Factorizing Laplace exponents in a spectrally positive Lévy process (Q1194601)

If \(X\) is a spectrally positive Lévy process, then the restriction of \(\Psi(s)=\log\mathbb{E} \exp\{-sX_ 1\}\) to \([z,\infty)\) is a bijection with inverse \(\Phi\), where \(z\) is the largest real zero of \(\Psi\); and \(\mathbb{E} \exp(-s\tau_ t)=\exp(-t\Phi(s))\), where \(\tau_ t=\inf\{s: -X>t\}\) is the first passage-time process for \(-X\). Besides \(\Phi\), there are the related Laplace exponents of subordinators: \[ K_ 1(s)=s/\Phi(s),\;K_ 2(s)=\Psi(s+{\mathfrak z})/(s+{\mathfrak z}),\;K_ 3(s)=\Phi(s)-{\mathfrak z}. \] The starting motivation of this paper is to provide a pathwise explanation for the factorization \(K_ 1=K_ 2\circ K_ 3\). This is achieved by a transformation of Pitman's type, introduced by the same author. Important by-products are interesting properties for the ladder point sets of \(X\). Define the past-supreme process \(\bar X\) of \(X_ 1\), \(\bar X_ t=\sup\{X_ 1: s\in t\}\), with continuous part \(\bar X^ c\), and the increasing process \(\tilde X_ t=\bar X^ c_ t|\sum(\bar X_{s- }-X_{s-}|_{\{\bar X_ s>\bar X_{s-}\}}\) (over \(s\leq t\)). The ladder time set \({\mathcal L}=\{t: X_ t=\bar X_ t\}\) is regenerative and, having a local time process \(L\), \({\mathcal L}\) is the closure of the range of \(L^{-1}\). Also introduce \(Y=\tilde X+\bar X-X\), \(\sigma_ t=\sup\{s: Y_ s>t\}\), the last-passage-time process for \(Y\), \(\tilde S=\tilde X\circ L^{-1}\), \(\bar S=\bar X\circ L^{-1}\). Then \((L^{- 1},\tilde S)_ t\) is almost surely \((\sigma_ x,x)_ x\), \(\{(\sigma_ x,x): x\geq 0\}=((\tau_ t,t): t\geq 0\}\), in distribution. The closure of the latter set is called the ladder point set.