  We consider a discrete-time process adapted to some filtration which lives on
a (typically countable) subset of $\mathbb{R}^d$, $d\geq 2$. For this process,
we assume that it has uniformly bounded jumps, is uniformly elliptic (can
advance by at least some fixed amount with respect to any direction, with
uniformly positive probability). Also, we assume that the projection of this
process on some fixed vector is a submartingale, and that a stronger additional
condition on the direction of the drift holds (this condition does not exclude
that the drift could be equal to 0 or be arbitrarily small). The main result is
that with very high probability the number of visits to any fixed site by time
$n$ is less than $n^{1/2-\delta}$ for some $\delta>0$. This in its turn implies
that the number of different sites visited by the process by time $n$ should be
at least $n^{1/2+\delta}$.
