14 problems

The delayed Poissonian mean-field model is an interacting-particle system with Poisson-like attributes, and its corresponding mean-field PDE is intended to describe the large-parti…

Poisson Hypothesis. As $M,N\to\infty$, every fixed $k$-tuple of servers becomes asymptotically mutually independent and identically distributed; the flow $\mathcal{F}_{M,N}$ to a g…

Uniform LSI conjecture. Under suitable additional assumptions, uniqueness of the stationary solution of the mean-field limit, equivalently uniqueness of the solution of the associa…

Let $(x_k^1,\dots,x_k^N)$ be the GA particle system, and let $(\overline{x}_k^1,\dots,\overline{x}_k^N)$ be $N$ independent copies of the mono-particle process. Coupling conjecture…

Cubic-term removal conjecture. The cubic term

Let $\phi\colon \mathbf R^d\to\mathbf R$ be such that $\operatorname{e}^{-\phi}$ is integrable, and let $\overline X_t$ denote the limiting particle process with law…

Limiting log-Sobolev constant conjecture. Under assumptions A1 and A2, the limit of the particle-system log-Sobolev constants is uniquely characterised by the mean-field dissipatio…

Let $N$ be the number of neurons and let $\mathfrak{Dim}$ denote the dimension parameter governing the model's technical difficulty. Assume that the quantity $L$ from the source re…

Detailed-chaos propagation conjecture. The Kac process on $S_{n,\epsilon_n}$ propagates detailed chaos according to Definition$2$.

Exponential-gap propagation conjecture. The process defined here propagates chaos with exponential gap distribution.

Exchangeability conjecture. The convergence result for the empirical measures should still hold if the independence assumption on the initial data is replaced by exchangeability of…

Assume that the collision kernel is the hard-sphere kernel … for $(v,v_{*})\in\mathbb R^{3}\times\mathbb R^{3}$ and $\omega\in\mathbb S^{2}$, and let $\mathcal{Q}$ be the correspon…

Relaxation conjecture. Condition $(\text{S})$ is purely technical and can be relaxed when the signal-drift matrix $A$ is stable.

Optimal convergence-speed conjecture. The fastest possible convergence has speed $1/\sqrt{N}$, achieved when