Small Noise Perturbations in Multidimensional Case

TL;DR

This paper investigates the behavior of multidimensional ODEs perturbed by alpha-stable noise with irregular vector fields, extending previous one-dimensional results to higher dimensions and analyzing their zero-noise limits.

Contribution

It generalizes existing one-dimensional zero-noise limit results to multidimensional cases with irregular vector fields and introduces a novel analytical approach based on SDE transformations.

Findings

Extended zero-noise limit results to multidimensional systems.

Analyzed SDEs for large time behavior related to perturbed ODEs.

Provided a framework for studying irregular vector fields in stochastic perturbations.

Abstract

In this paper we study zero-noise limits of $\alpha -$stable noise perturbed ODE's which are driven by an irregular vector field $A$ with asymptotics $% A(x)\sim \overline{a}(\frac{x}{\left\vert x\right\vert })\left\vert x\right\vert ^{\beta -1}x$ at zero, where $\overline{a}>0$ is a continuous function and $\beta \in (0,1)$. 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\longrightarrow \infty $, which is obtained through a transformation of the perturbed ODE.