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We propose random walks on suitably defined graphs as a framework forfinescale modeling of particle motion in an obstructed environment where theparticle may have interactions with the obstructions and the mean path lengthof the particle may not be negligible in comparison to the finescale. Thismotivates our study of a periodic, directed, and weighted graph embedded in${\mathbb R}^d$ and the scaling limit of the associated continuous-time randomwalk $Z(t)$ on the graph's nodes, which jumps along the graph's edges with jumprates given by the edge weights. We show that the scaled process $\varepsilon^2Z(t/\varepsilon^2)$ converges to a linear drift $\bar{U}t$ and the case ofinterest 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. Thediffusivity of the limiting Brownian motion can be computed by solving a set oflinear algebra problems. As we allow for jump rates to be irreversible, ourframework allows for the modeling of very general forms of interactions such asattraction, repulsion, and bonding. We provide some sufficient conditions fornull drift that include certain symmetries of the graph. We also provide aformal asymptotic derivation of the effective diffusivity in analogy withhomogenization theory for PDEs. For the case of reversible jump rates, wederive an equivalent variational formulation. This derivation involvesdeveloping notions of gradient for functions on the graph's nodes, divergencefor ${\mathbb R}^d$-valued functions on the graph's edges, and a divergencetheorem.

Homogenization of a Random Walk on a Graph in $\mathbb{R}^d$: An  approach to predict macroscale diffusivity in media with finescale  obstructions and interactions