11 problems

A planar Brownian path has local pivoting points of angle $\alpha$ when two portions of the path meet locally at such an angle; let the maximal angle be the supremum of the angles…

Let $V$ and $W$ be complementary Jordan domains, meaning that they share the same boundary. A domain is informally called "small" or "large" according to the finiteness or infinitu…

For two independent planar Brownian motions $X$ and $X'$, let $\theta_X(z)$ and $\theta_{X'}(z)$ denote their winding numbers around $z$, and define … Let $\ell$ be the intersectio…

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…

Let $A_a$ be the set of $a$-thick points of planar Brownian motion. The known lower bound is $\dim(A_a)\geq 2-a$ almost surely for $a\in(0,1/2)$. Bass, Burdzy and Khoshnevisan's th…

Let $D$ be a planar domain, let $x_0\in D$, and let $\nu_\varepsilon^\gamma$ denote the regularised random measure associated with the local times of planar Brownian motion started…

Common critical parameter conjecture. The two constants coincide,

Asymptotic expansion conjecture. As $r\to\infty$,