Krylov's conjecture: existence of strong solutions of Itô equations with a small parabolic-Morrey drift and an $L^2$-in-time bounded drift component

N. V. Krylov's 2026 paper on the parabolic Adams theorem states, as its third main result, an explicitly unproven conjecture: the Itô equation $x_s=x+\int_0^s\sigma(t+r,x_r)\,dw_r+\int_0^s b(t+r,x_r)\,dr$ with merely Borel coefficients, uniformly nondegenerate $a=\sigma\sigma^*$, $D\sigma\in L_{1,\mathrm{loc}}$, and drift decomposed as $b=b_M+b_B$ — a Morrey part $b_M$ that is small in a parabolic cylinder Morrey norm together with $D\sigma$, plus a part $b_B$ bounded in space with $\int\bar b_B^2(t)\,dt<\infty$ — should have a solution adapted to the Wiener filtration $\{\mathcal{F}_t^w\}$, i.e. a strong solution. The conjecture is stated verbatim, without proof, in the source paper; in the subsequent literature, Krylov's own 2026 book proves strong solvability for the same drift decomposition only under additional Morrey bounds uniform in time and a structural assumption on $\sigma$. The conjecture in its stated generality — allowing genuine time singularities of the Morrey part and an arbitrary Borel nondegenerate diffusion — remains open.

Background

For $d\ge2$ and $d_1\ge d$ consider the Itô stochastic differential equation $x_s=x+\int_0^s\sigma(t+r,x_r)\,dw_r+\int_0^s b(t+r,x_r)\,dr,$ where $t\in\mathbb{R}$ and $x\in\mathbb{R}^d$ are parameters, $w$ is a $d_1$-dimensional Wiener process (Brownian motion), $\sigma$ is a Borel $d\times d_1$ matrix-valued function on $\mathbb{R}^{d+1}$, and $b$ is a Borel $\mathbb{R}^d$-valued function. The diffusion is assumed uniformly nondegenerate: $a:=\sigma\sigma^*$ takes values in $S_\delta$, the set of symmetric $d\times d$ matrices with eigenvalues in $[\delta,\delta^{-1}]$ for a fixed $\delta\in(0,1]$. A weak solution is a solution on some filtered probability space carrying a suitable Wiener process; a strong solution is a solution $\{x_s\}$ adapted to the filtration of the driving noise, i.e. $x_s$ is $\mathcal{F}_s^w$-measurable for each $s\ge0$, where $\mathcal{F}_s^w$ is the completion of $\sigma\{w_r,r\le s\}$, so that the solution is a functional of the observed Wiener path. Whether singular drift still admits strong solutions is a central well-posedness question of stochastic analysis, studied intensively since the 1970s.

The sharp scale-invariant conditions on the drift are expressed through parabolic Morrey norms. For $p_0\in(2,2+d]$ and $\rho>0$, let $\mathbb{C}_r$ be the collection of parabolic cylinders $C_r(t,x)=[t,t+r^2)\times B_r(x)$ in $\mathbb{R}^{d+1}$, with $B_r(x)$ the Euclidean ball, and set $\hat f_{p_0,\rho}=\sup_{r\le\rho}\,r\sup_{C\in\mathbb{C}_r}\big(|C|^{-1}\int_C|f|^{p_0}\big)^{1/p_0}$. The prefactor $r$ makes the norm invariant under the parabolic scaling $(t,x)\mapsto(\lambda^2t,\lambda x)$, so the class contains drifts behaving like $c/|x|$ with $c$ small — singularities far beyond any Lebesgue class. The analytic engine for handling such drifts is the parabolic Adams theorem, a mapping property of parabolic Riesz potentials (potential operators built from the heat kernel) weighted by Morrey functions, which lets integration-by-parts estimates be iterated with all constants kept under control. Here $D\sigma$ denotes the matrix of first-order spatial partial derivatives of $\sigma$.

Krylov's paper (arXiv:2602.08130) uses this engine to prove, for smooth coefficients, $L_{2n}$-estimates for the evolution family (the two-parameter family of solution operators) of a divergence-form parabolic operator (an operator $\partial_tu+\frac12D_i(a^{ij}D_ju)+b^iD_iu$ with the second-order term in divergence form) with Morrey lower-order terms, and moment estimates for the derivatives of solutions of the above SDE with respect to the initial data — the estimates from which strong solvability follows by a classical analytic criterion. The strongest adjacent results are: Röckner–Zhao (Probab. Theory Related Fields 2025), strong well-posedness for $a=I$ and drifts in the critical Ladyzhenskaya–Prodi–Serrin scale $L^{q,p}$ (i.e. $L^q$ in time, $L^p$ in space) with $d/p+2/q\le1$, the endpoint $d/p+2/q=1$ being the critical case; and Kinzebulatov–Madou (Stoch. Dyn. 2024), strong solutions for form-bounded drifts (drifts whose quadratic-form bound against test functions is dominated by the Dirichlet form of the Laplacian), again with $a=I$, a class that however excludes drifts with $L^2$-in-time singularities. Neither covers measurable non-constant diffusion together with time-singular drift.

The paper then states, as its third main result and explicitly without proof, the conjecture that strong solvability should hold for drifts decomposed as $b=b_M+b_B$, where the Morrey part $b_M$ is small in the above parabolic cylinder norm together with $D\sigma$, and the bounded part $b_B$ is bounded in $x$ for each $t$ with only $\int\bar b_B^2(t)\,dt<\infty$ — the latter allowing rather strong singularities in time, unlike the form-bounded class. The author indicates a proof strategy via the paper's Theorem 5.3 but does not supply it. In the subsequent book-length account (arXiv:2605.03236), Krylov proves strong existence for exactly this drift decomposition (Theorem 6.8.2) but under additional hypotheses: the Morrey part and $D\sigma$ must also satisfy spatial Morrey bounds uniform in $t$ (which excludes time singularities of the Morrey part), and $\sigma$ must satisfy a structural condition under which spatial mollification preserves uniform nondegeneracy. The conjecture in its stated generality is what remains.

Problem Statement

Fix $d\ge2$, $d_1\ge d$, $\delta\in(0,1]$, $p_0\in(2,2+d]$, and $\rho_0\in(0,\infty)$. Let $\mathbb{C}_r$ denote the collection of parabolic cylinders $C_r(t,x)=[t,t+r^2)\times B_r(x)$ in $\mathbb{R}^{d+1}$, where $B_r(x)$ is the Euclidean ball. Let $\sigma$ be a Borel $d\times d_1$ matrix-valued function on $\mathbb{R}^{d+1}$ such that $a=\sigma\sigma^*$ is $S_\delta$-valued (symmetric matrices with eigenvalues in $[\delta,\delta^{-1}]$) and $D\sigma\in L_{1,\mathrm{loc}}(\mathbb{R}^{d+1})$, and let $b=b_M+b_B$ with Borel $\mathbb{R}^d$-valued summands. Define $\widehat{D\sigma}_{p_0,\rho_0}=\sup_{r\le\rho_0}\;r\sup_{C\in\mathbb{C}_r}\Big(\frac{1}{|C|}\int_C|D\sigma|^{p_0}\,dxdt\Big)^{1/p_0},\qquad\hat b_{M,p_0,\rho_0}=\sup_{r\le\rho_0}\;r\sup_{C\in\mathbb{C}_r}\Big(\frac{1}{|C|}\int_C|b_M|^{p_0}\,dxdt\Big)^{1/p_0},$ $\bar b_B(t)=\sup_{x\in\mathbb{R}^d}|b_B(t,x)|.$ Conjecture (Krylov, arXiv:2602.08130): there exists a constant $N_0=N_0(d,\delta,p_0)$ such that, if $N_0\big(\widehat{D\sigma}_{p_0,\rho_0}+\hat b_{M,p_0,\rho_0}\big)\le1\qquad\text{and}\qquad\int_{\mathbb{R}}\bar b_B^2(t)\,dt<\infty,$ then for every $t\in\mathbb{R}$ and $x\in\mathbb{R}^d$ the equation $x_s=x+\int_0^s\sigma(t+r,x_r)\,dw_r+\int_0^s\big(b_M+b_B\big)(t+r,x_r)\,dr$ has a solution $\{x_s\}$ adapted to the filtration of $w$, i.e. $x_s$ is $\mathcal{F}_s^w$-measurable for every $s\ge0$, where $\mathcal{F}_s^w$ is the completion of $\sigma\{w_r,r\le s\}$ and $w$ is the driving $d_1$-dimensional Wiener process. Prove this conjecture, or refute it by exhibiting coefficients satisfying all the stated assumptions for which no $\{\mathcal{F}_s^w\}$-adapted solution exists. The resolution must use only the stated hypotheses: no smoothness of $\sigma$ or $b$, no Morrey bounds uniform in time, and no structural assumption on $\sigma$ beyond $a=\sigma\sigma^*\in S_\delta$ and $D\sigma\in L_{1,\mathrm{loc}}$.

References

1. N. V. Krylov, On the parabolic Adams theorem and its applications to diffusion processes, arXiv:2602.08130 (2026), DOI 10.48550/arXiv.2602.08130, https://arxiv.org/abs/2602.08130
2. N. V. Krylov, Stochastic Itô Equations and Parabolic Second-Order Equations with singular Drift, arXiv:2605.03236 (2026), DOI 10.48550/arXiv.2605.03236, https://arxiv.org/abs/2605.03236
3. N. V. Krylov, On strong solutions of time inhomogeneous Itô's equations with Morrey diffusion gradient and drift. A supercritical case, Stochastic Process. Appl. 185 (2025), Paper No. 104619, DOI 10.1016/j.spa.2025.104619, arXiv:2211.03719, https://arxiv.org/abs/2211.03719
4. M. Röckner, G. Zhao, SDEs with critical time dependent drifts: strong solutions, Probab. Theory Related Fields 192 (2025), no. 3-4, 1071-1111, DOI 10.1007/s00440-025-01390-9, arXiv:2103.05803, https://arxiv.org/abs/2103.05803
5. D. Kinzebulatov, K. R. Madou, Strong solutions of SDEs with singular (form-bounded) drift via Röckner-Zhao approach, Stoch. Dyn. 24 (2024), no. 7, Paper No. 2450050, DOI 10.1142/s0219493724500503, arXiv:2306.04825, https://arxiv.org/abs/2306.04825
6. D. Kinzebulatov, C. Yue, Non-local SDEs, critical drifts and local blow-ups, arXiv:2606.02786 (2026), DOI 10.48550/arXiv.2606.02786, https://arxiv.org/abs/2606.02786
7. N. V. Krylov, On the heat equation with singular drift, arXiv:2604.11960 (2026), DOI 10.48550/arXiv.2604.11960, https://arxiv.org/abs/2604.11960