Pathwise uniqueness for Itô SDEs with drift in the endpoint Krylov–Röckner class $L^2_t L^\infty_x$

For the Itô stochastic differential equation $dX_t = b(t,X_t)\,dt + dW_t$ in $\mathbb{R}^d$ with identity diffusion coefficient, weak existence is known for drifts $b \in L^2(0,T; L^\infty(\mathbb{R}^d))$, the endpoint case $q=2$, $p=\infty$ of the Krylov–Röckner / Ladyzhenskaya–Prodi–Serrin critical condition $2/q + d/p = 1$. Whether pathwise (strong) uniqueness holds for this endpoint class is a long-standing open problem, explicitly flagged as unresolved by Wei–Hu–Yuan (arXiv:2310.00421, SIAM J. Math. Anal. 2025). Krylov's monograph (arXiv:2605.03236) records the endpoint as covered only by weak existence plus conditional weak uniqueness. A resolution is either a proof of unconditional pathwise uniqueness for the full class or a counterexample drift in the class with two distinct solutions driven by the same Brownian motion.

Background

Consider the Itô stochastic differential equation

$$dX_t = b(t, X_t)\,dt + dW_t, \quad X_0 = x_0 \in \mathbb{R}^d,$$

where $W$ is a standard $d$-dimensional Brownian motion and the diffusion coefficient is the identity matrix. A weak solution is a pair $(X, W)$ on some filtered probability space satisfying the equation; a strong solution is one adapted to the filtration generated by the given Brownian motion; pathwise uniqueness means that any two solutions driven by the same Brownian motion on the same probability space coincide almost surely. By the Yamada–Watanabe theorem, weak existence together with pathwise uniqueness is equivalent to the existence of a unique strong solution. For merely measurable (non-Lipschitz) drifts, the landmark result of Krylov and Röckner (2005) gives a unique strong solution when $b \in L^q(0,T; L^p(\mathbb{R}^d))$ with $\frac{2}{q} + \frac{d}{p} < 1$ — the subcritical regime with respect to parabolic scaling, under which the heat semigroup smooths faster than the drift can transport. This condition is the probabilistic counterpart of the Ladyzhenskaya–Prodi–Serrin (LPS) regularity criteria for the 3D Navier–Stokes equations: stochastic Lagrangian representations of a Navier–Stokes velocity field lead exactly to SDEs whose drift belongs to critical LPS classes, so the borderline line $\frac{2}{q} + \frac{d}{p} = 1$ is physically canonical. (Wei–Hu–Yuan arXiv:2310.00421 encode drift regularity by a 'degree' $\mathrm{deg}(b)$; the critical line corresponds to $\mathrm{deg}(b) = -1$.) Veretennikov's classical theorem yields pathwise uniqueness for drifts bounded in both time and space, but the critical Lebesgue classes are unbounded in general. On the critical line itself, progress in the last decade covers only finite spatial exponents: Nam (2020) treats the Orlicz-refined critical classes $L^{q,1}(0,T; L^p(\mathbb{R}^d))$ with $p, q \in (1,\infty)$ — the Lorentz-type, sharper-integrability refinement of $L^q_t L^p_x$, a strictly smaller space; Röckner–Zhao proved unique weak solvability for critical Lebesgue drifts (Bernoulli 2023) and unique strong solvability for $p \in (d,\infty)$, $q \in (2,\infty)$, $d \ge 3$, plus $b \in C([0,T]; L^d)$ (PTRF 2025) — all of which require $p < \infty$ and so exclude the endpoint. The remaining endpoint is $q = 2$, $p = \infty$: drifts essentially bounded in space for almost every time but only square-integrable in time, so $\|b(t,\cdot)\|_{L^\infty}$ may blow up like $t^{-1/2+\varepsilon}$ near $t = 0$. This class sits exactly at parabolic criticality, where the standard machinery — the Zvonkin transform (removing the drift through a parabolic PDE with right-hand side built from $b$) and Krylov-type estimates on the occupation of the solution — loses precisely the integrability it would need. Wei, Hu and Yuan state: 'it is still unknown whether the pathwise uniqueness is true or not under the critical case' $q = 2$, $p = +\infty$, 'and this problem is a long-standing open problem.'

Problem Statement

Let $d \ge 1$, $T \in (0,\infty)$, and let $b \in L^2(0,T; L^\infty(\mathbb{R}^d))$ be a Borel measurable drift, i.e. $\int_0^T \|b(t,\cdot)\|^2_{L^\infty(\mathbb{R}^d)}\,dt < \infty$, so that $b$ is essentially bounded in space for almost every time but may blow up in time like $t^{-1/2+\varepsilon}$. Consider the Itô stochastic differential equation

$$dX_t = b(t, X_t)\,dt + dW_t, \quad X_0 = x_0 \in \mathbb{R}^d, \quad t \in [0,T],$$

with identity diffusion coefficient and $W$ a standard $d$-dimensional Brownian motion; weak existence is already known for this class. Does pathwise uniqueness hold, i.e. must any two weak solutions defined on the same probability space and driven by the same Brownian motion coincide almost surely for every such drift and every initial point? Equivalently, by the Yamada–Watanabe theorem, does a unique strong solution exist? A negative answer — a drift in this class together with two distinct solutions driven by the same Brownian motion — would equally resolve the problem. This is the endpoint case $q = 2$, $p = \infty$ of the Krylov–Röckner critical condition $\frac{2}{q} + \frac{d}{p} = 1$, which Wei–Hu–Yuan explicitly identify as a long-standing open problem.

References

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