On range and local time of many-dimensional submartingales

Abstract

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 $n^{1/2-δ}$ for some $δ>0$. This in its turn implies that the number of different sites visited by the process by time $n$ should be at least $n^{1/2+δ}$.