Non-Uniqueness for Nonlinear Fokker--Planck Equations and Their Associated Distribution-Dependent SDEs
A talk in the Bielefeld Stochastic Afternoon series by
Huaxiang Lv
| 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*}
\mathrm{d}X_t % statt {\rm d}X_t
=
v(t,X_t,\rho_t)\,\mathrm{d}t % statt {\rm d}t
+
\sqrt{2}\,
\sigma(t,X_t,\rho_t)\,\mathrm{d}W_t, % statt {\rm d}W_t
\qquad
\rho_t:=\frac{\mathrm{d}\mu_t}{\mathrm{d}x}, % statt {\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 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. Within the CRC this talk is associated to the project(s): B1 |