  We propose a probabilistic derivation of the so-called chemical diffusion
master equation (CDME) and describe an infinite dimensional moment generating
function method for finding its analytical solution. CDMEs model by means of an
infinite system of coupled Fokker-Planck equations the probabilistic evolution
of chemical reaction kinetics associated with spatial diffusion of individual
particles; here, we focus an creation and mutual annihilation chemical
reactions combined with Brownian diffusion of the single particles. Our
probabilistic approach mimics the derivation of backward Kolmogorov equations
for birth-death continuous time Markov chains. Moreover, the proposed infinite
dimensional moment generating function method links certain finite dimensional
projections of the solution of the CDME to the solution of a single linear
fourth order partial differential equation containing as many variables as the
dimension of the aforementioned projection space.
