Refinements of asymptotics at zero of Brownian self-intersection local times

TL;DR

This paper refines the asymptotic analysis at zero of Brownian self-intersection local times, providing estimates and asymptotics for measures related to Brownian motion's self-intersections in higher dimensions.

Contribution

It introduces new estimates for Gaussian densities and Hermite polynomials, and derives asymptotics and Wasserstein distance bounds for measures associated with Brownian self-intersections.

Findings

Almost sure estimates for Itô-Wiener expansion terms

Asymptotic behavior of measures on Wiener space in dimension ≥ 4

Bounds on Wasserstein distance between measures and Wiener measure

Abstract

In this article we establish some estimates related to the Gaussian densities and to Hermite polynomials in order to obtain an almost sure estimate for each term of the It\^{o}-Wiener expansion of the self-intersection local times of the Brownian motion. In dimension $d\geqslant 4$ the self-intersection local times of the Brownian motion can be considered as a family of measures on the classical Wiener space. We provide some asymptotics relative to these measures. Finally, we try to estimate the quadratic Wasserstein distance between these measures and the Wiener measure.