On a Selection Problem for Small Noise Perturbation in Multidimensional Case

TL;DR

This paper investigates the limiting behavior of multidimensional differential equations with discontinuous drift under small noise perturbations, focusing on how the drift's jump discontinuity influences the limit process.

Contribution

It provides a comprehensive analysis of the limit behavior for multidimensional systems with hyperplane discontinuities in the drift, a topic previously underexplored.

Findings

Behavior depends on signs of the normal component of drift near the hyperplane

All cases of sign configurations are analyzed

Results clarify the limit process structure in multidimensional discontinuous systems

Abstract

The problem on identification of a limit of an ordinary differential equation with discontinuous drift that perturbed by a zero-noise is considered in multidimensional case. This problem is a classical subject of stochastic analysis. However the multidimensional case was poorly investigated. We assume that the drift coefficient has a jump discontinuity along a hyperplane and is Lipschitz continuous in the upper and lower half-spaces. It appears that the behavior of the limit process depends on signs of the normal component of the drift at the upper and lower half-spaces in a neighborhood of the hyperplane, all cases are considered.