Ladder costs for random walks in L\'evy random media

Abstract

We consider a random walk $Y$ moving on a L\'evy random medium, namely a one-dimensional renewal point process with inter-distances between points that are in the domain of attraction of a stable law. The focus is on the characterization of the law of the first-ladder height $Y_{\mathcal{T}}$ and length $L_{\mathcal{T}}(Y)$, where $\mathcal{T}$ is the first-passage time of $Y$ in $\mathbb{R}^+$. The study relies on the construction of a broader class of processes, denoted Random Walks in Random Scenery on Bonds (RWRSB) that we briefly describe. The scenery is constructed by associating two random variables with each bond of $\mathbb{Z}$, corresponding to the two possible crossing directions of that bond. A random walk $S$ on $\mathbb{Z}$ with i.i.d increments collects the scenery values of the bond it traverses: we denote this composite process the RWRSB. Under suitable assumptions, we characterize the tail distribution of the sum of scenery values collected up to the first exit time $\mathcal{T}$. This setting will be applied to obtain results for the laws of the first-ladder length and height of $Y$. The main tools of investigation are a generalized Spitzer-Baxter identity, that we derive along the proof, and a suitable representation of the RWRSB in terms of local times of the random walk $S$. All these results are easily generalized to the entire sequence of ladder variables.