Limit behavior of random walk in a non-symmetric random medium
The paper is motivated by physically important modeling of random medium attempts, especially in connection with the so-called anomalous (diffusive) transport exhibiting close links with Lévy stable distributions and processes. The English translation utilizes ``steady'' instead of ``stable'' as an equivalent of ``ustoičivi'' in the Russian original. The non-symmetric random medium model is here characterized by two independent sequences of random variables, considered as parameters in the transport equation. Each sequence is composed of independent equally distributed random variables, whose distribution function lies in the domain of attraction of a concentrated in \((0,+\infty)\) stable law with the characteristic exponent in \((0,1]\). One departs from the general system of differential equations for transition probabilities (on a one- dimensional non-homogeneous lattice). The main theorem of the paper states that if transition parameters are stable random variables, then the random walk in a random medium has a well-defined limit distribution. The previous research was based on the assumption that transition parameters are random with finite moments of the first and second order, or in the relaxed form -- with finite second moments for stable laws, these moments are known not to exist.
convergence of probability distributions
scientific article; zbMATH DE number 536636