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The clock $A_t={\int }_0^t{e}^{2{\beta }_s}ds$ is continuous, strictly increasing and unbounded. Define ${\tau }_r=\inf \{t:A_t\ge r\}$. Part (a), with the initial vector retained, proves thatEvery finite $r$ has finite ${\tau }_r$, and $|{\mathcal{B}}_r|=e^{{\beta }_{{\tau }_r}}>0$. On the common event where the clock is a continuous bijection of $[0,\infty )$, this holds simultaneously for all times. Thus the origin is a polar point for planar Brownian motion started at $(1,0)$:To prove unbounded-time visits near zero, let $t_n=\inf \{t\ge n:{\beta }_t=-n\}$. Brownian recurrence makes each $t_n$ finite almost surely, and $t_n\ge n\to \infty$. Set $r_n=A_{t_n}$. Since $A_t\to \infty$, $r_n\to \infty$, whileConsequentlyFor every fixed neighbourhood of zero there are visits at arbitrarily large times. Since $\mathcal{B}$ has the law of planar Brownian motion with this initial value, both conclusions hold for that Brownian law. This complex exponential construction of planar Brownian motion proves the two properties together without confusing recurrence near a point with hitting the point.