We study the symmetric simple exclusion process on the Sierpinski gasket ($SG$) driven by the action of particle reservoirs attached to boundary vertices of $SG$.

We establish three hydrodynamic limit theorems for the empirical current: the law of large numbers, the large deviations principle, and the large deviations principle for the mean current on a long-time interval. On $\mathbb{Z}^d$ these results were established assuming translational invariance and Gaussian space-time diffusive estimates. But on $SG$ neither assumption is valid, and it is unclear how to make sense of the resulting reaction-diffusion PDE.

In this work we overcome all the aforementioned obstacles. First, we establish the "moving particle lemma" on weighted graphs, using the "octopus inequality" of Caputo, Liggett, and Richthammer in their seminal proof of Aldous' spectral gap conjecture. This enables us to prove a local version of the two-blocks estimate on $SG$, thereby answering a question posed by Jara. Second, we actively use the theory of differential $1$-forms on $SG$ developed by Teplyaev and collaborators, which allows us to characterize the speed of convergence of discrete $1$-forms on $SG$, and prove uniqueness of solutions to the resulting reaction-diffusion PDE.

This is joint work with Alexander Teplyaev.