Stochastic Stability for Flows with Smooth Invariant Measures

TL;DR

This paper investigates the concept of stochastic stability in smooth invariant measure flows, focusing on non-singular flows on the circle and volume-preserving flows under diffusive perturbations, providing theoretical insights into their stability properties.

Contribution

It offers a comprehensive analysis of stochastic stability for flows with smooth invariant measures, including specific results for circle flows and volume-preserving flows under diffusive noise.

Findings

Non-singular flows on the circle are fully analyzed for stochastic stability.

Volume-preserving flows are stochastically stable under homogeneous diffusion perturbations.

The paper establishes conditions under which flows maintain stability with respect to stochastic perturbations.

Abstract

We study the notion of stochastic stability with respect to diffusive perturbations for flows with smooth invariant measures. We investigate the question fully for non-singular flows on the circle. We also show that volume-preserving flows are stochastically stable with respect to perturbations that are associated with homogeneous diffusions.