Ladder costs for random walks in L\'evy random media
Alessandra Bianchi, Giampaolo Cristadoro, Gaia Pozzoli

TL;DR
This paper studies the first-ladder height and length of a random walk in a Lévy random medium, introducing a new class of processes called RWRSB, and characterizes their distributions using advanced probabilistic tools.
Contribution
It introduces the RWRSB process and provides a detailed characterization of ladder variables in Lévy media, extending existing theories with new identities and representations.
Findings
Characterization of the tail distribution of the sum of scenery values
Derivation of a generalized Spitzer-Baxter identity
Representation of RWRSB in terms of local times of the random walk S
Abstract
We consider a random walk 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 and length , where is the first-passage time of in . 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 , corresponding to the two possible crossing directions of that bond. A random walk on 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…
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Taxonomy
TopicsStochastic processes and statistical mechanics · Random Matrices and Applications · Diffusion and Search Dynamics
