Right inverses of Lévy processes and stationary stopped local times (Q1590678)

Call inverse of a real-valued process \(X\) any increasing process \(H\) such that \(X(H_x)=x\) for all \(x\geq 0\). The paper is concerned with inverses of symmetric Lévy processes. It is first shown that inverses exist if and only if \(X\) is recurrent and has a nontrivial Brownian component. In that case, there is a minimal inverse \(K\) which is a subordinator distributed as the inverse local time of \(X\) at \(0\), up to some linear time-change. This is used in the second part to derive results in the same vein as those in the classical fluctuation theory. More precisely, it is pointed out that if \(G_t:=\sup\{K_x: K_x\leq t\}\) and \(L_t:=X_{G_t}\), then \(Z_t:=X_t-L_t\) is a Markov process with local time at zero \(L_t\). This yields the joint law of \((Z_{\tau},L_{\tau})\) when \(\tau\) is an independent exponential time, and then a nice description of the excursion measure of \(Z\).