Small Noise Perturbations in Multidimensional Case

Abstract

In this paper we study zero-noise limits of $\hat{I}\pm -$stable noise perturbed ODE's which are driven by an irregular vector field $A$ with asymptotics at zero, where $\overline{a}>0$ is a continuous function and ${\hat{I}}^2\in (0,1)$. The results established in this article can be considered a generalization of those in the seminal works of Bafico \cite{Ba} and Bafico, Baldi \cite{BB} to the multi-dimensional case. Our approach for proving these results is inspired by techniques in \cite% {PP_self_similar} and based on the analysis of an SDE for $t⟶\infty$, which is obtained through a transformation of the perturbed ODE.

30 pages