Excursions of a spectrally negative Lévy process from a two-point set

Abstract: Let $a\in (0,\infty)$. For a spectrally negative L\'evy process $X$ with infinite variation paths the resolvent of the process killed on hitting the two-point set $V={-a,a}$ is identified. When further $X$ has no diffusion component the Laplace transforms of the entrance laws of the excursion measures of $X$ from $V$ are determined. This is then applied to establishing the Laplace transform of the amount of time that elapses between the last visit of $X$ to a given point $x$, before hitting some other point $y>x$, and the hitting time of $y$. All the expressions are explicit and tractable in the standard fluctuation quantities associated to $X$.