract

In this paper, we investigate some geometric properties of non-smooth random curves within a stochastic flow. We consider a polygonal line $\Gamma(\vec{u}_{1},\cdots,\vec{u}_{n})$, 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.