Well-posedness by noise for SDEs and stochastic transport equations with random Hölder drift (Flandoli–Gubinelli–Priola's random-drift problem)

The Flandoli–Gubinelli–Priola (FGP) regularization-by-noise program shows that adding a nondegenerate transport noise restores well-posedness of linear transport equations and of the associated stochastic differential equations (SDEs) when the drift is deterministic but merely Hölder continuous in space. Remark 3 of FGP's 'Remarks on the stochastic transport equation with Hölder drift' (arXiv:1301.4012) poses as a main open problem the case of a genuinely random drift $b=b(\omega,t,x)$. Since FGP's own counterexample $b=|x-W_t|^{1/2}$, in its bounded form $|x-W_t|^{1/2}\wedge1$, shows that the unrestricted extension is false, the open problem is the conditional theory: identify structural conditions on the $\omega$-dependence of an adapted Hölder drift — beyond the known Malliavin-regularity, one-dimensional one-sided-bound, and deterministic-plus-smooth-random decomposition results — under which the noise still yields pathwise uniqueness and a differentiable stochastic flow for the SDE, and uniqueness of weak $L^\infty$-solutions, preservation of $C^1$ regularity, and weak* stability for the stochastic transport equation; and delimit the sharpness of such conditions against the known counterexample. The problem remains open: only partial results under restrictive structural assumptions are known.

Background

The linear transport equation $\partial_t u + b\cdot\nabla u = 0$ describes a passive quantity $u$ advected by a vector field (drift) $b$. For irregular drifts the method of characteristics breaks down: DiPerna and Lions (1989) established existence and uniqueness of weak solutions for $b$ in Sobolev spaces, but for drifts that are merely Hölder continuous in space — $b\in L^{\infty}(0,T;C_b^\alpha(\mathbb{R}^d;\mathbb{R}^d))$ with $\alpha\in(0,1)$, i.e. bounded with a Hölder-continuous spatial modulus — characteristics can split and weak solutions can lose uniqueness. Flandoli, Gubinelli and Priola (FGP) discovered that a suitable noise restores well-posedness. For the Stratonovich-form stochastic transport equation $\partial_t u + b\cdot\nabla u\,dt + \sigma\nabla u\circ dW = 0$, where $W$ is a $d$-dimensional Brownian motion, $\sigma\neq0$ is a constant, and the Stratonovich (circle) integral — the limit of integrals against smooth approximations of the noise, which preserves the ordinary chain rule — represents a transport-type perturbation, FGP proved, for deterministic-in-$\omega$ Hölder drifts: (1) the SDE $dX_t=b(t,X_t)dt+\sigma dW_t$ admits a stochastic flow of diffeomorphisms $\varphi_t$, i.e. a random map $x\mapsto\varphi_t(x)$ such that each trajectory solves the SDE, $x\mapsto\varphi_t(x)$ is a bijection of $\mathbb{R}^d$, and both its spatial derivative and that of its inverse are Hölder continuous; (2) uniqueness of weak $L^{\infty}$-solutions of the SPDE (bounded adapted processes solving the equation in distributional sense) when $\mathrm{div}\,b\in L^p([0,T]\times\mathbb{R}^d)$ for some $p>d\wedge2$ (relaxed to $\mathrm{div}\,b\in L^1_{\mathrm{loc}}$ when $\alpha>1/2$); (3) preservation of $C^1$ spatial regularity: the solution with $C^1$ initial datum $u_0$ stays $C^1$ in space, given by $u(t,x)=u_0(\varphi_t^{-1}(x))$; and (4) weak* stability: solutions depend continuously on the initial datum in the weak-* topology of $L^{\infty}$. The classical baseline for strong uniqueness of the SDE with additive nondegenerate noise and deterministic drift is the Krylov–Röckner result for $b\in L^q_tL^p_x$ with $d/p+2/q<1$, a class which (after the standard localization) contains bounded Hölder drifts. The proofs rest on the Zvonkin–Itô–Tanaka trick: one solves a backward parabolic equation with coefficient $b$ to build a change of variables that absorbs the irregular drift; the required parabolic regularity is available precisely because $b$ is a deterministic function of $(t,x)$. This is exactly what fails for random drifts: FGP showed (Section 6.2 of their Inventiones paper) that the extension to random drifts is false in general — for the progressively measurable, spatially Hölder(1/2) drift $b(\omega,t,x)=|x-W_t(\omega)|^{1/2}$ (FGP's original form; the truncated, bounded variant $|x-W_t|^{1/2}\wedge1$ is the restatement quoted by Zhao (2021) and Holden–Karlsen–Pang (2022)), $X_t=W_t$ solves $dX=b(\omega,t,X)dt+dW$, and so does the process built from the non-trivial ODE trajectory: with $Y=X-W$, the ODE $\dot Y=|Y|^{1/2}$ (respectively $\dot Y=|Y|^{1/2}\wedge1$) admits both $Y\equiv0$ and $Y=t^2/4$ (for the truncated drift, $Y=t^2/4$ up to $t=2$, continued as $Y=t-1$ for $t\ge2$). Hence a blanket extension cannot hold, and the open problem posed in Remark 3 of the 2013 'Remarks' paper is necessarily conditional: for which structural assumptions on the $\omega$-dependence of an adapted Hölder drift does the noise still regularize? The known sufficient structures are restrictive. Malliavin differentiability of the drift in the sample point (the Malliavin derivative $Db_t(x)$ measures the response of the coefficient to perturbations of the driving noise path), exploited through backward stochastic PDEs, yields well-posedness for uniformly nondegenerate Itô SDEs whose random coefficients are Hölder in $x$ with Malliavin derivative also Hölder in $x$ (Zhao). A decomposition $b=b_1(t,x)+b_2(t,x,\omega)$ with Borel measurable $b_1$ of spatial linear growth and adapted $b_2$ that is $C^1$ in $x$ with $|b_2|+|\partial_x b_2|\le M_2(\omega)$ where $M_2$ has exponential moments, and Malliavin differentiable, works in dimension one (Menoukeu-Pamen–Tangpi; the $x$-independent case $b_2=b_2(\omega,t)$ is a special case treated separately there), as does the one-dimensional one-sided gradient bound $\partial_x u\le K(\omega,t)$ with exponential moments for $u\in L^p(\Omega;L^\infty([0,T];\dot H^1(\mathbb{R})))$ (the homogeneous Sobolev space of functions with one square-integrable derivative) (Holden–Karlsen–Pang). Nothing is known for general adapted Hölder drifts in $d\ge2$ without such structure, and none of the partial results transfers to the transport-equation regularity questions.

Problem Statement

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},P)$ be a filtered probability space carrying a $d$-dimensional Brownian motion $W$, let $\sigma\neq0$ be a fixed real constant, and let the drift $b:\Omega\times[0,T]\times\mathbb{R}^d\to\mathbb{R}^d$ be progressively measurable (adapted; the case of interest is a genuinely random, path-dependent drift) with $b\in L^\infty(\Omega\times[0,T];C_b^\alpha(\mathbb{R}^d;\mathbb{R}^d))$ for some $\alpha\in(0,1)$. This is the regime posed as a main open problem in Remark 3 of Flandoli–Gubinelli–Priola's 'Remarks on the stochastic transport equation with Hölder drift' (arXiv:1301.4012). Because the unrestricted extension is refuted by the counterexample $b(\omega,t,x)=|x-W_t|^{1/2}\wedge1$, the problem is to build the conditional regularization-by-noise theory for random drifts, with one overarching objective — well-posedness by noise for genuinely random Hölder drifts — and three supporting objectives:

(i) Identify verifiable structural conditions on the $\omega$-dependence of $b$, strictly weaker than or incomparable with the known sufficient conditions (Zhao's Malliavin–Hölder regularity of $b$ in $\omega$; the one-dimensional one-sided gradient bound with exponential moments of Holden–Karlsen–Pang; the deterministic-plus-smooth-random decomposition $b=b_1(t,x)+b_2(t,x,\omega)$ of Menoukeu-Pamen–Tangpi), under which the SDE $dX_t=b(\omega,t,X_t)\,dt+\sigma\,dW_t$ admits pathwise uniqueness (any two solutions with the same initial condition driven by the same Brownian motion coincide) and a differentiable stochastic flow of diffeomorphisms with the same Hölder regularity of the flow and of its inverse as in the deterministic-in-$\omega$ case.

(ii) Under such conditions, prove for the Stratonovich stochastic transport equation $\partial_tu+b\cdot\nabla u\,dt+\sigma\nabla u\circ dW=0$ with random drift: uniqueness of weak $L^\infty$-solutions under the same divergence integrability as in the deterministic-in-$\omega$ case, understood pathwise for almost every $\omega$ (that is, $\mathrm{div}\,b\in L^p([0,T]\times\mathbb{R}^d)$ for some $p>d\wedge2$, or $\mathrm{div}\,b\in L^1_{\mathrm{loc}}$ when $\alpha>1/2$); preservation of $C^1$ spatial regularity for $C^1$ initial data; and weak* stability of solutions with respect to the initial datum.

(iii) Delimit sharpness: show which hypotheses are necessary, separating any proposed condition from the known failure case $b=|x-W_t|^{1/2}\wedge1$, and clarify whether the condition covers natural classes of adapted Hölder drifts arising in applications (e.g. drifts driven by functionals of the noise path with dissipative structure).

Any answer must reduce to the FGP deterministic-in-$\omega$ theory when $b$ does not depend on $\omega$.

References

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