Escape from bounded domains driven by multi-variate $\alpha$-stable noises

TL;DR

This paper analyzes how multivariate alpha-stable noises influence the mean first passage time of a random walker escaping a bounded domain, revealing complex dependencies on the stability index and differences between noise types.

Contribution

It provides a detailed analysis of escape times under multivariate alpha-stable noises and introduces a method to distinguish noise types based on escape behavior.

Findings

Mean first passage time shows non-monotonous dependence on alpha.

Escape behavior differs between spherical and Cartesian Le9vy flights.

Higher-dimensional escape processes can identify noise types.

Abstract

In this paper we provide an analysis of a mean first passage time problem of a random walker subject to a bi-variate $\alpha$-stable L\'evy type noise from a 2-dimensional disk. For an appropriate choice of parameters the mean first passage time reveals non-trivial, non-monotonous dependence on the stability index $\alpha$ describing jumps' length asymptotics both for spherical and Cartesian L\'evy flights. Finally, we study escape from $d$-dimensional hyper-sphere showing that $d$-dimensional escape process can be used to discriminate between various types of multi-variate $\alpha$-stable noises, especially spherical and Cartesian L\'evy flights.