Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 203 / 4 / d

The statement is false; the displayed probability is zero. Let $\tau$ be the first exit from the fixed domain $\mathbb{D}$ and choose $x_n>0$ tending to zero. The Bessel scaling in part b givesHence $T_{x_n}\to 0$ in probability, while continuity of the trace gives $\tau >0$ almost surely. ThereforeSwallowing $x_n$ before leaving $\mathbb{D}$ forces a boundary contact away from zero before $\tau$. Taking the limit shows that such a contact occurs almost surely, so