“…In fact, there is a broad area of possible applications where the mathematical use of graphs and random dynamics stated on them, play a crucial role, as in the case, e.g., of quantum mechanics, see, e.g. [36], the books [24,31] and references therein; in neurobiology, as an example concerning the study of stochastic system of the FitzHugh-Nagumo type, see, e.g., [1,2,3,8,10]; or in finance, see, e.g., [6,14,15,26] and references therein, particularly in the light of numerical applications, see, e.g., [18] Concerning the aforementioned ambit, a possible approach which has shown to be particularly useful, is to introduce a suitable infinite dimensional space of functions that takes into account the underlying graph domain and then tackle the diffusion problem exploiting both functional analytic tools and infinite dimensional analysis. This technique had led to a systematic study of Stochastic Partial Differential Equations (SPDEs) on networks, showing that it is in general possible to rewrite a diffusion problem defined on a network in a general abstract form, see, e.g., [8,10,11,19], and the monograph [33] for a detailed introduction to the subject.…”