[Paper Review] Correction to the paper "Some remarks on Davie's uniqueness theorem"

A. V. Shaposhnikov|arXiv (Cornell University)|Mar 20, 2017

Navier-Stokes equation solutions10 references8 citations

TL;DR

This paper corrects a technical gap in the proof of pathwise uniqueness for stochastic differential equations with irregular drifts, as originally claimed in Shaposhnikov (2017). By replacing a flawed uniform Hölder continuity assumption with a refined application of a modified Kolmogorov continuity theorem, the author restores the validity of the main result: for bounded, Borel measurable drifts satisfying a critical integrability condition ($\frac{d}{p} + \frac{2}{q} < 1$), pathwise uniqueness holds for almost all Brownian paths.

ABSTRACT

The property 4 in Proposition 2.3 from the paper "Some remarks on Davie's uniqueness theorem" is replaced with a weaker assertion which is sufficient for the proof of the main results. Technical details and improvements are given.

Motivation & Objective

• To address a technical gap in the proof of uniform Hölder continuity of the flow in Shaposhnikov (2017), specifically in property 4 of Proposition 2.3.
• To restore the validity of the main result on pathwise uniqueness for SDEs with bounded, Borel measurable drifts.
• To provide a corrected, rigorous proof of the Hölder continuity of the solution flow under the same integrability condition as in the original paper.
• To ensure the core conclusion of pathwise uniqueness for almost all Brownian paths remains valid despite the initial flaw in the continuity argument.

Proposed method

• Replaces the original claim of uniform Hölder continuity with a modified version of the Kolmogorov continuity theorem tailored to discrete time grids.
• Applies a refined moment estimate for the difference of solutions to the transformed SDE, showing $\sup_{s} \mathbb{E} \sup_{t} |X_{s,t}^x - X_{s,t}^y|^a \leq C(|x-y|^a + |x-y|^{a-1})$.
• Uses a probabilistic covering argument with a sequence $S_n$ of finite sets with $|S_n| \leq 2^{\eta n}$ to control the Hölder norm over dyadic times.
• Applies Lemma 2.1, a probabilistic version of the Kolmogorov continuity theorem, to deduce almost sure Hölder continuity of the flow on the set $S_n$.
• Imposes a localization condition $|x| < N$ and uses a truncation argument to handle the bounded drift case.
• Combines the corrected flow regularity with a Borel–Cantelli argument and a perturbation estimate from Lemma 3.6 to prove pathwise uniqueness.

Experimental results

Research questions

• RQ1 Does the original proof of pathwise uniqueness for SDEs with irregular drifts in Shaposhnikov (2017) hold under the stated conditions, particularly regarding the uniform Hölder continuity of the flow?
• RQ2 Can the flawed assumption of uniform Hölder continuity (property 4 in Proposition 2.3) be replaced with a weaker, provable condition that still supports the main result?
• RQ3 Is the pathwise uniqueness result for SDEs with bounded, Borel measurable drifts still valid when the continuity of the flow is established via a corrected application of the Kolmogorov continuity theorem?
• RQ4 Can the proof be made rigorous using only moment estimates and probabilistic covering arguments without requiring stronger a priori regularity?
• RQ5 What is the minimal regularity condition on the drift $b$ that still guarantees pathwise uniqueness for almost all Brownian paths?

Key findings

• The original claim of uniform Hölder continuity of the solution flow (property 4 in Proposition 2.3) is replaced with a weaker, provable statement using a modified Kolmogorov continuity theorem.
• The corrected proof establishes that for any $\alpha \in (0,1)$ and $\eta > 0$, there exists a full-probability set $\Omega'$ such that the flow satisfies $|\varphi_{s,t}(x) - \varphi_{s,t}(y)| \leq C|x-y|^\alpha$ for $s \in S_n$, $|x-y| \leq 2^{-n}$, and $x,y$ bounded.
• The moment estimate $\sup_s \mathbb{E} \sup_t |X_{s,t}^x - X_{s,t}^y|^a \leq C(|x-y|^a + |x-y|^{a-1})$ is sufficient to imply the required Hölder continuity on the discrete time grid $S_n$.
• The pathwise uniqueness result for SDEs with bounded, Borel measurable drifts satisfying $\frac{d}{p} + \frac{2}{q} < 1$ is restored and now rigorously proven.
• The proof relies on a Borel–Cantelli argument over dyadic partitions and a perturbation estimate from Lemma 3.6, which bounds the difference of solutions by $C l^{4/3}$ for intervals of length $l$.
• The final argument shows that the difference between two solutions vanishes in the limit of refining the dyadic grid, implying uniqueness via continuity and the $M^{-1/15}$ decay bound.