  The chemical diffusion master equation (CDME) describes the probabilistic
dynamics of reaction--diffusion systems at the molecular level [del Razo et
al., Lett. Math. Phys. 112:49, 2022]; it can be considered the master equation
for reaction--diffusion processes. The CDME consists of an infinite ordered
family of Fokker--Planck equations, where each level of the ordered family
corresponds to a certain number of particles and each particle represents a
molecule. The equations at each level describe the spatial diffusion of the
corresponding set of particles, and they are coupled to each other via reaction
operators --linear operators representing chemical reactions. These operators
change the number of particles in the system, and thus transport probability
between different levels in the family. In this work, we present three
approaches to formulate the CDME and show the relations between them. We
further deduce the non-trivial combinatorial factors contained in the reaction
operators, and we elucidate the relation to the original formulation of the
CDME, which is based on creation and annihilation operators acting on
many-particle probability density functions. Finally we discuss applications to
multiscale simulations of biochemical systems among other future prospects.
