A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

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 the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by √ N and show that they converge in finite-dimensional distributions as N → ∞ 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.

Asymptotic error in the eigenfunction expansion for the Green's function of a Sturm-Liouville problem

“…We show later that [11,Theorem 1.5] which quantifies an integrated version of the completeness and orthogonality property for Legendre polynomials can be deduced from Theorem 1.5. Moreover, we have the following convergence results for the Legendre polynomials {P n : n ∈ N 0 }, the Chebyshev polynomials of the first kind {T n : n ∈ N 0 } and the Chebyshev polynomials of the second kind {U n : n ∈ N 0 }, which all are scalar multiplies of Jacobi polynomials.…”

“…One base case is covered by [10, Theorem 1.2], another one by Proposition 2.3 below, and the remaining two cases can be deduced from these results. When proving Proposition 2.3, we employ a similar proof strategy as was used for [10, Theorem 1.2] and [11,Theorem 1.5], that is, we split the analysis into an on-diagonal and an off-diagonal part, with the pointwise convergence on the diagonal being a consequence of a convergence of moments and a locally uniform convergence, which allows for an application of the Arzelà-Ascoli theorem.…”

“…Using characteristic functions as in the proof of [11,Theorem 1.6], Theorem 1.2 is a consequence of Proposition 2.3 since the fluctuation processes (F N t ) t∈[0,1] are zero-mean Gaussian processes whose covariance functions, due to (2.8), are exactly given by…”

“…with the one caveat that in the off-diagonal convergence for the associated Laguerre polynomials we can only deal with scaling exponents strictly smaller than 1 2 . In our analysis, we use the same approach which already proved powerful in [10,11] and which was demonstrated in the previous section, that is, we split our considerations into an on-diagonal part, consisting of a moment argument and local uniform bounds feeding into the Arzelà-Ascoli theorem, and an off-diagonal part. While we are able to establish in general that, under suitable boundary conditions, in the limit as N → ∞ the moments on the diagonal satisfy the same recurrence relation as the desired limit function and to derive a general Christoffel-Darboux type formula used for the off-diagonal convergence, it seems difficult to develop a full general analysis.…”

“…The main advantage compared to [11] is that we have a streamlined way to deal with the moment convergence. Proposition 3.1 and Proposition 3.2 imply that for a family of classical orthogonal polynomials which is subject to suitable boundary conditions the limit moments on the diagonal satisfy the same recurrence relation as the moments of the desired limit function.…”