Diffusion processes, metric graphs and boundary value problems for reaction diffusion systems

We consider a mathematical model of a class of first order reaction-diffusion system, known as TAP systems, in which gas reactants and products diffuse in a domain and reaction takes place at a single relatively small catalytic site in the domain. The central problem is to determine the probability of reaction, or, equivalently, the yield of the reaction, in terms of the geometric parameters of the system and the chemical reaction constant. This is shown to be solved by a boundary value problem for the time-independent Feynman-Kac equation. Our focus here is on network-shaped (reactor) domains. The main result of the paper is a factorization formula for reaction yield that separates the purely geometric from the chemical kinetic characteristics of the process. The formula is shown to hold exactly for systems described by metric graphs.