Large deviations of local times of Lévy processes
Let \((X(t))_{t\in[0,\infty)}\) be a symmetric Lévy process on \(\mathbb R\) with characteristic function \(E(e^{i\lambda X(t)}) =e^{-t\psi(\lambda)}\). Assume that \(\psi\) is regularly varying at infinity with parameter \(\beta\in(1,2]\), and let \(\overline\beta\) be the conjugate number of \(\beta\), i.e., \(1/\beta+1/\overline\beta=1\). By earlier results of Kesten resp. Barlow, there is a jointly continuous version of the local time process \((L_t^x)_{t\in[0,\infty),x\in\mathbb R}\) of the process \((X(t))_{t\in[0,\infty)}\). Define \(L_t^*=\max_{x\in\mathbb R}L_t^x\). The main result is the proof of the existence of \(\lim_{y\to\infty} \psi(y)^{-1}\log P(L_t^0>yt^{1/\overline\beta})\) and the same quantity with \(L_t^0\) replaced by \(L_t^*\), and the explicit identification of the two limits (they are actually equal to each other) in terms of the Gamma function.
scientific article; zbMATH DE number 1552711