8 problems

Continuity conjecture. The map

Let $W$ be an interaction potential satisfying Assumption, and let $k_{\min}=\operatorname*{arg\,min}_{k\in\mathbb N}\hat W_k$. The single-component resonance condition is … where…

Let $\bm X^P$ be the continuous-time particle-system approximation and let $Z(t)$ be the true solution of the corresponding McKean–Vlasov equation. Let…

Let $g:\mathbb{R}\to\mathbb{R}$ be smooth with bounded derivatives, let $G^P(x)=\frac{1}{P}\sum_{j=1}^P g(x_j)$, and let $\bm X_N^P$ be a realization of the particle system with co…

Let $(\mathcal{V}^i_t,\mathcal{W}^i_t,\mathcal{X}_0^i)$, $1\leq i\leq \mathsf{N}+1$, be independent copies of $(\mathscr{V}_t,\mathscr{W}_t,\mathscr{X}_0)$, and let…

Let $N\in\mathbb{N}$ and let $(L^N(t))_{t\in[0,\infty)}$ be the empirical distribution process of the mean-field interacting diffusions, defined by … The process…

Numerical-scheme convergence conjecture. This algorithm gives the correct approximation in the limit as the mesh sizes vanish.

Global uniqueness conjecture. Solutions satisfying the physical jump condition are unique. Between jump times, the solution is continuously differentiable, and for every $t_0>0$ wi…