Extrapolation and sampling for processes on spatial graphs

The paper studies processes defined on time domains structured as oriented spatial graphs (or metric graphs, or oriented branched 1-manifolds). This setting can be used, for example, for forecasting models involving branching scenarios. For these processes, a notion of the spectrum degeneracy that takes into account the topology of the graph is introduced. The paper suggests sufficient conditions of uniqueness of extrapolation and recovery from the observations on a single branch. This also implies an analog of sampling theorem for branching processes, i.e., criterions of their recovery from a set of equidistant samples, as well as from a set of equidistant samples from a single branch.