Global uniqueness conjecture for physical solutions of the McKean–Vlasov equation

Let $(\nu_t,L_t)_{t\geq 0}$ be a solution of the McKean–Vlasov equation in

$$satisfying the physical jump condition$$

. A jump time $t_0$ is a time with $\Delta L_{t_0}>0$.

Global uniqueness conjecture. Solutions satisfying the physical jump condition are unique. Between jump times, the solution is continuously differentiable, and for every $t_0>0$ with $\Delta L_{t_0}>0$,

$$L'_{t_0+h}=O(|h|^{-1/2})\quad\text{as }h\to 0.$$

Thus the conjecture predicts square-root singularities immediately before and after jumps.

The conjecture concerns global well-posedness and regularity of physical solutions, including solutions restarted after blow-ups from initial laws whose densities need not vanish at the origin. The source states that it is currently far from being proved, with the main obstruction arising from such non-vanishing post-jump initial densities.