On occupation time functionals for diffusion processes and birth-and-death processes on graphs

Occupation time functionals for a diffusion process or a birth-and-death process on the edges of a graph . depending only on the values of the process on a part .... of . are closely related to so-called eigenvalue depending boundary conditions for the resolvent of the process. Under the assumption that the connected components of .... are trees, we use the special structure of these boundary conditions to give a procedure that replaces each of the trees by only one edge and that associates this edge with a speed measure such that the respective functional for the appearing process on the simplified graph coincides with the original one.