41 problems

Let $D_\lambda$ denote the diffusion coefficient measured at scale length $\lambda$ in turbulent air. Richardson's conjecture. The diffusion coefficient $D_\lambda$ depends on the…

An Einstein relation concerns the response of a reversible random motion to a small bias and the variance of its unbiased diffusive scaling limit. Einstein relation conjecture. The…

Algebraic-tail conjecture. There is an algebraic tail in space after the sound wave.

Let $d\geq 1$, let $U\in L^\infty(\mathbb{T}^d)$, and let $p(t,x,y)$ be the heat kernel associated with the periodic diffusion described by the Dirichlet form in the source. Let…

Let the asymmetric simple exclusion process be the Markov process on $\{0,1\}^{\mathbb Z^d}$ in which particles jump with asymmetric rates, subject to the exclusion rule that each…

Diffusion-pole conjecture. One has

Let $d\geq 2$ and let $\beta>0$. Consider a translation-invariant Coulomb interacting Brownian motion in $d$ dimensions, and denote its self-diffusion coefficient by…

Let $N$ denote the truncation size used in the Lanczos-based approximation of the diffusion constant. Convergence-rate conjecture. The rate of convergence goes as … The conjecture…

Let $\mathcal{L}_{\mathcal{D}^{\star}}$ be the operator associated with the optimal diffusion coefficient in a one-dimensional setting, and let…

Let $G=(V,E,w)$ be a graph with features given by a map $f:V\to M$ into a Riemannian manifold, and suppose the graph's features have a convex bounding ball. Convergence conjecture.…

Under the assumptions of the invariance principle, let $\sigma^2$ be the diffusion coefficient in the stationary-process invariance principle. Positivity conjecture. The diffusion…

Let $\sigma^2$ be the diffusion coefficient appearing in the invariance principle for a simple symmetric random walk under the path measure $P_t^\gamma$ among a Poisson system of i…

Let $\sigma_N$ be the cover time for $N$ searchers, let $\mathbb{E}[\sigma_1]$ be the mean cover time for a single searcher, and let $L$ and $D$ denote the characteristic length an…

Let $m\ge2$ small spherical targets have radii $\varepsilon r_0,\dots,\varepsilon r_{m-1}$ in a bounded three-dimensional reflecting domain, and let $K_N$ be the index of the targe…

Let $M\subset\mathbb{R}^3$ be a bounded domain with reflecting boundary, and let $V_k$ be small spherical targets centered at $v_k$ with radii $\varepsilon r_k$, where the starting…

Consider diffusion with diffusivity $D>0$ between an inner spherical target of radius $R_0$ and an outer spherical target of radius $R_1>R_0$, with starting radius $\|X(0)\|$ satis…

Let $F_1(t)$ denote the probability density or hitting-time contribution associated with the target $V_1$ in the one-dimensional diffusion problem. Let $L_1$ be the starting distan…

The diffusion coefficient $D(t)$ describes the growth of the mean square displacement through $\mathbf E[|X(t)|^2]$ of order $tD(t)$. In the analogous self-interacting diffusions,…

Consider the free boundary position $s(t)$ in the free boundary model for the diffusion of solvent molecules into EPDM rubber, with the numerical instance corresponding to Figure (…

Let $s(t)$ denote the penetration front at time $t$, and let $\gamma$ be a positive number such that the front behaves like $t^{\gamma}$ for sufficiently large $t$. Under suitable…

Diffusion-limit conjecture. The limit $\xi=(\xi_1,\xi_2)$ is a Gaussian random variable with zero mean, with

Anderson's conjecture. Every initial condition can be almost surely decomposed into two parts: a low-energy part that remains dynamically localized and a bulk-energy part that prop…

Ma's conjecture. As $\lambda\to\infty$, where $\lambda$ is the mortality parameter, the conditional mean first passage times satisfy

Yuste et al.'s conjecture. The mean fastest first passage time has the approximation

Weiss–Shuler–Lindenberg conjecture. The expected $k$-th extreme first-passage time satisfies