Strong solutions and regularity for multidimensional SDEs with random and unbounded drifts

Menoukeu-Pamen and Tangpi (SIAM J. Math. Anal., 2019) proved in dimension one the existence and uniqueness of strong solutions, Malliavin differentiability with an explicit representation of the derivative, and Sobolev differentiability of the stochastic flow, for SDEs with additive noise whose drift is the sum of a deterministic, merely measurable part of linear growth and a random, adapted, bounded Lipschitz part with Malliavin-derivative bounds. They explicitly left open the extension to multidimensional state space — their compactness estimates rest on an explicit one-dimensional random ODE for the Malliavin derivative — and asked whether PDE (Kolmogorov-equation) methods survive the non-Markovian setting. Later work partially addressed both issues: Zhao (Rev. Mat. Iberoam., 2022) proved strong well-posedness in arbitrary dimension for random coefficients that are bounded and Hölder in space via backward stochastic Kolmogorov equations, and one-dimensional refinements were obtained by Holden–Karlsen–Pang and Bogso et al. The multidimensional extension of the original measurable/unbounded-drift theorems, including the Malliavin and Sobolev regularity of the solution and its flow, remains open and is stated here precisely.

Background

A stochastic differential equation (SDE) $dX_t=b(t,X_t)\,dt+\sigma\,dB_t$ couples a drift coefficient, a diffusion coefficient and a driving Brownian motion $B$. A strong solution is a solution adapted to the filtration generated by $B$ itself, i.e. a pathwise functional of the driving noise, whereas a weak solution may live on another probability space; classically, weak existence combined with pathwise uniqueness implies existence of a strong solution. For deterministic drifts that are merely bounded and measurable, Zvonkin (1974) and Veretennikov (1981) proved strong existence and uniqueness by solving a backward Kolmogorov partial differential equation and applying a transformation of state space that removes the drift (the Zvonkin transform), and Krylov and Röckner (2005) extended well-posedness to unbounded drifts in mixed $L^q_t L^p_x$ spaces with $n/p+2/q<1$. These PDE arguments hinge on the Markovian structure created by a deterministic drift.

When the drift $b(t,x,\omega)$ is itself random and adapted to the driving noise — as in diffusions in random potentials, path-dependent switching models, or forward–backward systems — the associated Kolmogorov equation has random, non-adapted coefficients and the classical route breaks down. Weak existence still follows from Girsanov theory, but strong existence becomes delicate. Randomness is a genuine impediment: Flandoli, Gubinelli and Priola (2010) observed that the one-dimensional drift $b_t(x)=\sqrt{|x-W_t|}\wedge1$, which is bounded, adapted and Hölder-1/2 in the space variable, admits two distinct adapted solutions, so spatial regularity of the random part below Lipschitz level cannot suffice for pathwise uniqueness.

An alternative probabilistic route avoids Markovianity: it constructs approximating solutions that are smooth in the sense of the Malliavin calculus of variations — the Malliavin derivative $D_tX$ measures the sensitivity of $X$ to perturbations of the Brownian path at time $t$ — and exploits a compactness criterion in $L^2(\Omega)$ for Malliavin-differentiable random variables to pass to a strong limit, deriving pathwise uniqueness from uniqueness in law; this program was developed by Menoukeu-Pamen, Meyer-Brandis, Nilssen, Proske and Zhang (2013) for bounded deterministic drifts. Menoukeu-Pamen and Tangpi (2019) carried it out in dimension one for $dX_t=(b_1(t,X_t)+b_2(t,X_t,\omega))\,dt+\sigma\,dB_t$ with additive noise, where $b_1$ is deterministic, Borel measurable and of linear growth, and $b_2$ is adapted, bounded and continuously differentiable in $x$ with a random Lipschitz constant $M_2(\omega)$ carrying an exponential moment, together with bounds on the Malliavin derivative of $b_2$ uniform in $x$. They proved: (Theorem 1.1) existence and uniqueness of a global strong solution; (Theorem 1.2) Malliavin differentiability of $X_t$ with moment bounds and an explicit representation of $D_tX_s$; (Theorem 1.3) for small horizons, weak differentiability of the flow $x\mapsto X_t^x$ in weighted Sobolev spaces $W^{1,p}(\mathbb{R},w)$ with weights $w$ satisfying $\int_{\mathbb{R}}e^{c|x|^{2}}w(x)\,dx<\infty$ for every $c\ge0$. The compactness estimates rest on an explicit solution of a scalar random ordinary differential equation satisfied by the Malliavin derivative, and the authors state that this — and hence the extension of all three theorems to state dimension $n\ge2$ — does not carry over; they also record, as an open methodological question, whether PDE methods could survive the non-Markovian setting by treating the random part of the drift as a forcing term.

Subsequent literature partially filled the gap. Zhao (2022), developing the PDE route, proved unique solvability of $n$-dimensional Itô equations with random coefficients by solving a backward stochastic Kolmogorov equation — a backward stochastic partial differential equation handled with the Itô–Wentzell formula, which extends Itô's formula to random fields that are semimartingales in time — and applying a modified Zvonkin transformation; his assumptions require the drift to be bounded and Hölder in $x$, and its Malliavin derivative to be Hölder in $x$ as an $L^{2p}(\Omega)$-valued field. One-dimensional refinements with random drift of Sobolev or bounded-variation type were obtained by Holden, Karlsen and Pang (2022) and by Bogso, Likibi Pellat, Kuissi Kamdem and Menoukeu-Pamen (2025). What no later work provides is the multidimensional extension of the original theorems under the original assumption structure — a deterministic, merely measurable, unbounded (linear-growth) part combined with a random bounded Lipschitz part — together with Malliavin differentiability of the solution and Sobolev differentiability of the stochastic flow.

Problem Statement

Let $n,d\in\mathbb{N}$, $T>0$, and let $\sigma\in\mathbb{R}^{n\times d}$ have full row rank (the source paper treats $n=1$ with $\sigma\in\mathbb{R}^{d}$, $|\sigma|^{2}>0$). Let $B$ be a $d$-dimensional Brownian motion on its completed Wiener space with the natural filtration, and consider on $[0,T]$

$dX_{t}=\bigl(b_{1}(t,X_{t})+b_{2}(t,X_{t},\omega)\bigr)\,dt+\sigma\,dB_{t},\qquad X_{0}=x\in\mathbb{R}^{n}.$

Assume the multidimensional analogue of conditions (A1)–(A2) of Menoukeu-Pamen and Tangpi (2019):

1. $b_{1}:[0,T]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is Borel measurable and $|b_{1}(t,y)|\le k(1+|y|)$ for some $k\ge0$;
2. $b_{2}:[0,T]\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$ is predictable, $b_{2}(t,\cdot,\omega)$ is continuously differentiable, and there is a random variable $M_{2}(\omega)\ge0$ with $|b_{2}(t,y,\omega)|+\|\nabla_{y}b_{2}(t,y,\omega)\|\le M_{2}(\omega)$ for all $(t,y,\omega)$ and $\mathbb{E}[\exp(C|M_{2}(\omega)|^{2})]<\infty$, where $C>0$ is a fixed constant determined by the data $T$, $n$, $d$, $k$ and $\sigma$ (in the one-dimensional source the value $C=48Td\max_i\sigma_i^{2}/(\sigma_1^{2}+\cdots+\sigma_d^{2})^{2}$ of this constant is fixed, its dependence on the data — including the dependence on the linear-growth constant $k$ — being as recorded in Lemma 3.1 of the source);
3. $b_{2}(s,y,\cdot)$ is Malliavin differentiable for every $(s,y)$ and $|D_{t}b_{2}(s,y,\omega)|\le\widetilde{M}_{2}(s,t,\omega)$ for all $y$, where $\sup_{0\le s\le T}\mathbb{E}\bigl[\bigl(\int_{0}^{T}|\widetilde{M}_{2}(s,t,\omega)|^{2}\,dt\bigr)^{4}\bigr]<\infty$ and $\mathbb{E}[|D_{t}b_{2}(s,y,\cdot)-D_{t'}b_{2}(s,y,\cdot)|^{4}]\le C'|t'-t|^{\alpha}$ for some constants $C',\alpha>0$.

The problem is to extend the three one-dimensional theorems of the source to arbitrary state dimension $n\ge2$:

(a) existence and uniqueness of a global strong solution $X\in S^{2}(\mathbb{R}^{n})$, the space of continuous processes adapted to the Brownian filtration with $\mathbb{E}\bigl[\sup_{0\le t\le T}|X_{t}|^{2}\bigr]<\infty$ (the analogue of Theorem 1.1);

(b) Malliavin differentiability $X_{t}\in\mathbb{D}^{1,2}(\mathbb{R}^{n})$ — the Sobolev space of random variables whose Malliavin derivative is square-integrable — for every $t$, together with moment bounds and an explicit representation of the Malliavin derivative generalizing the one-dimensional formula obtained from the scalar random ODE for $D_{t}X_{s}$ (the analogue of Theorem 1.2);

(c) for $T$ small enough and every $p\ge2$, weak differentiability of the stochastic flow $x\mapsto X_{t}^{x}$ as an element of $L^{2}(\Omega;W^{1,p}(\mathbb{R}^{n},w))$ for weights $w$ with $\int_{\mathbb{R}^{n}}e^{c|y|^{2}}w(y)\,dy<\infty$ for all $c\ge0$ (the analogue of Theorem 1.3).

The question does not prescribe a method. The target is the conjunction of (a)–(c) under the assumptions above, with no assumption replaced by a stronger one (for example boundedness or Hölder spatial regularity of $b_{1}$, or Hölder regularity in $y$ of $D_{t}b_{2}$). A refutation is equally admissible: an explicit coefficient set satisfying the assumptions above in some dimension $n\ge2$ for which strong existence or pathwise uniqueness fails resolves the question negatively.

References

1. Olivier Menoukeu-Pamen and Ludovic Tangpi (2019). Strong Solutions of Some One-dimensional SDEs with Random and Unbounded Drifts. SIAM Journal on Mathematical Analysis 51(5):4105–4141. DOI 10.1137/18M1218662, arXiv:1810.01314, https://arxiv.org/abs/1810.01314
2. Guohuan Zhao (2022). SDEs with random and irregular coefficients. Revista Matemática Iberoamericana 38(3):947–980. DOI 10.4171/RMI/1313, arXiv:2003.04436, https://arxiv.org/abs/2003.04436
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