Non-selection of Lagrangian trajectories in the zero-noise limit for a class of stochastic regularizations

Abstract

We prove the lack of selection in the zero-noise limit for solutions to SDEs driven by a divergence-free, H\"older continuous vector field with exponent $\alpha\in(0,1)$, arbitrarily close to $1$ but fixed. The result applies to a broad class of regularizing additive noises, including fractional Brownian motion and stable L\'evy processes. The proof combines pathwise Lagrangian arguments, based on the analysis of the deterministic flows associated to mixing velocity fields, with probabilistic estimates coming from the stochastic sewing lemma. This allows to show that lack of selection happens simultaneously on a large set of initial data, whose complement has arbitrarily small Lebesgue measure.