In this paper, we study large and moderate deviation principles for stochastic partial differential equations (SPDEs) on metric graphs and their associated multiscale models via the weak convergence approach, providing a refined characterization of the probabilities of rare events. Several challenges unique to the graph setting are encountered, including operator degeneracy near vertices and the lack of compactness on non-compact graphs. To address these difficulties, we introduce novel weighted Sobolev spaces on graphs, and prove compact embedding results specifically adapted to the degeneracy structure. Our analysis is particularly applicable to SPDEs on graphs arising as limits of stochastic reaction-diffusion systems on narrow domains and from fast-flow asymptotics of stochastic incompressible fluids, yielding new deviation results for these models.
Cui et al. (Sat,) studied this question.