We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle may not be negligible in comparison to the finescale. This motivates our study of a periodic, directed, and weighted graph embedded in ${\mathbb R}^d$ and the scaling limit of the associated continuous-time random walk $Z(t)$ on the graph's nodes, which jumps along the graph's edges with jump rates given by the edge weights. We show that the scaled process $\varepsilon^2 Z(t/\varepsilon^2)$ converges to a linear drift $\bar{U}t$ and the case of interest to us is that of null drift $\bar{U}=0$. In this case, we show that $\varepsilon Z(t/\varepsilon^2)$ converges weakly to a Brownian motion. The diffusivity of the limiting Brownian motion can be computed by solving a set of linear algebra problems. As we allow for jump rates to be irreversible, our f 查看全文>>