I've been taken by the concise result1
that (roughly!), on a $2$-dimensional torus $\mathbb{T}^2$, the time it takes
to visit nearly every point (within $\epsilon$, as $\epsilon \to 0$) is: $\frac{2}{\pi}$.


My question is:



> 
> ***Q***. Is the same situation known for the $d$-dimensional 
>  torus, $\mathbb{T}^d$? What is the time it takes for a Brownian-motion particle to visit within $\epsilon \to 0$ of
>  every point of $\mathbb{T}^d$?
> 
> 
> 


This is probably known—or known to be unknown—so this is just
a reference request.




---


1
Dembo, Amir, Yuval Peres, Jay Rosen, and Ofer Zeitouni. "Cover times for Brownian motion and random walks in two dimensions." *Annals of Mathematics* (2004): 433-464.
[Annals link](https://www.jstor.org/stable/3597219?casa_token=Xsb4bweNvrEAAAAA:5xPV9W42VU_6bBbiW9eFIAGIpqvjuMWMtFpMLAEVVUCVqW8foeGRxMFKSx-pFW64b0FJIkpONRbGXoi45Y_Gd5bZlBywqVgqAra2fupzlfHBkYXrYwj7&seq=1#metadata_info_tab_contents).



How long for Brownian motion to "fill-out" a torus in d-dimensions?
A very general answer, in dimension $d\geq 3$, 
 is in the following paper of Dembo, Peres and Rosen
<https://projecteuclid.org/euclid.ejp/1464037588>:
for compact $d$-dimensional manifolds,
$$C\_\epsilon(M) /\epsilon^{2-d}\to V(M)$$
where $V(M)$ is the volume.


The $d$ dimensional case for $d\geq 3$ is much simpler than $d=2$ because in the latter, you have 
long range (=logarithmic) correlations arising from the recurrence of BM. 


In passing, let me mention that Belius's paper mentioned in Josiah Park's answer gives a more refined answer, namely a limit law for a centered version of
$\sqrt{C\_\epsilon}$ (with no scaling), 
for the random walk case. Similar results can be written 
for the manifold case in $d\geq 3$. For the
critical case of $d=2$, higher precision than the law of large numbers is available but as of this writing, the limit law for $\sqrt{C\_\epsilon}-E\sqrt{C\_\epsilon}$ has not been derived.


