The zero-noise limit of SDEs with $L^\infty$ drift

TL;DR

This paper investigates the behavior of one-dimensional stochastic differential equations with discontinuous drifts as noise vanishes, showing convergence to a unique distribution among possible deterministic solutions.

Contribution

It establishes conditions under which solutions of SDEs with discontinuous drifts converge to a unique limit distribution in the zero-noise limit.

Findings

Convergence to a unique distribution in the zero-noise limit

Tools for computing the limit distribution

Applicable to equations with discontinuous drifts

Abstract

We study the zero-noise limit for autonomous, one-dimensional ordinary differential equations with discontinuous right-hand sides. Although the deterministic equation might have infinitely many solutions, we show, under rather general conditions, that the sequence of stochastically perturbed solutions converges to a unique distribution on classical solutions of the deterministic equation. We provide several tools for computing this limit distribution.