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

This paper establishes the non-uniqueness of stationary solutions for distribution-dependent stochastic differential equations and their associated nonlinear Fokker-Planck equations on $\mathbb T^d$ or $\mathbb R^d$ ( $d\geq 2$ ) by constructing divergence-free drifts with critical regularity that yield infinitely many or arbitrarily many distinct equilibrium states.

Original authors: Huaxiang Lü

Published 2026-07-01

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CC BY 4.0

Original authors: Huaxiang Lü

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a giant, invisible crowd of people moving around in a city. In physics and math, we often use equations to predict how this crowd will behave. Usually, we assume that if we know exactly where everyone starts and how the wind (or external forces) is blowing, the crowd will move in one specific, predictable way. This is called "well-posedness."

This paper, written by Huaxiang Lü, challenges that assumption. It shows that for certain types of crowds (specifically, those where the movement of each person depends on the density of the whole crowd), the rules of predictability can break down. Even if you start with a perfectly calm, stationary crowd and apply a tiny, barely noticeable push, the crowd might split into many different possible futures instead of following just one path.

Here is a breakdown of the paper's main ideas using simple analogies:

1. The Setup: The Crowd and the Wind

The paper studies a mathematical model called a Distribution-Dependent Stochastic Differential Equation (DDSDE).

The Analogy: Think of a crowd of people walking in a park. Each person's movement is influenced by two things:
1. The Wind: An external force pushing them (like a breeze).
2. The Crowd: How crowded it is around them. If it's packed, they move differently than if it's empty.
The Equation: This is the "Nonlinear Fokker-Planck Equation." It's a complex recipe that tells us how the density of the crowd changes over time.

2. The Problem: The "Smooth" vs. The "Rough"

In the real world, wind is usually smooth. In math, we prefer "smooth" forces because they make the equations easy to solve, and we get a unique answer (one specific future).

However, the author asks: What if the wind is "rough"?

The Metaphor: Imagine the wind isn't a gentle breeze but a chaotic, jagged gust that changes direction instantly and erratically. In math terms, this is a "low regularity" or "rough" drift.
The Discovery: The paper proves that if this rough wind is just a tiny bit "jagged" (specifically, in a mathematical category called $L^d-$ ), the system loses its ability to predict a single future.

3. The Main Result: One Start, Many Endings

The paper's biggest claim is Non-Uniqueness.

The Scenario: Imagine the crowd is standing perfectly still (a "stationary state"). The wind is calm.
The Twist: The author introduces a tiny, invisible, rough wind that is so small you can barely measure it.
The Outcome: Instead of the crowd staying still or moving in one direction, the math shows that infinitely many different scenarios can happen from that exact same starting point.
- In one scenario, the crowd swirls left.
- In another, they swirl right.
- In a third, they split into two groups.
- All of these are mathematically valid solutions to the same equation with the same starting conditions.

4. The Method: "Convex Integration" (The Architect's Trick)

How did the author prove this? They used a technique called Convex Integration.

The Analogy: Imagine you are an architect trying to build a tower that looks like a straight line from a distance, but up close, it's actually made of thousands of tiny, jagged zig-zags.
The Process:
1. Start with a simple, smooth solution (the straight line).
2. Add a tiny, high-frequency "wiggle" (a rough perturbation) to the wind.
3. This wiggle creates an error in the prediction.
4. The author then adds another wiggle to cancel out that error, but this new wiggle creates a new tiny error.
5. They repeat this process infinitely many times.
The Magic: By carefully controlling the size and frequency of these wiggles, they can make the final result look like a valid solution, but because they added so many different "wiggles," they can construct many different valid solutions from the same starting point.

5. Two Specific Findings

The paper presents two main types of chaos:

Finding A: Infinite Evolutions from a Stationary State
Even if the crowd starts perfectly still, a tiny rough wind can cause the system to evolve into infinitely many different patterns. This happens at a "critical threshold"—meaning the wind is just rough enough to break the rules, but not so rough that the math breaks completely.
Finding B: Multiple Stationary States (Multistability)
The paper also shows that a system can have multiple different "resting states."
- The Analogy: Think of a ball in a landscape. Usually, a ball rolls to the bottom of a single valley (one stable state).
- The Result: The author shows that with the right rough wind, you can create a landscape with many valleys. The ball could settle in any of them, and the system has no way of "knowing" which one it should pick. This mimics phase transitions in physics (like water turning into ice or steam), where a system can exist in multiple stable forms.

6. Where This Applies

The author tested this on two types of "cities":

The Torus (A Donut Shape): A finite, repeating space (like a video game world where you walk off the left edge and appear on the right).
The Whole Space (The Infinite Plane): An open, infinite world.

In both cases, the result holds: a tiny, rough disturbance destroys the uniqueness of the solution.

Summary

In simple terms, this paper says: "If you have a crowd of interacting particles and you push them with a very specific kind of rough, jagged force, you cannot predict their future. They could go in infinitely many different directions, or settle into many different resting positions, even if they all started in the exact same place."

This is a fundamental discovery about the limits of predictability in complex systems, showing that "smoothness" is often the only thing keeping our mathematical models from falling apart into chaos.

Technical Summary: Non-Uniqueness for Nonlinear Fokker–Planck Equations and Their Associated Distribution-Dependent SDEs

Problem Statement
The paper investigates the well-posedness of distribution-dependent stochastic differential equations (DDSDEs) and their associated nonlinear Fokker–Planck equations (nFPEs). Specifically, it addresses the phenomenon of non-uniqueness in law for these systems. While classical theory establishes well-posedness for DDSDEs with sufficiently regular coefficients (e.g., drifts in $L^r_t L^p_x$ with $\frac{d}{p} + \frac{2}{r} < 1$ ), this work explores the critical and supercritical regimes where the drift is rough. The central question is whether an arbitrarily small, sufficiently rough, divergence-free perturbation to the drift field can destroy the uniqueness of solutions, both for time-evolution starting from a stationary state and for the stationary states themselves.

The study considers two primary settings:

The Periodic Setting: The domain is the torus $\mathbb{T}^d$ ( $d \ge 2$ ).
The Whole-Space Setting: The domain is $\mathbb{R}^d$ ( $d \ge 2$ ), focusing on equations with confining potentials and nonlinear diffusion.

The equations under consideration take the general form:
$dX_t = v(t, X_t) dt + (V * \rho_t)(X_t) dt + \sqrt{2}\sigma(t, X_t, \rho_t) dW_t,$
where $\rho_t$ is the probability density of $X_t$ , $v$ is an external drift, $V$ represents interaction kernels, and $\sigma$ is a distribution-dependent diffusion coefficient. The associated nFPE is:
$\partial_t \rho - \text{div}\text{div}((\sigma\sigma^T)(\rho)\rho) + \text{div}(v\rho) + \text{div}((V*\rho)\rho) = 0.$

Methodology
The primary tool employed is the convex integration method, originally developed for fluid dynamics (e.g., the Euler and Navier-Stokes equations) to construct wild solutions. The authors adapt this technique to the context of nonlinear, distribution-dependent parabolic equations.

Key technical innovations and adaptations include:

Handling Nonlinearity: Unlike linear Fokker–Planck equations where convex combinations of solutions remain solutions, the nFPE is genuinely nonlinear. The authors cannot simply superimpose solutions. Instead, they construct infinitely many distinct solutions by iteratively eliminating stress terms in a system of coupled equations.
Infinite System Construction: To generate infinitely many solutions (Theorem 1.3), the authors construct an increasing sequence of integers $\{N_q\}$ and apply the convex integration scheme only to the first $N_{q+1}$ equations at step $q+1$ . They introduce a decay factor $2^{-i}$ for the stress terms of equations with indices $i > N_{q+1}$ to ensure the total perturbation converges while maintaining the necessary estimates for the active equations.
Building Blocks: The construction utilizes specific "intermittent spatial-time jets" (in the time-dependent case) and "Mikado-type flows" (in the stationary case). These building blocks are designed to have disjoint supports and specific oscillatory properties to cancel out error terms (oscillation, nonlinear, linear, and commutator errors) while controlling the regularity of the drift.
Regularity Thresholds: The construction targets drifts in the space $C_t L^{d-}$ (continuous in time, $L^p$ in space with $p < d$ ), which lies at the natural critical regularity threshold. The paper demonstrates that well-posedness fails in any neighborhood of a smooth drift within this low-regularity topology.
Whole-Space Adaptation: For the $\mathbb{R}^d$ case, the authors utilize the Bogovskii operator to handle the inverse divergence problem while preserving compact support. This allows the construction to be localized to a fixed domain, ensuring the perturbation does not affect the behavior of the system at infinity or the confining potential.

Key Contributions and Results

Non-Uniqueness of Time Evolution (Theorem 1.3 & 1.6):
- The authors prove that for $d \ge 2$ , there exist divergence-free drift fields $v \in C_t L^{d-}$ (arbitrarily close to a smooth drift) such that the nFPE admits infinitely many distinct probability density solutions starting from the same stationary initial density (e.g., the uniform measure on the torus or a specific stationary state in $\mathbb{R}^d$ ).
- By the superposition principle, this implies the existence of infinitely many distinct martingale solutions to the corresponding DDSDE.
- This result is shown to be essentially sharp, as well-posedness is expected for drifts in $C_t L^{d+}$ .
Non-Uniqueness of Stationary States (Theorem 1.5 & 1.7):
- For $d \ge 3$ , the paper constructs divergence-free drifts for which the stationary nFPE admits at least $N$ distinct stationary probability density solutions for any prescribed $N \in \mathbb{N}$ .
- This multiplicity of equilibrium states is analogous to multistability and phase-transition phenomena in physical systems.
- The result applies to both regular interaction kernels and more singular cases (e.g., Green function interactions where $V \sim |x|^{1-d}$ ).
Extension to Singular Interactions (Theorem 1.4):
- The non-uniqueness mechanism is extended to cases with more singular interaction kernels (e.g., $V = \text{div } b$ with $b \in L^{\frac{d}{d-\gamma}, \infty}$ ).
- In this regime, the authors obtain $N$ distinct solutions with improved time regularity estimates ( $C_t$ bounds) to justify the nonlinear interaction terms involving singular kernels.
Specific Physical Models:
The framework is shown to apply to a broad class of models, including:
- Distribution-independent SDEs with rough drifts.
- Convolution-type McKean–Vlasov equations.
- Porous-medium type equations.
- Keller–Segel equations (chemotaxis).
- 2D vorticity Navier–Stokes equations.
- Landau-type equations.

Significance and Claims
The paper claims to establish the first non-uniqueness results for a broad class of nonlinear and distribution-dependent Fokker–Planck equations. The authors emphasize that the construction of infinitely many solutions is fundamentally different from the linear case, as convex combinations of solutions are not generally solutions in the nonlinear setting.

The results highlight the fragility of well-posedness in the presence of rough drifts, even when the drift is divergence-free and arbitrarily small in low-regularity norms. The findings suggest that the critical regularity threshold for uniqueness in these systems is sharp. Furthermore, the construction of multiple stationary states provides a mathematical mechanism for multistability in mean-field systems with rough external fields, extending the understanding of phase transitions beyond classical models with constant diffusion.

The work does not propose new experimental applications but rather provides a rigorous theoretical counterexample to the uniqueness conjecture in the critical regularity regime for a wide class of interacting particle system limits.