Brownian-like motion covering a set of positive measure

Question

I've been wondering about this question, in $\mathbb{R}^2$ under what conditions does a Brownian-like motion $BL_t$ covers a set of positive measure or, what are the odds that, for some initial conditions and $\epsilon>0$, as $t\to\infty$ there is a chance that $BL_t$ covers a set of measure $\epsilon$.