Sharp pathwise nonuniqueness for additive SDEs

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 $α<0$, there exists a velocity field $u \in L^\infty_t C^α_x$ that admits a unique weak solution but does not satisfy pathwise uniqueness (and hence has no strong solutions). This contrasts with the case $α\geq 0$, 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 Hölder regularities and an analogous sharpness of the strong well-posedness by noise regime for fractional Brownian motions.

• The paper establishes that additive SDEs with irregular drift exhibit unique weak solutions while failing to maintain pathwise uniqueness at sharp regularity thresholds.
• It employs a multiscale random shear flow construction that induces explosive separation of trajectories, effectively challenging classical regularization-by-noise expectations.
• The study precisely identifies the critical regularity boundary (α=0 for Brownian noise and 1–1/(2H) for fractional noise) that separates regimes of strong and weak well-posedness.

Sharp Pathwise Nonuniqueness at the Threshold for Additive SDEs

Introduction and Main Results

This work establishes sharp thresholds for pathwise (strong) nonuniqueness of solutions to additive stochastic differential equations (SDEs) with irregular drift. The authors construct explicit velocity fields $u$ in subcritical regularity spaces that admit unique weak solutions but fail to satisfy pathwise uniqueness, thus separating the regimes of weak and strong well-posedness for SDEs with additive noise.

The central SDE under study is: $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$ where $W_t$ is Brownian motion (or, more generally, a suitable Hölder-continuous process) and the drift $u$ is time-dependent and spatially rough. Historically, Zvonkin--Veretennikov [1974, 1981] and Davie [2007] established strong well-posedness under nonnegative Hölder regularity ($C^\alpha_x$, $\alpha \geq 0$), while recent works have extended weak uniqueness down to negative regularity $C^{-\rho}_x$ ($\rho \in (0,1/2)$) [Flandoli et al., 2017].

The primary result fills a longstanding gap by constructing, for each $\alpha < 0$, random (divergence-free) velocity fields $u \in L^\infty_t C^\alpha_x$ for which:

• The SDE has a unique weak solution,
• Pathwise uniqueness fails: There exist weak solutions with the same driving noise but distinct solution paths,
• There are explicit lower bounds on the spatial separation of trajectories started arbitrarily close together ("explosive separation").

Moreover, the result is shown to be sharp: for $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$0, pathwise uniqueness always holds, and for $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$1 it can always fail.

Definitions, Thresholds, and Context

Regularity Regimes and Notions of Uniqueness

The analysis is made precise through function space hierarchy:

• The drift $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$2 lives in negative or nonnegative Hölder ($dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$3) or Besov classes.
• SDE solutions are considered in three notions: weak uniqueness (marginals agree), pathwise uniqueness (solutions coupled with the same Brownian path agree), and path-by-path uniqueness (for almost every realization of noise $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$4, there is at most one solution).

Crucially, weak uniqueness does not imply pathwise uniqueness in this setting, and the constructed examples precisely exploit this gap.

Summary of Main Theorem

• Theorem: For each $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$5, there exists a random, divergence-free $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$6 such that for the SDE with additive Brownian noise,
  • Weak uniqueness holds,
  • Pathwise (and thus path-by-path) uniqueness fails maximally.

Further extensions handle fractional Brownian motion (fBm) with Hurst parameter $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$7, showing the corresponding threshold for $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$8.

Structural Insights and Proof Techniques

Mechanism: Instability via Random Shear Flow Construction

The essential idea is to construct velocity fields that induce explosive separation of trajectories, even in the presence of a shared noise path. The construction leverages:

• Random alternating shear flows at multiple spatial and temporal scales,
• Careful balance of time and space scales so that the drift can "overcome" the regularizing effect of the noise on arbitrarily short scales,
• Independence of the velocity field across non-overlapping time intervals, which enables application of the Berry--Esseen theorem to quantify separation.

In detail, the proof uses an iterative, multiscale argument:

• At each scale, the drift induces a spatial shift in one direction while the noise, being Hölder regular, cannot mix particles across the active direction on the relevant short timescales.
• By alternating horizontally and vertically, and choosing parameters appropriately, one obtains a non-vanishing lower bound on the separation of particles started infinitesimally close.
• The summability of probabilities of insufficient separation facilitates a Borel--Cantelli argument, yielding almost sure nonuniqueness.

Probabilistic Framework

To move from quantitative "explosive separation" to qualitative "nonuniqueness", a general framework is developed:

• Definition of explosive separation and regular driving noises,
• Disintegration of weak solution laws conditioned on noise,
• Application of measurable selection theory for non-atomic conditional laws,
• Limiting arguments using compactness of weak solutions (tightness via Arzelà-Ascoli-type arguments).

This framework rigorously constructs nontrivial couplings where conditional on the noise, the law of solutions is not a Dirac mass—hence, the SDE lacks strong solutions but maintains weak uniqueness.

Contrasts and Implications

Contrast with Classical Theory

• Regularization by noise is not universal in negative regularity. Previous positive results for weak uniqueness do not extend to pathwise uniqueness below the $dX_t = u(t,X_t)\,dt + dW_t, \qquad X_0 = y,$9 threshold.
• The pathwise nonuniqueness persists even for divergence-free velocity fields, in contrast to weak uniqueness theory where divergence-free structure is beneficial.
• One-dimensional SDEs maintain pathwise uniqueness further into the negative regularity regime, as shown by [Butkovsky, 2025].

Relation to Spontaneous Stochasticity

The analysis is fundamentally different from the anomalous dissipation/spontaneous stochasticity phenomenon in passive scalars with vanishing noise—here, nonuniqueness is already present for fixed (nonzero) noise amplitude.

Implications for Theory and Future Directions

The paper defines the precise transition point from strong to weak well-posedness for a broad class of SDEs with additive noise and low-regularity drift, closing a central open problem in probabilistic ODE/SDE regularization by noise.

Several directions for further research emerge:

• Generalization to $W_t$0-stable (pure jump) noise processes,
• Extension of weak uniqueness for general time-dependent drifts and fBm with $W_t$1,
• Understanding if similar "maximal" nonuniqueness can be established for SDEs outside the additive regime.

Conclusion

The paper "Sharp pathwise nonuniqueness for additive SDEs" (2604.23883) constructs explicit divergence-free, low regularity vector fields for which SDEs with additive Brownian or fractional Brownian noise exhibit weak uniqueness but fail to have pathwise uniqueness. The threshold $W_t$2 ($W_t$3-regularity for Brownian noise, $W_t$4 for fBm) is shown to be sharp. The proof introduces a new technique leveraging central limit theorem scaling in the advection of random shear flows, and the implications decisively clarify the limitations of regularization by noise in low regularity regimes for additive SDEs. The results provide a blueprint for analyzing strong versus weak well-posedness in further stochastic PDE and ODE models with critical drift regularity.