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

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,ρ_t)\,{\rm d}t + \sqrt{2}\, σ(t,X_t,ρ_t)\,{\rm d}W_t, \qquad ρ_t:=\frac{{\rm d}μ_t}{{\rm d}x}, \end{align*} where $μ_t=\operatorname{Law}(X_t)$. 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.

• The paper establishes that nonlinear Fokker–Planck equations with drifts in critical low-regularity spaces lead to an infinite multiplicity of weak solutions via convex integration.
• It demonstrates that even small or singular perturbations in distribution-dependent SDEs can generate multiple stationary measures and phase transitions.
• The methodology innovatively uses space-time intermittent building blocks and hierarchical convex integration to precisely match known uniqueness thresholds.

Non-Uniqueness Phenomena in Nonlinear Fokker–Planck Equations and Distribution-Dependent SDEs

Problem Setting and Motivation

This work rigorously addresses the question of (non-)uniqueness for nonlinear Fokker–Planck equations (nFPEs) and their associated distribution-dependent stochastic differential equations (DDSDEs), both over the $d$-dimensional torus $\mathbb{T}^d$ and in $\mathbb{R}^d$, for $d \geq 2$. Such DDSDEs represent McKean–Vlasov-type SDEs, where both drift and diffusion may depend nonlinearly and nonlocally on the evolving law of the process itself. Formally, given a process $X_t$ with $\mu_t = \text{Law}(X_t)$, the governing SDE is

$$\mathrm{d}X_t = v(t, X_t, \rho_t) \,\mathrm{d}t + \sqrt{2}\,\sigma(t, X_t, \rho_t)\,\mathrm{d}W_t, \quad \rho_t := \frac{\mathrm{d}\mu_t}{\mathrm{d}x},$$

which semigroup-theoretically leads to a nonlinear Fokker–Planck equation for the density $\rho_t$ due to the law-dependence in $v, \sigma$.

The motivation lies in the mathematical and physical relevance of these models, which describe mean-field limits for interacting particle systems and arise in numerous applications, including fluid mechanics (e.g., 2D Navier–Stokes vorticity), chemotaxis (Keller–Segel), kinetic theory (Landau equation), porous media, and other systems exhibiting self-interaction, phase transitions, and multistability. A central and largely open issue is the fine boundary between well-posedness regimes and, conversely, the existence of non-uniqueness phenomena—either for the transient evolution or for the equilibrium states—especially for non-smooth coefficient regimes.

Main Results and Methodological Innovations

Sharp Non-Uniqueness in Weak Solutions and Stationary States

The paper establishes rigorous and essentially sharp non-uniqueness for both evolution and stationary problem settings for nFPEs and associated DDSDEs. The core contributions are as follows:

• Infinitely Many Evolutions from a Stationary State: For each $d \geq 2$ and in both periodic and whole-space settings, the authors construct divergence-free drifts $\mathbb{T}^d$0 with critical low regularity ($\mathbb{T}^d$1) such that infinitely many distinct evolution solutions arise, all starting from a stationary density—often the natural invariant measure (e.g., uniform on $\mathbb{T}^d$2, or a confining equilibrium density on $\mathbb{T}^d$3). Crucially, the uniqueness regime is known or conjectured to begin at $\mathbb{T}^d$4.
• Multiplicity of Stationary Measures (Phase Transitions): Similarly, for $\mathbb{T}^d$5, and for every prescribed $\mathbb{T}^d$6, there exist “bad” drifts producing at least $\mathbb{T}^d$7 distinct stationary probabilistic solutions, even for highly regular or small perturbations of otherwise unique models (e.g., periodic nonlinear diffusions or singular mean-field interactions).
• Extension to Singular Nonlocal Interactions: The construction adapts to singular mean-field kernels (e.g., Green's function for Keller–Segel or Coulomb-type for Landau equations), provided precise integrability and spatial regularity thresholds are observed.
• Analysis in Whole Space: The approach is made robust to settings without a canonical invariant measure by localizing the convex integration procedure using the Bogovskii operator, thereby producing compactly supported non-uniqueness anomalies in both evolution and stationary problems.

Technical Approach: Convex Integration for Infinite Systems

An essential methodological novelty lies in the adaptation of convex integration—a technique originally developed for non-uniqueness in finite systems (e.g., Euler, Navier–Stokes, continuity equations)—to infinite systems of coupled PDEs, as occurs when constructing infinitely many solutions for nFPEs. The main elements include:

• Hierarchical Convex Integration: At each iterative stage, only finitely many equations (corresponding to $\mathbb{T}^d$8 initial perturbations) are “fixed” while decaying bounds are enforced on the tail, ensuring the total perturbation series is convergent and stress errors remain controlled for every index.
• Space-Time Intermittent Building Blocks: The construction employs $\mathbb{T}^d$9-based intermittent jets with localized support, combining geometric rigidity (divergence-free, zero-mean spatial structure) and temporal localization for sharper regularity.
• Controlling Regularity Thresholds: The perturbations are designed to optimize regularity in both space and time, achieving explicit sharpness in the Lebesgue indices, precisely matching uniqueness thresholds known from the linear theory.

Rigorous Analytical Framework

The authors ensure all constructed densities are true probability solutions with physically meaningful properties: positivity, conservation of mass, appropriate integrability and regularity, and, in the stationary case, nontriviality and non-coincidence. For each case, they verify all solution properties and employ the superposition principle to lift PDE solutions back to pathwise (martingale) solutions of the underlying DDSDEs.

Numerical Thresholds and Claims

The key numerical findings are:

• Critical Drift Regularity for Non-Uniqueness: For $\mathbb{R}^d$0-dimensional settings, non-uniqueness arises as soon as $\mathbb{R}^d$1 enters $\mathbb{R}^d$2, and uniqueness is expected for better-integrated drifts ($\mathbb{R}^d$3).
• Arbitrarily Large Multiplicity: For any $\mathbb{R}^d$4 (and $\mathbb{R}^d$5), at least $\mathbb{R}^d$6 nontrivial stationary solutions can be constructed.
• Optimality: These thresholds and multiplicity properties are proven to be essentially sharp, matching known (or conjectured) well-posedness boundaries for both linear and nonlinear Fokker–Planck-type equations.

Implications and Future Directions

Theoretical Ramifications

This work provides a complete resolution to the existence of multiple solutions—both in time evolution and stationary regimes—for nFPEs and DDSDEs with critical and supercritical drifts, extending even to degenerate or strongly nonlinear/singular interaction mechanisms. It offers a direct analog in the mean-field/stochastic setting to the renowned non-uniqueness results for Euler and Navier–Stokes via convex integration, with sharp probabilistic interpretations (invariant measures, mean-field phase transitions, multiplicity of martingale solutions).

The construction generalizes Dawson’s classical “phase transition” behavior in McKean–Vlasov models to higher dimensions, nontrivial diffusion and singular interactions, and distribution-dependent noise (e.g., in the Landau kinetic regime). Physical interpretations include the breakdown of classical propagation of chaos, multistability, and the potential failure of the $\mathbb{R}^d$7-theorem.

Practical and Modeling Consequences

In practical terms, the sharpness and flexibility of the results indicate that caution must be exercised in the modeling and simulation of mean-field particle systems even for seemingly benign low-regularity perturbations. For instance, in high-dimensional population dynamics, turbulence, or anomalous diffusion, the introduction of small, rough perturbations—otherwise undetectable by regularity-based heuristics—can fundamentally alter the long-time and statistical behavior, generating a profusion of statistically distinct invariant or stationary regimes.

Prospects for Future Research

This framework is expected to be adaptable to even broader classes of equations, including degenerate or fractional diffusion operators, nonlinear drift dependence, or domains with boundaries (using the Bogovskii operator). Further, the method paves the way for addressing questions about ergodicity, selection principles (e.g., entropy minimality, statistical selection), and the impact of numerical discretization in particle and stochastic approximations.

It also raises new questions about the structure of attractors, long-time statistical properties, and the applicability of convex integration mechanisms in infinite-dimensional or controlled mean-field systems, potentially affecting stochastic control, mathematical finance, and large agent-based systems.

Conclusion

The paper rigorously demonstrates that weak non-uniqueness and multiplicity of stationary measures are pervasive phenomena for nonlinear Fokker–Planck equations and the associated law-dependent SDEs, as soon as the drift enters certain low-regularity ($\mathbb{R}^d$8) regimes—even under arbitrarily small perturbations. This outcome is achieved by extending and innovatively applying the convex integration method to infinite coupled PDE systems, accommodating general nonlinear and singular interaction structures.

The findings have significant theoretical implications for the understanding of stochastic mean-field dynamics, challenge assumptions in modeling and simulation frameworks, and open multiple directions for future research in the analysis of PDEs, stochastic processes, and statistical mechanics.

Reference:

"Non-Uniqueness for Nonlinear Fokker--Planck Equations and Their Associated Distribution-Dependent SDEs" (2606.31500)