12 problems

Let $U$ and $R$ be domains in the setting of Theorem, let $z$ be the specified starting point, and write $\tau_U$, $\tau_R$, $\tau_{U^+}$, and $\tau_{R^+}$ for the corresponding fi…

Exit-time conjecture.

Davis's conjecture. For all $t>0$,

Polynomial tail conjecture. For every $p>{\rm H}(W)$, there exists a constant $C>0$ such that

Let $U,W\neq\mathbb C$ be simply connected planar domains containing $0$, let $T^V$ be the first exit time of planar Brownian motion from $V$, and define … A domain $W$ is spiral-l…

Long-stay limit conjecture. If ${\rm H}(U)>{\rm H}(W)$, then

Let $Z_t=X_t+iY_t$, $t\geq 0$, be standard planar Brownian motion starting at the origin, and let $T^D$ denote its first exit time from a domain $D$ containing $0$. Let…

For $epsilon>0$, let $X^\epsilon$ solve … in $\mathbb{R}^d$, where $0$ is an unstable critical point of $b$, $\lambda_1>0$ is the leading simple eigenvalue of $Db(0)$, and…

Let a perturbed multi-strip ${\Theta}$ satisfy Assumption A, and let $u(x,t)$ have the representation … where ${\mathcal A}=\int_{{\Theta}}v_0$, and where…

Dimension-only constant conjecture. In the case of isotropic Lévy processes, the lower-bound constant $c$ can always be chosen to depend only on $d$, rather than on $r$ or $X$. Thi…

Let ${\cal S}$ be the class of normalized univalent functions on the unit disk, let ${\cal K}$ denote the Koebe functions, and let $K$ be the Koebe function. For $f\in {\cal S}$, w…

Let $\tau_x$ be the exit time of the ordered region for three random walks started from an ordered point $x$. The universal tail conjecture. The tail of the distribution of…