Well-posedness of martingale solutions and Fokker–Planck equations beyond Figalli's hypotheses: sub-Sobolev spatially dependent degenerate diffusion, time-measurable uniformly elliptic coefficients, and stability without uniform density bounds

Figalli (2008) coupled the Fokker–Planck equation with stochastic Lagrangian flows to prove existence, uniqueness, stability, and the Chapman–Kolmogorov property for martingale solutions of stochastic differential equations with rough or degenerate coefficients, under two sets of hypotheses: uniformly elliptic diffusion matrices with divergence bounds and time regularity $\partial_t a_{ij}\in L^\infty$, and degenerate but space-independent diffusion with bounded-variation drift. The paper explicitly signals generalizations of this theory. The convex-class generalization has been superseded by the Ambrosio–Figalli–Trevisan superposition principle, now available for general probability solutions. The spatially dependent degenerate case has been resolved when the diffusion matrix is Sobolev-regular, through the program of Le Bris–Lions completed by Luo, Trevisan, Röckner–Zhang, Champagnat–Jabin, and Grube. The surviving open core is threefold: (i) uniqueness in Figalli's density classes — equivalently, almost-everywhere well-posedness of martingale solutions — for spatially dependent degenerate diffusion with $\sigma$ of spatial regularity below Sobolev (bounded variation or bounded measurable), where the commutator technique provably breaks down; (ii) $L^2$-uniqueness for uniformly elliptic equations without the time-regularity hypothesis $\partial_t a_{ij}\in L^\infty$; (iii) stability of stochastic Lagrangian flows without the uniform density bound of Figalli's stability theorem. Adjacent frontiers — the driftless equation with a scalar measurable diffusion coefficient (Belaribi–Russo), diffusion matrices of vanishing mean oscillation degenerating only at points (Lee–Trutnau), the stationary degenerate problem (Bogachev–Shaposhnikov–Shatilovich), and sharp non-uniqueness in law for supercritical divergence-free drifts with non-degenerate noise (Lü–Röckner) — stop on either side of this core without entering it.

Background

A stochastic differential equation (SDE) $dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,dW_t$ on $\mathbb{R}^d$ describes a random trajectory driven by a Brownian motion $W$; the drift $b$ and the diffusion coefficient $\sigma$ are given bounded measurable functions (matrix-valued for $\sigma$), and $a=\sigma\sigma^{*}/2$, with $\sigma^{*}$ the transpose, is the diffusion matrix. Classical existence and uniqueness theory requires Lipschitz-continuous coefficients and fails for merely bounded measurable ones. The appropriate weak notion is that of a martingale solution (equivalently, a solution of the Stroock–Varadhan martingale problem): a probability law on the path space $C([0,T];\mathbb{R}^d)$ under which, for every smooth compactly supported test function $f$, the process $f(X_t)-f(X_0)-\int_0^t (b\cdot\nabla f+\frac12 a:D^2f)(s,X_s)\,ds$ is a martingale with respect to the filtration generated by the coordinate process. Well-posedness means existence together with uniqueness in law. The diffusion matrix is called uniformly elliptic when $\langle\xi,a(t,x)\xi\rangle\ge\alpha|\xi|^2$ for some $\alpha>0$, and degenerate when such a bound fails, i.e. the noise may vanish in some directions or regions.

The one-dimensional time marginals of a martingale solution solve, in the sense of distributions, the Fokker–Planck equation (also called the forward Kolmogorov equation)

$$\partial_t\mu_t+\sum_i\partial_i(b_i\mu_t)-\frac12\sum_{ij}\partial_{ij}(a_{ij}\mu_t)=0,$$

a second-order equation in conservative (divergence) form. For rough coefficients the central analytic question is uniqueness of distributional solutions within classes of densities — the Eulerian counterpart of the probabilistic question.

The deterministic analogue ($\sigma=0$) is the transport equation and the ordinary differential equation $\dot X=b(t,X)$, for which DiPerna and Lions (1989), extended by Ambrosio (2004) to vector fields of bounded variation, introduced renormalized solutions (distributional solutions $u$ for which $\beta(u)$ again solves the equation for every smooth real function $\beta$) and regular Lagrangian flows: flows well defined and unique for Lebesgue-almost every initial point, obtained as limits of classical flows along smooth approximations. A function has bounded variation (BV) when its distributional derivative is a finite measure; $W^{1,p}$ and $W^{2,p}$ denote the Sobolev spaces of functions with weak first, respectively second, derivatives in $L^p$. The technique rests on commutator estimates, which quantify how nearly mollified transport operators commute with the equation; the singular part of a BV derivative is precisely what can break them.

Figalli (2008) developed the stochastic counterpart of this program. A stochastic Lagrangian flow (SLF) for coefficients $(b,a)$ is a measurable family $\{\nu_x\}_{x\in\mathbb{R}^d}$ of probability measures on path space such that $\nu_x$ is a martingale solution starting from $x$ for Lebesgue-almost every $x$, and the averaged time marginals admit densities, uniformly bounded in $L^\infty$, for every initial density. His convex class $\mathscr{L}_+$ consists of the non-negative functions in $L^\infty([0,T];L^1(\mathbb{R}^d))\cap L^\infty([0,T];L^\infty(\mathbb{R}^d))$ that are weak-star continuous in time; its key feature is the monotonicity property $0\le\tilde\mu_t\le\mu_t\in\mathscr{L}_+\Rightarrow\tilde\mu_t\in\mathscr{L}_+$, which powers localization by cut-off functions. Figalli proved a superposition principle (probability solutions of the Fokker–Planck equation lift to martingale solutions); an equivalence between uniqueness for the equation in $\mathscr{L}_+$ and uniqueness of the SLF — hence well-posedness of the martingale problem for almost every initial datum; a stability theorem for sequences of approximating coefficients; and the Chapman–Kolmogorov property (consistency of the solution family under restarting it at intermediate times, as with Markov transition kernels). On the PDE side he proved uniqueness in $\mathscr{L}_+$ in two regimes: (a) uniformly elliptic $a$ with divergence bounds $\sum_j\partial_ja_{ij}\in L^\infty$, a one-sided $L^\infty$ bound on $(\operatorname{div}b-\frac12\sum_{ij}\partial_{ij}a_{ij})^-$, and time regularity $\partial_ta_{ij}\in L^\infty$; (b) degenerate $a$ independent of $x$, with $b$ of BV type and $(\operatorname{div}b)^-\in L^1_tL^\infty_x$. The hypothesis $\partial_ta_{ij}\in L^\infty$ is what upgrades uniqueness from the energy class $L^2([0,T];H^1)\cap W^{1,2}([0,T];H^{-1})$ to all of $L^2([0,T]\times\mathbb{R}^d)$, the space needed for the probabilistic application; space-independence of $a$ is what makes the renormalization argument work in the degenerate case. The stability theorem requires, besides convergence of the coefficients, a uniform $L^\infty$ bound on the densities of the approximating time marginals (assumption (ii)), which Figalli verifies through the Feynman–Kac formula (a representation of solutions of linear parabolic equations as expectations along the associated diffusion) in his applications.

Subsequent work reshaped this landscape. Le Bris and Lions (2008) studied the same equations from the PDE side and posed, as an open question, the case of Sobolev-regular, possibly degenerate and spatially dependent $\sigma$ with BV drift. Luo (2013) answered it and noted explicitly that this generalizes Figalli's degenerate uniqueness theorem, whose diffusion matrix was space-independent. Trevisan (2016) recovered and extended the whole probabilistic picture for weakly differentiable coefficients: uniqueness for spatially dependent degenerate $a\in L^1(W^{2,p})$ with $b\in L^1(W^{1,p})$, the superposition principle under minimal integrability, and equivalence with well-posedness of the martingale problem; he explicitly lists the case of BV-regular coefficients as one he does not address. The superposition principle itself — now called the Ambrosio–Figalli–Trevisan principle — was extended to general probability solutions by Bogachev, Röckner, and Shaposhnikov (2021), to nonlocal operators by Röckner, Xie, and Zhang (2020), and to Hilbert spaces by Dieckmann (2022), while Rehmeier (2021, 2022) constructed flow selections (a measurable, consistent choice of a Markovian flow inside the solution set of the nonlinear equation) connecting well-posedness of the equation to uniqueness of flows. Röckner and Zhang (2010), Luo (2014), Champagnat and Jabin (2018), and Grube (2025) pursued restricted pathwise uniqueness (pathwise uniqueness holding only among solutions that spend zero time at the degeneracy set of the diffusion) and uniqueness for the Fokker–Planck equation under one-sided monotonicity (an inequality of the form $\langle b(t,x)-b(t,y),x-y\rangle\le C|x-y|^2$, or its analogue involving the diffusion) or Sobolev-type conditions, Grube extending the bounded-coefficient results of Röckner–Zhang to unbounded coefficients. The monograph of Bogachev, Krylov, Röckner, and Shaposhnikov (2015) surveys the Fokker–Planck–Kolmogorov theory. What no work in the literature covers is the regime below Sobolev regularity of the diffusion: Luo documents that for BV diffusion coefficients the singular part of the derivative makes the commutator estimates ill-defined, and Bogachev, Shaposhnikov, and Shatilovich (2026) show that even the stationary degenerate problem with measurable coefficients is only partially understood, with explicit non-uniqueness examples. The closest results stop on either side of this regime: Belaribi and Russo (2012) prove uniqueness only in the driftless case of the equation $\partial_t z=\Delta(az)$ with a scalar (non-matrix) nonnegative measurable coefficient $a$, and only in a class whose solution difference $z$ satisfies $z\in L^2$ and $az\in L^2$; Lee and Trutnau (2026) treat diffusion matrices of vanishing mean oscillation (VMO — locally continuous on average, a class that excludes the jump-type singular part of BV) that degenerate only at points, with uniqueness in law only among solutions spending zero time at those points; and Lü and Röckner (2025) construct divergence-free drifts in $L^r_tL^p_x\cap C_tL^{d-}$ with $d/p+1/r>1$, above the known uniqueness thresholds, for which the SDE with non-degenerate additive noise admits two distinct weak solutions — so in the non-degenerate problem, too, weak uniqueness fails below the known regularity thresholds.

Problem Statement

The problem is to extend Figalli's stochastic Lagrangian flow / Fokker–Planck coupling theory past its remaining hypotheses. Three components remain open.

(1) Degenerate, spatially dependent diffusion of sub-Sobolev regularity. Let $a=\sigma\sigma^{*}/2$ be bounded, measurable, symmetric and non-negative definite, degenerate (not uniformly elliptic) and genuinely dependent on the space variable, with $\sigma$ of spatial regularity strictly below Sobolev — in particular $\sigma$ of bounded variation, or merely bounded measurable. Let $b$ be bounded with $b\in L^1([0,T];BV_{loc})$ and $(\operatorname{div}b)^-\in L^1([0,T];L^\infty)$. Decide whether the Fokker–Planck equation for $(b,a)$ has a unique solution in Figalli's class $\mathscr{L}_+$ — the non-negative functions in $L^\infty([0,T];L^1(\mathbb{R}^d))\cap L^\infty([0,T];L^\infty(\mathbb{R}^d))$ that are weak-star continuous in time — equivalently, whether the associated stochastic Lagrangian flow is unique, i.e. whether the martingale problem is well posed for Lebesgue-almost every initial datum.

(2) Time regularity in the uniformly elliptic regime. In the setting of Figalli's Theorem 4.3 — bounded measurable $a,b$ with $\sum_j\partial_ja_{ij}\in L^\infty$ and $(\operatorname{div}b-\frac12\sum_{ij}\partial_{ij}a_{ij})^-\in L^\infty$ and $a$ uniformly elliptic — decide whether uniqueness in $L^2([0,T]\times\mathbb{R}^d)$, and consequently uniqueness in $\mathscr{L}_+$ and almost-everywhere well-posedness of the martingale problem (Figalli's Theorems 1.3 and 5.1), survive without the hypothesis $\partial_ta_{ij}\in L^\infty$, that is, for diffusion matrices merely measurable in time.

(3) Stability without uniform density bounds. Identify intrinsic conditions on bounded approximating coefficients $(b_n,a_n)\to(b,a)$ in $L^1_{loc}$ under which the conclusion of Figalli's stability Theorem 3.7 holds without assumption (ii) — the uniform $L^\infty$ bound on the densities of the approximating time marginals, currently established only through Feynman–Kac representations — or show by counterexample that some substitute for (ii) is necessary.

The three components are jointly posed, each in the generality stated above. The further direction signaled in the source — extension of the coupling theory to other convex classes of time-marginals — is superseded by the Ambrosio–Figalli–Trevisan superposition principle and is not posed here.

References

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