9 problems

Let $\mathcal{A}(M,P,N)$ be the single-level estimator, and let $\bm X_N^P$ denote the particle-system realization. For $M$ samples and $P$ particles, variance-decay conjecture. Th…

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 $N\geq 1$, let $\tilde{X}_N$ be the random number of particles remaining in the physical perfectly plastic collision process, and let $\tilde{Z}_N$ be the corresponding random…

Let $N\geq 1$ be the initial number of particles, and let $\tilde{X}_N$ denote the random number of particles remaining after the perfectly plastic collision process. A random vari…

Tableau-process weight conjecture. This weight is proportional to the weight derived from the tableau process of Ayyer et al.

Let $f^n(\cdot,t)$ denote the empirical distribution of the locations of an $n$-particle stochastic system, and let $f^n_*(\cdot,t)$ be its recentered version whose median is at…

Uniform particle-distance conjecture. There exist $T>0$ and $a<b$, depending only on $W$, $\phi_0^x$, and $\phi_0^\tau$, such that

Let $\boldsymbol{X}(n)$ be the particle configuration at time $n$, let $\boldsymbol{0}$ denote the densely packed initial configuration, and let…