Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots, S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.

Let $\tau_{N}$ be the first time $S_{n}$ exits the box $[-N,N] \times \dots \times [-N,N]$. For $d \geq 2$, is anything known about the behaviour of $E(\tau_{N})$ as $N\rightarrow \infty$ ?

Pointers about the corresponding results for Brownian motion also very welcome.