An individual, henceforth called the runner starts at the center of an open two dimensional ball $B$ of radius $r > 1$.

At each turn, a vector $x \in S^1$ is chosen uniformly at random, and the runner may choose to move one unit in the direction $x$ or $-x$ - that is, his new position is just his old position, translated by $x$ or $-x$. The runner’s goal is to stay within $B$ for as long as possible, hence he shall always make the choice so as to minimise the distance to the center.

Denote by $P_t$ the runner’s position at time $t$, and define the exit time $\tau$ by

$$\tau := \inf \{t \in \mathbb Z_+ \, | \,P_t \in B^c\}.$$

It is true that $\tau$ is finite almost surely.

Question: Can we compute explicitly $\mathbb E[\tau]$?

As a bonus question, what is the optimal strategy for a square?