 Solution

ID: past-exam-of-the-continuous-martingales-and-stochastic-calculus-course-of-the-university-of-oxford/2023/3/b/ii/solution

Fix an outer radius

R > ρ = ∣ z_{0} ∣

. By continuity, reaching zero before radius

R

requires first reaching every radius

ε \in (0, ρ)

0 \leq P (τ_{0} < τ_{R}) \leq P (τ_{ε} < τ_{R}) = \frac{l o g ( R / ρ )}{l o g ( R / ε )} ⟶ 0 (ε ↓ 0) .

Thus

P (τ_{0} < τ_{R}) = 0

for every fixed

R > ρ

On any continuous sample path that reaches zero in finite time, the radius has a finite maximum before that time. There is therefore some integer

m > ρ

for which zero is reached before radius

m

. Consequently

{τ_{0} < \infty} = ⋃_{m \in N, m > ρ} {τ_{0} < τ_{m}} .

Every event on the right has probability zero, so

P (τ_{0} = \infty) = 1.

This proves that planar Brownian motion avoids a fixed point even though it can reach arbitrarily small positive radii.

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