Brownian intersection local times: Upper tail asymptotics and thick points
type: Item
instance of: scholarly article
title: Brownian intersection local times: Upper tail asymptotics and thick points
zbMATH Open document ID: 1032.60073
author: Peter Mörters
author: W. König
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Let \(p<d/(d-2)\) be positive integers, \(U\subset B\subset {\mathbb R}^{d}\) a domain where \(B=B(0,R)\) is a ball \(\{x;|x|\leq R\}\) (for \(d\geq 3\), \(R\) may be \(\infty\)), \(G\) be the Green function in \(B\), \(({\mathcal U}f)(x)= \int_{_{U}}G(x,y)f(y)dy\), considered as \({\mathcal U}:L^{2p/(2p- 1)}(U)\rightarrow L^{2p}(U)\), and \(l\) be the projected intersection local time of \(p\) independent Brownian motions started in points in \(U\) and stopped at the first exit from \(B\) (the rigorous constructions are reviewed in the paper). First result: \[ \lim_{a\to \infty }a^{-1/p}\log P(l(U)>a)= -p/\varrho^{\ast}, \] where \(\varrho^{\ast }=\sup\langle g^{2p-1},{\mathcal U}g^{2p-1}\rangle\) over \(g\in L^{2p}(U)\), \(\|g\|_{2p}=1\). Second result: The \(\sup\) is attained, every \(g\) realizing it has \(\sup_{U}<\infty\) and \(\inf_{U}>0\), is in \({\mathcal C}^{2}(U)\) and in \({\mathcal C}^{1}(\overline {U})\), satisfies \({\mathcal U}g^{2p-1}= \varrho^{\ast }g\) on \(U\), extends continuously on \(B\), null on \(\partial B\), with \((1/2)\Delta g= -\varrho^{\ast -1} g^{2p-1}1_{U}\) on \(B\backslash \partial U\). If \(p=1\), then \(g\) is unique. Third result: If \(d\geq 3\), \(R=\infty\) and \(U=B(0,1)\), then \(g\) is unique, rotationally symmetric around \(0\), \(g(x)=cz(a|x|)\), \(z\) is a solution of \(z''(x)+(d-1)x^{-1}z'(x)+z(x)^{2p-1}=0\), \(z(0)=1\), \(z'(0)=0\), on a maximal \([0,\xi)\) and \(a\) is the minimal (in \((0,\xi)\)) with \((d-2)z(a)= -az'(a)\). Also \(p^{2}/\varrho^{\ast 2}\) is \(a^{4}/4\) for \(p=1\), \(d\geq 3\) and \(4\pi a\int_{0}^{a}z^{4}(s)s^{2}ds\) for \(p=2\), \(d=3\). Fourth result: \[ \sup_{x\in {\mathbb R}^{d}}\limsup_{r\downarrow 0}l(B(x , r))r^{-1}\log^{-2}(1/r)= \limsup_{r\downarrow 0}\sup_{x\in S}l(B(x,r))r^{-1}\log^{-2}(1/r)=\varrho^{\ast 2}/4 \] and the Hausdorff dimension of the set of all \(x\in S\) with \(\limsup_{r\downarrow 0}l(B(x,r))r^{-1}\log^{- 2}(1/r)=a\) is \(1-2a^{1/2}\varrho^{\ast -1}\); \(S\) is the int…
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 1906096
zbMATH Keywords: intersection local time
zbMATH Keywords: Green function
zbMATH Keywords: thick point
zbMATH Keywords: variational problem
zbMATH Keywords: Hausdorff dimension
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1039548368
OpenAlex ID: W2054889038
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DOI: 10.1214/AOP/1039548368
description: scientific article; zbMATH DE number 1906096