Sharp pathwise nonuniqueness for additive SDEs

TL;DR

This paper constructs velocity fields demonstrating the limits of strong well-posedness for additive SDEs driven by Brownian noise, showing nonuniqueness in certain regularity regimes.

Contribution

It provides explicit examples of velocity fields with low regularity where weak solutions exist but pathwise uniqueness fails, highlighting the sharpness of classical well-posedness results.

Findings

Existence of velocity fields in $L^ abla_t C^ abla_x$ with nonuniqueness

Nonuniqueness extends to non-Brownian noises with certain regularities

Sharpness of strong well-posedness by noise regime is established

Abstract

We construct a family of velocity fields demonstrating the sharpness of the classical Zvonkin--Veretennikov--Davie strong well-posedness by noise regime. We consider stochastic differential equations driven by Brownian noise with drift $u$ and show that for any $\alpha<0$, there exists a velocity field $u \in L^\infty_t C^\alpha_x$ that admits a unique weak solution but does not satisfy pathwise uniqueness (and hence has no strong solutions). This contrasts with the case $\alpha \geq 0$, for which the existence of a unique strong solution is guaranteed. The velocity field construction is random, and the proof essentially uses central limit theorem scaling through the Berry--Esseen theorem. We also give natural extensions to non-Brownian driving noises, including nonuniqueness for arbitrary driving noises with certain H\"older regularities and an analogous sharpness of the strong well-posedness by noise regime for fractional Brownian motions.