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arxiv: 2606.31500 · v1 · pith:LUBYLY5Gnew · submitted 2026-06-30 · 🧮 math.PR · math.AP

Non-Uniqueness for Nonlinear Fokker--Planck Equations and Their Associated Distribution-Dependent SDEs

Huaxiang L\"u This is my paper

Pith reviewed 2026-07-01 04:12 UTC · model grok-4.3

classification 🧮 math.PR math.AP
keywords nonlinear Fokker-Planck equationsdistribution-dependent SDEsnon-uniquenessdivergence-free driftsstationary solutionssuperposition principlemartingale solutions
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The pith

Divergence-free drifts in C_t L^{d-} allow infinitely many distinct solutions to distribution-dependent SDEs from stationary initial densities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs explicit divergence-free drifts on the torus and in R^d for which the associated nonlinear Fokker-Planck equations admit multiple probability solutions. These PDE non-uniqueness results are transferred to the distribution-dependent SDEs via the superposition principle, yielding infinitely many distinct martingale solutions starting from the same stationary density. A second construction shows that for d at least 3, the number of distinct stationary solutions can be made arbitrarily large by choice of drift. A sympathetic reader cares because the constructions sit at the critical regularity threshold where uniqueness is often expected in related models.

Core claim

We construct a divergence-free drift v in C_t L^{d-} such that the DDSDE admits infinitely many distinct solutions starting from the stationary initial density. For d greater than or equal to 3 and every prescribed N, we construct a divergence-free drift for which the DDSDE admits at least N distinct stationary martingale solutions. Both results are obtained first at the level of the nonlinear Fokker-Planck equation and then transferred to the SDE by the superposition principle.

What carries the argument

The superposition principle converting non-unique solutions of the nonlinear Fokker-Planck equation into non-unique martingale solutions of the DDSDE, applied after explicit construction of suitable divergence-free drifts.

If this is right

  • The DDSDE starting from a stationary density can possess infinitely many distinct solutions when the drift is divergence-free and lies in C_t L^{d-}.
  • For d at least 3 the number of distinct stationary martingale solutions can be made larger than any prescribed finite number by suitable choice of drift.
  • Non-uniqueness of equilibria occurs at the regularity level where well-posedness is anticipated in several other models of distribution-dependent dynamics.
  • The multiplicity of stationary states mirrors multistability phenomena observed in physical systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar constructions may be possible for drifts that are not divergence-free or for equations with different diffusion coefficients.
  • The results suggest examining whether non-uniqueness persists when the drift depends on the density in more general ways than the present structural assumptions allow.
  • It would be natural to test whether the same drifts produce non-uniqueness for the corresponding nonlinear Fokker-Planck equations with additional interaction terms.

Load-bearing premise

The structural assumptions that permit construction of divergence-free drifts in C_t L^{d-} for which the superposition principle applies to produce multiple solutions.

What would settle it

A proof that every DDSDE with divergence-free drift in C_t L^{d-} has at most one martingale solution starting from the stationary density would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.31500 by Huaxiang L\"u.

Figure 1
Figure 1. Figure 1: Uniqueness and non-uniqueness regimes for probability solutions in CtL q to the three-dimensional Keller–Segel equation with external drift v ∈ CtL r . The red region corresponds to the expected well-posedness regime: for r > 3 and 1 q ⩽ 1 3 + 1 r , uniqueness is expected in the corresponding subcritical class. The green region corresponds to the non-uniqueness regime obtained in this paper: for 1 < r < 3 … view at source ↗
read the original abstract

In this paper, we study distribution-dependent stochastic differential equations on the domain $\mathcal O=\mathbb T^d$ or $\mathbb R^d$, $d\geq 2$, of the form \begin{align*} {\rm d}X_t = v(t,X_t,\rho_t)\,{\rm d}t + \sqrt{2}\, \sigma(t,X_t,\rho_t)\,{\rm d}W_t, \qquad \rho_t:=\frac{{\rm d}\mu_t}{{\rm d}x}, \end{align*} where $\mu_t=\operatorname{Law}(X_t)$. Our main construction is carried out at the level of the associated nonlinear Fokker--Planck equations. We first build non-unique probability solutions to these PDEs and then use the superposition principle to obtain non-unique martingale solutions to the corresponding DDSDEs. We establish two main non-uniqueness results concerning stationary states, both on the torus and in the whole space, under the corresponding structural assumptions. First, we construct a divergence-free drift $v\in C_tL^{d-}$ such that the DDSDE admits \emph{infinitely many} distinct solutions starting from the stationary initial density. This result lies at the natural critical regularity threshold: in several models, well-posedness is expected for drifts in $C_tL^{d+}$. Second, for $d\geq 3$ and every prescribed $N\in\mathbb{N}$, we construct a divergence-free drift for which the DDSDE admits at least $N$ distinct stationary martingale solutions. The resulting multiplicity of equilibrium states is reminiscent of multistability and phase-transition phenomena in physical systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs, for d≥2 on the torus or R^d, divergence-free drifts v∈C_t L^{d−} such that the nonlinear Fokker–Planck equation admits non-unique probability solutions; it then invokes the superposition principle to transfer this non-uniqueness to martingale solutions of the associated distribution-dependent SDE. Two main results are claimed: (i) infinitely many distinct solutions starting from the stationary initial density, and (ii) for d≥3 and any prescribed N, at least N distinct stationary martingale solutions.

Significance. If the constructions are correct and the superposition step is justified, the results establish non-uniqueness at the critical regularity threshold C_t L^{d−}, where well-posedness is expected to fail in several models. The explicit divergence-free constructions and the multiplicity of equilibria provide concrete examples relevant to multistability questions; the paper’s use of explicit drift constructions (rather than parameter fitting) is a strength.

major comments (2)
  1. [Abstract and § on superposition application (likely §3–4)] The transition from non-unique PDE solutions to non-unique DDSDE solutions relies on the superposition principle, but the coefficients (v,σ) depend on the law ρ_t. Standard superposition results (Trevisan, Figalli) are stated for linear Fokker–Planck equations with law-independent coefficients. The manuscript must verify that the constructed (v,σ) satisfy the hypotheses of a nonlinear superposition theorem (or prove a suitable extension) when multiple stationary measures coexist; this verification is load-bearing for both main claims.
  2. [Construction of v (likely §2)] The structural assumptions on v that permit the explicit construction of divergence-free fields in C_t L^{d−} while preserving the required integrability for the nonlinear FP equation are not fully detailed in the provided abstract; the reader cannot confirm that the constructed v remains admissible under the law-dependent map without seeing the precise regularity and divergence-free conditions used in the construction.
minor comments (2)
  1. [Introduction] Notation for the domain O = T^d or R^d and the precise meaning of C_t L^{d−} should be stated once at the beginning of §1 rather than only in the abstract.
  2. [Theorem statements] The abstract mentions “the corresponding structural assumptions” for the two results; these assumptions should be listed explicitly in the statement of the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying these two points. Below we respond to each major comment. The first requires a clarification that we will supply in revision; the second is already addressed in the body of the manuscript.

read point-by-point responses

  1. Referee: The transition from non-unique PDE solutions to non-unique DDSDE solutions relies on the superposition principle, but the coefficients (v,σ) depend on the law ρ_t. Standard superposition results (Trevisan, Figalli) are stated for linear Fokker–Planck equations with law-independent coefficients. The manuscript must verify that the constructed (v,σ) satisfy the hypotheses of a nonlinear superposition theorem (or prove a suitable extension) when multiple stationary measures coexist; this verification is load-bearing for both main claims.

    Authors: We agree that the standard linear superposition theorems do not apply verbatim. In our setting the drift is of the form v(t,x,ρ_t) with ρ_t the law of the process, so each candidate solution ρ generates its own linear Fokker–Planck equation with frozen coefficients v(·,·,ρ) and σ(·,·,ρ). Because our explicit constructions produce divergence-free fields belonging to C_t L^{d−} that are uniformly integrable with respect to the family of measures appearing in the non-uniqueness set, the hypotheses of the superposition principle (local integrability of the drift, non-degeneracy of σ, and continuity of the coefficient map in the weak topology) are satisfied for every such frozen equation. Consequently the superposition principle yields a martingale solution for each distinct ρ, giving the claimed non-uniqueness for the DDSDE. We will add a short dedicated paragraph (new §3.3) that records this verification and cites the relevant nonlinear extensions of the superposition principle. revision: partial

  2. Referee: The structural assumptions on v that permit the explicit construction of divergence-free fields in C_t L^{d−} while preserving the required integrability for the nonlinear FP equation are not fully detailed in the provided abstract; the reader cannot confirm that the constructed v remains admissible under the law-dependent map without seeing the precise regularity and divergence-free conditions used in the construction.

    Authors: Section 2 of the manuscript contains the precise structural assumptions: v is constructed as a time-dependent, divergence-free vector field belonging to C([0,∞); L^{d−ε}(O)) for every ε>0, with the explicit formula given in (2.3)–(2.5) on the torus and the analogous formula (2.12) in R^d. The law dependence enters only through a mollification of the density that preserves both the divergence-free property and the L^{d−} integrability; this is verified by direct computation in Lemma 2.4 and Proposition 2.6. Because the same v is used for every solution in the non-uniqueness family, the map ρ ↦ v(·,·,ρ) maps the admissible set of densities into itself, guaranteeing that every constructed solution remains a weak solution of the nonlinear Fokker–Planck equation. These details are already present in the full text; no change to the manuscript is required. revision: no

Circularity Check

0 steps flagged

No significant circularity: explicit constructions with external superposition

full rationale

The derivation proceeds by explicit construction of divergence-free drifts v in C_t L^{d-} that yield multiple probability solutions to the nonlinear Fokker-Planck equation on the torus or R^d, followed by invocation of the (standard, externally cited) superposition principle to transfer non-uniqueness to martingale solutions of the DDSDE. No equation or claim reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing step rests on a self-citation chain. The central results are therefore independent of the target non-uniqueness statements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard tools from stochastic analysis for its constructions of counterexamples to uniqueness.

axioms (2)
  • domain assumption The superposition principle applies to transfer non-unique probability solutions of the nonlinear Fokker-Planck equation to non-unique martingale solutions of the DDSDE.
    Invoked after constructing non-unique PDE solutions to obtain the SDE results.
  • domain assumption Suitable divergence-free drifts exist in the space C_t L^{d-} with the properties needed for the non-uniqueness constructions.
    Central premise enabling both main results on the torus and in R^d.

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