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

Planar Brownian motion is recurrent. More explicitly, the planar Brownian annulus hitting probability giveswhen $|x|>1$. Thus the unit disc is hit almost surely from every starting point. Applying the Strong Markov property after each departure and return shows that such returns occur after arbitrarily large times. Therefore

By the Tonelli theorem and the planar Brownian transition density,For $s\ge 1$,while for $0\le s\le 1$ it is at most $\Vert f{\Vert }_{\infty }$. Hencefor $t\ge 1$, and consequently

Because $f$ is a continuous probability density, there are a point $z$, a radius $r>0$, and $ϵ>0$ such thatBy the recurrence of planar Brownian motion, the smaller disc $B(z,r/2)$ is visited at arbitrarily large times. Starting anywhere in that smaller disc, Brownian continuity and compactness give a uniform probability $\delta >0$ of staying in $B(z,r)$ for a fixed time $u>0$.

Apply the Strong Markov property at successive visits separated by at least $u$. The conditional probability of each stay event is at least $\delta$, so the conditional Borel-Cantelli lemma gives infinitely many successful stays almost surely. Every success adds at least $ϵu$ to $A_t$. Since $A_t$ is nondecreasing,