On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients

TL;DR

This paper investigates how small noise regularizes multidimensional ODEs with non-Lipschitz coefficients, showing that the drift's direction determines whether the limit process selects extreme solutions or satisfies an averaged ODE.

Contribution

It introduces a new averaging principle for multidimensional SDEs with non-Lipschitz coefficients and analyzes the selection mechanism of limit solutions based on drift behavior.

Findings

When the drift pushes away from the hyperplane, the limit process selects extreme solutions.

When the drift attracts to the hyperplane, the limit process satisfies an averaged ODE.

A new general averaging principle for such SDEs is formulated.

Abstract

In this paper we solve a selection problem for multidimensional SDE $d X^\varepsilon(t)=a(X^\varepsilon(t)) d t+\varepsilon \sigma(X^\varepsilon(t))\, d W(t)$, where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H$. It is assumed that $X^\varepsilon(0)=x^0\in H$, the drift $a(x)$ has a Hoelder asymptotics as $x$ approaches $H$, and the limit ODE $d X(t)=a(X(t))\, d t$ does not have a unique solution. We show that if the drift pushes the solution away of $H$, then the limit process with certain probabilities selects some extreme solutions to the limit ODE. If the drift attracts the solution to $H$, then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.