  Chemical master equation plays an important role to describe the time
evolution of homogeneous chemical system. In addition to the reaction process,
it is also accompanied by physical diffusion of the reactants in complex system
that is generally not homogeneous, which will result in non-exponential waiting
times for particle reactions and diffusion. In this paper we shall introduce a
chemical continuous time random walk under anomalous diffusion model based on
renewal process to describe the general reaction-diffusion process in the
heterogeneous system, where the waiting times are arbitrary distributed.
According to this model, we will develop the systematic stochastic theory
including generalizing the chemical diffusion master equation, deriving the
corresponding mass action law, and extending the Gillespie algorithm. As an
example, we analyze the monomolecular $A\leftrightarrow B$ reaction-diffusion
system for exponential and power-law waiting times respectively, and show the
strong fractional memory effect of the concentration of the reactants on the
history of the concentration in power-law case.
