28 problems

Mean-field equation conjecture. The dynamics of $\mu_t^*$ is given by $\mathcal L(X_0^*)=\lim_{N\rightarrow\infty}\mathcal L(X_0^{1,N})=\mu_0^*$, with $S_0^*$ distributed according…

The particle system is given by Systems and, while the corresponding kinetic model is given by Systems and. Mean-field convergence conjecture. The particle system converges to the…

Strong Schatten convergence conjecture. The convergence above is strong in every $\mathfrak{S}^p(\mathfrak{H}_k)$ with $p>1/s+1/2$.

Poisson Hypothesis. Spiking deliveries to distinct replicas shall be asymptotically distributed as independent Poisson (or compound) point processes.

Consider a closed network of $M$ servers and $N$ clients, with the ratio tending to a fixed value … For large $M$ and $N$, the network defines a complicated Markov process. Weak Po…

Let $(\zeta_i(k))_{i,k=1}^n$ be independent or identically distributed random variables, as appropriate, and let $V^n$ denote the optimal online vector-balancing cost. Let…

Lu–Lu–Nolen scaling conjecture. If the kernel is scaled proportionally to the number of particles as $N^\beta K(N^\beta\,\cdot)$, then the behavior of the particle system should re…

Rosenzweig's optimal scaling conjecture. This scaling assumption should be in general optimal: the incompressible Euler equation should not be the limiting evolution of the empiric…

Let $M$ be the state manifold, let $R>0$, let $v$ denote the velocity variable, and let ${\mathcal K}$ be the interaction kernel. Let $\mu_t^N$ be the empirical measure of the norm…

Let $F$ be the drift, $K$ the interaction kernel, and let $ut$ and $sigmat$ be the diffusion and common-noise coefficient matrices, with entries $nut^{ik}$ and $sigmat^{ik}$ for…

Rosenzweig's optimality conjecture. The scaling assumption above should be optimal in general: the incompressible Euler equation should not be the limiting evolution of the empiric…

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…

Macroscopic-scaling conjecture. In this case, there should exist $s>d$, a nontrivial measure $\mu$ on $\mathbb{R}^d$, and a sequence of minimizers $(X_N)_{N\geq 2}$ of $E_N$ such t…

Let $X_0^{i,N}$ denote the initial states of the neurons and let Assumption 4 be the condition specified in the paper for the initial data. The one-particle convergence conjecture.…

Let $\boldsymbol{x}_{i}^{(t)}$ evolve according to the finite-particle system defined by the Gaussian-Stein dynamics above, and let the corresponding mean-field limit be the associ…

Log-Sobolev constant limit conjecture. Under assumptions A1 and A2,

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…

Grazieschi et al.'s mean-field limit conjecture. The mean-field limit of this particle system should have the form of the limiting system obtained by taking $k=1$ and…

Consider a redundancy-$d$ system with $K$ servers, FCFS scheduling, exponentially distributed service times, and identical copies. Let $\bar\ell$ denote the quantity governing the…

For each species $j=1,cots,J$, let $\mu_t^{{\boldsymbol{\zeta}},j}$ denote its marginal distribution, and suppose the reaction network includes zeroth-order birth reactions or reac…

Let $v_k^n(t)$ be the random proportion of vertices at time $t$ that belong to clusters of size $k$ under the stationary law $\mathbb{P}_{\mathrm{stat}}^n$, and let…

The mean-field concentration PDE conjecture. The limiting concentration field is conjectured to satisfy

The mean-field weak PDE conjecture. The limiting measure $p_t$ is conjectured to satisfy

Let $N$ denote the scaling parameter for the network, let $W_c^{(N)}$ be the waiting time for class $c$, and let $\tilde W_c^{(N)}$ be its empirical counterpart. For complete inter…

Let $\theta$ be the bias parameter, $\theta<\text{?}$, and let $\theta$ and $\theta$ be the parameters appearing in the bias voter model on $\d^d$; for $p\a0in (0,1)$ and $t>0$, co…