Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 1 / c

Solution

Let

σ

be the first time Brownian motion started outside the unit half-disc reaches its semicircular boundary. A path that reaches

A

σ

gives

E_{z} [Im B_{τ}] = \int_{0}^{π} p (z, e^{i θ}) E_{e^{i θ}} [Im B_{τ}] d θ .

Take

z = i y

, multiply by

y

, and let

y \to \infty

hcap (A)

, while part b gives

y p (i y, e^{i θ}) ⟶ \frac{2}{π} sin θ .

Since the exit height is between zero and one, the dominated convergence theorem applies and yields

hcap (A) = \frac{2}{π} \int_{0}^{π} E_{e^{i θ}} [Im B_{τ}] sin θ d θ .

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