Let $\varphi$ be a normal positive linear functional on a von Neumann algebra $\mathcal{M}$. Let $p$ be the central support projection of $\varphi$. Define a linear functional $\psi$ on $p\mathcal{M}$ by $\psi(a)=\varphi(qa)$, where $q$ is a projection in the center of $\mathcal{M}$. Assume that ...

Suppose that $D$ is a subset of $\mathbb{R}^2$ with a smooth connected boundary $\partial D$. Let $U$ be an open subset of $\partial D$ and $x$ be a point in the interior of $D$. How do we know that the probability of exit of Brownian Motion, starting at $x$, through $U$ (first hitting time) is ...