Planar Brownian motion is recurrent. More explicitly, the <planar Brownian annulus hitting probability> gives
$$
\mathbb P_x(T_1<T_R)
=\frac{\log R-\log|x|}{\log R}
\longrightarrow1
\qquad(R\to\infty)
$$
when $|x|>1$. Thus the unit disc is hit almost surely from every starting point. Applying the after each departure and return shows that such returns occur after arbitrarily large times. Therefore
$$
\boxed{\{t\geq0:|X_t|\leq1\}\text{ is almost surely unbounded}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 5 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-201/5/a/solution