Brownian loops on non-smooth surfaces and the Polyakov-Alvarez formula
type: Item
instance of: scholarly article
title: Brownian loops on non-smooth surfaces and the Polyakov-Alvarez formula
author: Minjae Park
author: Joshua Pfeffer
author: Scott Sheffield
published in: Journal of Functional Analysis
publication date: 2024-08-01T00:00:00Z
review text: Consider the \(2\)-dimensional Brownian loop \((L(s))_{s\in [0,t]}\) with length \(t\) and center \(t^{-1}\int_0^t L(s) ds\), constructed by rescaling the unit-length Brownian loop \((l(s))_{s\in [0,1]}\) such that \(l(0)=l(1)\) and \(\int_0^1 l(s) ds =0\). Given \(\rho\) a Lipschitz function on a compact domain \(D\) in \({\mathbb R}^2\), the \(\rho\)-length of \((L(s))_{s\in [0,t]}\) is defined as \(\int_0^t e^{\rho (L(s))} ds\). The main result of the paper states that the Brownian loop measure of the set of loops with center in \(D\) and \(\rho\)-length at least \(\delta >0\) is given by \[ \frac{1}{2\pi \delta} \int_D e^{\rho (x)} dx + \frac{1}{48 \pi} \int_D | \nabla \rho (x)|^2 dx +o(1) \] as \(\delta\) tends to zero. In addition, it is shown that the \(o(1)\) error term can be bounded uniformly over families of functions \(\rho\) that are precompact in \(W^{1,1}(D)\) and have uniformly bounded Lipschitz constants. The above result is also extended to compact smooth two-dimensional Riemannian manifolds.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 81T40
zbMATH DE Number: 7890711
zbMATH Keywords: Brownian loop measure
zbMATH Keywords: Brownian loop soup
zbMATH Keywords: Polyakov-Alvarez formula
zbMATH Keywords: non-smooth surfaces
MaRDI profile type: Publication
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DOI: 10.1016/J.JFA.2024.110545
description: scientific article; zbMATH DE number 7890711