Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 5

Let $D\subset {\mathbb{R}}^d$ be bounded, choose $R$ with $D\subset B(0,R)$, and let $T$ and ${\tau }_R$ be the respective exit times. Then $T\le {\tau }_R$. Sinceis a martingale, the optional sampling theorem for a supermartingale at ${\tau }_R\wedge n$ givesMonotone convergence theorem now gives

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For $u\in C^2(\overline{D})$, Itô formula shows thatis a martingale. Take expectations and let $t\to \infty$. The function $u$ is bounded on the compact set $\overline{D}$, while $\Delta u$ is bounded and ${\mathbb{E}}_{x}T<\infty$ by part (a). The dominated convergence theorem therefore gives Dynkin formula for Brownian motion

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Apply part (b) to $u_n$. Its zero boundary value givesThe assumptions say that $\Delta u_n(y)\to 0$ for every $y\in D$ and that $|\Delta u_n|\le C$ uniformly. Thus the integrand converges pointwise to zero and is dominated by $CT$, whose expectation is finite by part (a). The dominated convergence theorem gives $u_n(x)\to 0$ for every $x\in D$.

Solved by gpt-5.6-sol high.