# A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

## TL;DR

This paper investigates the properties of Brownian motion conditioned on vanishing iterated integrals, revealing a polynomial decomposition, a semicircle law for fluctuations, and connections to Legendre polynomials and Catalan triangles.

## Contribution

It provides explicit representations of conditioned Brownian motion, proves convergence to zero process, and establishes a semicircle law for fluctuation variances, introducing new polynomial and probabilistic insights.

## Key findings

- Processes converge weakly to zero as N→∞
- Fluctuations scaled by √N converge to Gaussian variables
- Variances of fluctuations follow a scaled semicircle law

## Abstract

We consider a standard one-dimensional Brownian motion on the time interval $[0,1]$ conditioned to have vanishing iterated time integrals up to order $N$. We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as $N\to\infty$ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by $\sqrt{N}$ and show that they converge in finite dimensional distributions as $N\to\infty$ to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. The fluctuation result is a consequence of a limit theorem for Legendre polynomials which quantifies their completeness and orthogonality property. In the proof of the latter, we encounter a Catalan triangle.