Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 2 / b

The function $z\mapsto \log |z|$ is harmonic on the annulus $\epsilon <|z|<R$. Hence $\log |B_{t\wedge {\tau }_{\epsilon }\wedge {\tau }_R}|$ is a bounded martingale by Itô formula. Optional stopping givesSolving yields

For fixed $y\ne x$, part (i), translated by $y$, and then $\epsilon \downarrow 0$ show that planar Brownian motion has probability zero of ever hitting $y$ before leaving any fixed large disk. Letting the disk radius tend to infinity shows ${\mathbb{P}}_x(y\in B([0,1]))=0$. By Tonelli theorem,The nonnegative random area is therefore zero almost surely.