Lifschitz tail and Wiener sausage. II
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)].
scientific article; zbMATH DE number 4211220
scientific article