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 ${\mathbb{R}}^d$, $d\ge 2$, 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 ${\mu }_t=\mathrm{Law}(X_t)$. 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 $v\in C_t{L}^{d-}$ 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 $C_t{L}^{d+}$. Second, for $d\ge 3$ 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.

arXiv comment62 pages