Limit theorems for L\'evy flights on a 1D L\'evy random medium

TL;DR

This paper establishes limit theorems for a Le9vy flight on a random medium with heavy-tailed distributions, covering various parameter regimes and providing convergence results for rescaled processes.

Contribution

It proves annealed functional limit theorems for Le9vy flights on a Le9vy random medium across all parameter combinations, including fluctuation results in deterministic cases.

Findings

Proved annealed functional limit theorems for all b5, b2 parameter combinations.

Established convergence of finite-dimensional distributions when limits are not cb4adlb4ag.

Provided fluctuation theorems for deterministic limit processes.

Abstract

We study a random walk on a point process given by an ordered array of points $(\omega_k, \, k \in \mathbb{Z})$ on the real line. The distances $\omega_{k+1} - \omega_k$ are i.i.d. random variables in the domain of attraction of a $\beta$-stable law, with $\beta \in (0,1) \cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between $\omega_k$ and $\omega_\ell$ depend on $\ell-k$ and are given by the distribution of a $\mathbb{Z}$-valued random variable in the domain of attraction of an $\alpha$-stable law, with $\alpha \in (0,1) \cup (1,2)$. Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a L\'evy flight on a L\'evy random medium. For all combinations of the parameters $\alpha$ and $\beta$, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not c\`adl\`ag, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.