44 problems

Let $rho0: mathbb{R}rightarrow[0,1]$ be an initial density profile, and let ${nu^{epsilon}}{epsiloninmathbb{R}{0}}$ be probability measures on the configuration space…

Let $I_N$ be the lattice site set, let $\Omega_N$ be the configuration space, and let $\{\mu^N\}_{N\in\mathbb N}$ be probability measures on $\Omega_N$ satisfying…

Let $D(\rho)$ denote the diffusion coefficient of the Kob–Andersen model at density $\rho$, with parameters $k$ and $d$. In the case $k=d=2$, the high-density regime concerns…

Let $\beta\in(0,\frac12)$ parameterize an ${\cal O}(n^{-\beta})$ perturbation of a constant equilibrium profile in the hydrodynamic limit, with time rescaled by $n^{1+\beta}$ and s…

Seppäläinen's conjecture. Burgers' equation should govern the macroscopic propagation of disturbances of equilibria in this scaling regime for essentially all such interacting part…

Consider a particle system of branching Brownian motion with rank-based branching function $b:[0,1]\to[0,\infty)$ and increasing deletion function $D:[0,1]\to[0,1]$. Let $u$ denote…

Let $({\mathcal Y}_t^\varepsilon)_{t\ge 0}$ be the fluctuation field of the directed exclusion process, with $s_\varepsilon(t)$ determined by … The field … for test functions…

Let $u=(u_k)_{k\geq 0}$ denote the limiting particle-density profile, where $u_k=u_k(t,x)$ is the density of particles carrying exactly $k$ mutations at time $t>0$ and spatial posi…

Dirichlet hydrodynamic-limit conjecture. For $\theta<1$, the empirical density should be described in the hydrodynamic limit by this equation with Dirichlet boundary conditions.

Let $\mathcal{Q}^N$ be the law on $D([0,T],\mathcal{S}'(\mathbb{T}))$ of the fluctuation field $\{\mathscr{U}^N_t,t\in[0,T]\}_N$, with $N=2p$ for $p$ prime. For the stationary solu…

Let $p_1+\dots+p_N=1$, let $a_1,\ldots,a_N>0$, and for $s\geq 1/a_1$ let $X^{(s)}$ be the infinite bin model with move distribution…

Hydrodynamic-limit conjecture. The hydrodynamic limit is given by a diffusion equation with diffusion coefficient $D_\infty$.

Hydrodynamic-limit conjecture. The main result should remain valid for a fairly general class of initial distributions, beyond the supercritical ergodic configurations treated in t…

Let $u$ solve the hydrodynamic equation … with boundary parameters determined by the reservoir rates $(\alpha,\beta,\gamma,\delta)(t)$ through the relations defining $\nu_\pm(t)$.…

Let $(\xi_j)_{j\geq 1}$ be independent complex-valued random variables with distribution $\mu$, let $P_n(z)=\prod_{j=1}^n(z-\xi_j)$, and let $\mu\lsem P_n^{(k)}\rsem$ denote the em…

Reshuffling hydrodynamic-limit conjecture. Under these assumptions, the process $\phi^N_{\mathrm{RS}}(\cdot,\cdot)$ converges, in an appropriate sense, to the hydrodynamic model…

Let $\mathbf z_N(t)=(z_1(t),\dots,z_N(t))$ be the crystal surface process, and define … For $\boldsymbol\lambda_N(t)=(\lambda_1(t),\dots,\lambda_N(t))$, write…

Conjecture on the general-scaling derivation. The formal derivation toward the limiting system is wrong in general, because the part involving $f^{(1)}_\varepsilon$ in the expressi…

Conjecture on bounded domains with absorption. The results obtained on the torus should extend to this setting, yielding the derivation of the limiting Navier–Stokes equations, eit…

Let $T>0$, let $m$ be the porous-medium exponent, and let $g:{\mathbb T}\to[0,1]$ be measurable. For each $\kappa>0$, let…

Let a particle system have a general injection distribution, possibly injecting particles to the right of the free boundary, and let its hydrodynamic limit describe the macroscopic…

Let the $N$-server systems use the join-the-shortest-of-$d$-queues routing algorithm, and let their stationary distributions be denoted by $\pi_N$. Suppose that the hydrodynamic eq…

Let $g:[0,1]\to[0,1]$ be measurable, and consider the stationary solution obtained from the hydrodynamic limit in either of the regimes $\gamma\in(0,1)$ and $\theta>0$, or…

Let $\gamma\in(0,2)$ and let $\theta>0$ be sufficiently small. Consider the hydrodynamic equation for the boundary-driven super-diffusive symmetric exclusion process, with $\kappa$…

Let $Q^N$ be the law on $\mathcal D([0,T],\mathcal M_+)$ of the empirical process $\{\pi_t^N:t\in[0,T]\}$. Let $\rho:[0,T]\times[0,1]\to\mathbb R_+$ be a weak solution of the hydro…