On sojourn times, excursions and spectral measures connected with quasidiffusions
Let X be a quasidiffusion on the real line with infinitesimal operator \(D_ mD^+_ x\). The author considers excursions of X over some level b and sojourn times in an interval [0,b'] during such an excursion. The Laplace transforms are given in terms of eigenfunctions \(f_{\lambda}\) of \(D_ mD^+_ x-\lambda id.\) Let \(P_ t(b,b')\) be the time which X spends over b' during an excursion of X from zero to b occurring before the local time of X at zero equals t. This process turns out to be a compounded Poisson process. The Poisson measure with respect to the length and the maximum of an excursion is again given in dependence of eigenfunctions.
scientific article; zbMATH DE number 4015873