Small Noise Perturbations in Multidimensional Case
Andrey Pilipenko, Frank Norbert Proske
TL;DR
This work characterizes zero-noise limits for multidimensional singular ODEs driven by $\alpha$-stable noise with irregular vector fields $A$ whose near-origin behavior is $A(x)\sim \bar{a}(\varphi)|x|^{\beta-1}x$ for $\beta\in(0,1)$ and $\bar{a}>0$. By developing a space-time transformation and analyzing a model SDE with additive $\alpha$-stable noise, the authors identify a limiting exit-direction distribution $\bar{\Phi}(+\infty)$ that selects a unique exit path in the zero-noise limit. They prove that $X_\varepsilon(\cdot)$ converges in distribution to $X_0(\cdot,\bar{\Phi}(+\infty))$ in the Skorokhod space, where $X_0$ solves the unperturbed dynamics away from the origin and the exit time from a small neighborhood vanishes in the limit. In addition, they establish large-time asymptotics for solutions to SDEs with radial drift and prove a rigorous framework for zero-noise selection in the multidimensional, non-Lipschitz setting, with potential implications for singular ODEs and stochastic optimization models.
Abstract
In this paper we study zero-noise limits of $α-$stable noise perturbed ODE's which are driven by an irregular vector field $A$ with asymptotics $% A(x)\sim \overline{a}(\frac{x}{\left\vert x\right\vert })\left\vert x\right\vert ^{β-1}x$ at zero, where $\overline{a}>0$ is a continuous function and $β\in (0,1)$. 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.