A solution selection problem with small symmetric stable perturbations

TL;DR

This paper investigates the zero-noise limit of differential equations with singular coefficients driven by alpha-stable processes, demonstrating solution selection and computing associated probabilities through a jump decomposition approach.

Contribution

It introduces the first analysis of the zero-noise limit with alpha-stable noise for equations with singular coefficients, including solution selection and probability calculation.

Findings

Extremal solutions are selected in the zero-noise limit.

Probabilities of solution selection are explicitly computed.

A novel jump decomposition method is developed for the analysis.

Abstract

The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is an $\alpha $-stable process. It is proved that extremal solutions are selected and the respective probability of selection is computed. For this purpose an exit time problem from the half-line, which is of interest in its own right, is formulated and studied by means of a suitable decomposition in small and large jumps adapted to the singular drift.