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

Solution

Let

B

z \in D

, let

τ_{D}

be its exit time, and let

ϕ : D \to D^{'}

be conformal. Define

C_{s} = \int_{0}^{s \land τ_{D}} ∣ ϕ^{'} (B_{r}) ∣^{2} d r, σ_{t} = in f {s : C_{s} > t} .

B_{t} = ϕ (B_{σ_{t}}), 0 \leq t < C_{τ_{D}},

ϕ (z)

and stopped when it exits

D^{'}

. Thus conformal maps preserve Brownian paths after this random time-change.

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