22 problems

Let the Landau equation be the kinetic equation under consideration, and say that a solution decays to equilibrium slower than exponentially if its convergence to equilibrium is no…

Continuous-transfer-operator conjecture. The continuous version of $\widetilde T$ is, more precisely,

Let $f(t,x,p)$ solve the Fokker–Planck equation under consideration, with global equilibrium … where $M(p)=\frac{1}{(2\pi)^{d/2}}e^{-|p|^2/2}$ and $\beta_{\infty}$ is determined by…

Let $f_\infty(|{\bf x}|)$ be an isotropic probability density on ${\mathcal I}=\{ |{\bf x}|\leq i_+\}$, with $f_\infty>0$, and define the diffusion coefficient … For a constant $c$…

Convex-potential form conjecture. An optimal map should be of the form

Let $L$ have local minima $x_1,\dots,x_M$, and let $u_i$ be the solution of the localized equation around the local minimum $x_i$, for $1\leq i\leq M$. Let $\rho(t,x)$ denote the s…

Let $b$ and $\sigma$ be the drift and diffusion coefficients of the degenerate stochastic differential equation, and let $a:=\sigma\sigma^*/2$. For an initial measure $\nu$, consid…

Singularity conjecture for $\mathcal F_n$. For each $n\in\mathbb N$, the equation $\mathcal F_n$ is singular within the class $\mathcal U_n$ and has the properties asserted in the…

Let $J$ be a kernel that decays sufficiently fast, and let $F_\varepsilon$ denote the equilibrium of the corresponding nonlocal Fokker–Planck equation. Poisson-like tail conjecture…

Trend-to-equilibrium conjecture. The entropy $H[f(t)]$ decreases to its minimum, zero, as $t\to\infty$, and the solution converges to equilibrium, $f(t)\to f_{\infty}$, as…

Let $d$ be the spatial dimension and let $\gamma>0$ be the parameter in the Fokker–Planck equation. The regime considered in the paper is … The preceding discussion concerns the co…

Exponential-smallness conjecture. The coefficient $C^{\epsilon}$ has the form

Exponential-tail conjecture. When $x$ or $y$ is large, the solution $P(x,y)$ may behave like the exponential of a polynomial and decay exponentially; specifically,

Power-decay threshold conjecture. The power-like decay of the steady-state distribution appears immediately when $B$ exceeds $\sqrt{3}$.

Let $\rho_\text{fast}^{\delta,\gamma,p}$ denote the fast-regime distribution and let $\rho_0^{\delta}$ denote the corresponding free-mRNA distribution, with $\mathrm{CV}$ their coe…

Let the assumptions for the Fokker–Planck operator hold, and let an orthogonal decomposition based on such an operator be given. For smooth functions…

Let the moduli space of rank $n$ semistable lattices carry its universal family of tori, and let the associated infinite-dimensional Hilbert bundle have the smooth sections paramet…

Regularity conjecture. If the forcing $\vec{F}$ is nonzero but sufficiently regular, then the same regularity estimate holds as in the case $\vec{F}\equiv\vec{0}$; in particular, o…

Let $\widetilde{u}(t,z)$ solve the Ornstein–Uhlenbeck Fokker–Planck equation … obtained by replacing the coefficients in the Fokker–Planck equation (FP) with their linear approxima…

Regularity conjecture. The family $\mu_\epsilon$ is regular with respect to $\mathcal{A}$ for a much larger class of systems.

Finite-time blow-up conjecture. Solutions with $m\leq m_c$ should be globally defined for all $t\in[0,\infty)$, whereas solutions with $m>m_c$ should blow up at a finite time $T^*$…

Let $H_h$ be the operator describing one Euler–Maruyama step of size $h$ for the discretization of the discontinuous-drift equation, and let $f_h$ be a stationary density satisfyin…