Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 203 / 2 / c

Solution

By conformal invariance,

g_{A} (B)

H

, started at

g_{A} (i y) = x_{y} + i v_{y}, x_{y} \to 0, v_{y} / y \to 1.

Since

E \subset R ∖ [- 1, 1]

and

A \subset \overline{D}

, the boundary correspondence is regular there, and the exit event maps to

g_{A} (E)

H

therefore gives

P^{i y} (B_{τ_{A}} \in E) = \int_{g_{A} (E)} \frac{v _{y}}{( u - x _{y} ) ^{2} + v _{y}^{2}} \frac{d u}{π} .

y

makes the integrand converge uniformly to

1/ π

. Approximation by increasing bounded subsets and monotone convergence then gives

lim_{y \to \infty} y P^{i y} (B_{τ_{A}} \in E) = \frac{Leb ( g _{A} ( E ))}{π},

with both sides allowed to be infinite.

There are no discussions about this article yet.

There are currently no matching articles.