Planar <Brownian motion> and the initial law $\nu_{0,R}$ are invariant under every rotation about the origin. The hitting time $\tau_{0,r}$ is also rotation invariant, so the law of $B_{\tau_{0,r}}$ is invariant under every rotation of the circle of radius $r$. Planar Brownian motion hits that circle almost surely by <recurrence of planar Brownian motion>. The unique rotation-invariant probability measure on the circle is its uniform measure, and therefore
$$
B_{\tau_{0,r}}\sim\nu_{0,r}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 5 / a / Solution

Planar Brownian motion almost surely visits every neighborhood of every point. Equivalently, it hits every disc almost surely, although it does not hit any prescribed point with positive probability.


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 Incoming links: Recurrence of planar Brownian motion