Lifschitz tail and Wiener sausage. II
type: Item
instance of: scholarly article
title: Lifschitz tail and Wiener sausage. II
zbMATH Open document ID: 0732.60089
DOI: 10.1016/0022-1236(90)90013-B
author: Alain-Sol Sznitman
published in: Journal of Functional Analysis
publication date: 1990-01-01T00:00:00Z
review text: The author presents two situations where the methods developed in the first part (see the paper reviewed above) apply. The first situation is that of a periodic diffusion Z on \({\mathbb{R}}^ d\), \(d\geq 1\), with generator \(L={1\over 2} \partial_ ia_{ij}\partial_ j\), where \((a_{ij})\) is a \(Z^ d\)-periodic elliptic matrix with \(C^ 2\)- entries. The second situation is a hypoelliptic diffusion Z on a simply connected nilpotent Lie group with generator of the type \(L={1\over 2} \sum^{k}_{i=1}X^ 2_ i\), where the \(X_ i\) are left invariant vector fields. For both situations the author considers the Wiener sausage \(| W^ C_ t|\) of Z modeled on a nonpolar compact subset C of \({\mathbb{R}}^ d\) and gives an explicit expression of the limit \(\lim_{t\uparrow \infty}t^{-d/(d+2)}\log E_ 0[\exp (-\nu | W^ C_ t|)].\) The value of the limit does not change if \(E_ 0[...]\) is replaced by the Laplace transform of the vague limit \(\ell\) of the (normalized) empirical measure of the increasing sequence of eigenvalues of -L, when -L is considered in a ball \(B_ N\) (with radius N centered at the origin) with Dirichlet boundary conditions on \(\partial B_ N\) and on the obstacles intersecting \(B_ N\). By a Tauberian theorem the author obtains the limit of \(\ell ([0,\lambda])\) as \(\lambda \downarrow 0\), the Lifschitz tail behaviour of the density of states. Let us mention that other applications of these methods are treated in the author's article on the long-time asymptotics for the shrinking Wiener sausage [Commun. Pure Appl. Math. 43, No.6, 809-820 (1990; Zbl 0718.60091)].
reviewed by: M.Dozzi
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4211220
zbMATH Keywords: principal eigenvalue
zbMATH Keywords: hypoelliptic diffusion
zbMATH Keywords: invariant vector fields
zbMATH Keywords: Wiener sausage
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: Lifschitz tail behaviour
MaRDI profile type: Publication
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full work available at URL: 0022-1236(90)90013-b
OpenAlex ID: W4205643977
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description: scientific article; zbMATH DE number 4211220