245 problems

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_{\Omega}$. For $z_0\in\Omega$, let $c_{\beta}(z_0)$ denote its logarithmic capacity and let…

A Lévy process is a stochastic process with stationary and independent increments, starting at zero, and stochastically continuous. Hunt's hypothesis (H) is the assertion that ever…

Let $D$ be a bounded body in $\mathbb{R}^d$, and consider the Dirichlet problem with polynomial boundary data on $\partial D$. A polynomial is polynomial boundary data if it is the…

Widom's conjecture. The corresponding Szegő–Widom asymptotics holds for such sets, with a single arc behaving like an interval.

A Hunt process is a stochastic process with the strong Markov and regularity properties assumed in probabilistic potential theory. A set is semi-polar if it is contained in a count…

For $n\geq 3$, let $b_n>0$ be a universal constant such that every domain $\Omega\subset\mathbb{R}^n$ satisfies $\dim w_\Omega\leq n-b_n$, where $w_\Omega$ denotes harmonic measure…

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_\Omega$, let $u$ be harmonic on $\Omega$, and set $\rho=e^{-2u}$. Define $B_{\Omega,\rho}$ from an…

Let $P$ be a critical operator in a Riemannian manifold $\mathcal{M}$, and let $\varphi$ denote its ground state. Let $\mathcal{H}_P(\mathcal{M}\times\mathbb{R})$ be the cone of po…

Let a sequence of orthogonal polynomials be defined using a weight function on a contour connecting the endpoints of the orthogonality interval, and suppose that the weight functio…

Let $h^{\infty}$ denote the space of bounded harmonic functions on the unit ball $\mathbb{B}^n:=\{x\in\mathbb{R}^n:|x|<1\}$. For $x\in\mathbb{B}^n$ and $l\in\mathbb{S}^{n-1}$, let…

Let $A$ be a finite subset of $\mathbb{C}$, and let $f\in\mathcal{A}(\overline{\mathbb{C}}\setminus A)$ be a germ analytically continuable along every path in…

Let $K\subset\mathbb{C}$ be compact and connected with more than one point. Let $M_K$ be the smallest number such that, whenever $p$ has degree $n\geq1$ and $p=p_1\cdots p_m$, … He…

Let $\Sigma_0$ be the class of univalent meromorphic functions $F$ in the exterior of the unit disk, normalized by $F(z)=z+O(z^{-1})$ as $z\to\infty$. Let $K=F(\mathbb{B}^c)^c$ con…

Let $K\subset\mathbb{C}$ be compact and connected, with logarithmic capacity $\operatorname{cap}(K)=1$, $0\in K$, and conformal centroid at the origin. Let $\mu_K$ be the equilibri…

Let $K\subset\mathbb{C}$ be compact and connected, with logarithmic capacity $\operatorname{cap}(K)=1$, $0\in K$, and conformal centroid at the origin. Let $g$ be the Green's funct…

Let $D$ be a domain, let $0$ be a point whose accessibility from $D$ is considered, and consider the corresponding conditional symmetric $\alpha$-stable Lévy process at its lifetim…

Let $D$ be a domain, let $(X_t)$ be a Markov process of jump type, and let the representation formula referred to in the source be denoted by … generalizes to a wide class of Marko…

Let $I_{k,m}$ be horizontal intervals and $J_{k,m}$ the corresponding slits in the construction of the nowhere dense set $E\subset\mathbb R$. The Green function of…

Complex-charge equilibrium zero conjecture. The meromorphic function

Let $z_i$, $1\le i\le n$, be distinct points in $\mathbb{C}$ and let $a_i>0$, $1\le i\le n$, satisfy $\sum_{i=1}^{n}a_i=1$, where $n\ge 2$. Let $w_j$, $1\le j\le n-1$, be the zeros…

Let $\Gamma$ be a metrized graph, with total length $\ell(\Gamma)$ and tau constant $\tau(\Gamma)$. The ratio $\tau(\Gamma)/\ell(\Gamma)$ is invariant under scaling all edge length…

Let $(\Omega,S,B,\mu)$ be a tree-indexed process, and let $A(\Gamma;B)$ be the event that there is a witness in $\partial\Gamma$. Let $x(\omega)$ be any random element of…

Let $\Gamma$ be an infinite tree with root $\rho$. Consider a target percolation, and write … for the probability that the vertex $\sigma$ is connected to the root. Let…

Euclidean harmonic-measure dimension conjecture. The dimension of harmonic measure in $\mathbb{R}^n$ is conjectured not to exceed

Let $E\subset\mathbb{C}$ be a bounded non-degenerate continuum, and let $M_E$ denote the best constant associated with the norm inequality for polynomials. For the closed unit disk…