Being able to create and sustain robust, spatial-temporal inhomogeneity
is an important concept in developmental biology. Generally, the
mathematical treatments of these biological systems have used continuum
hypotheses of the reacting populations, which ignores any sources of
intrinsic stochastic effects. We address this concern by developing
analytical Fourier methods which allow us to probe the probabilistic
framework. Further, a novel description of domain growth is produced,
which is able to rigorously link the mean-field and stochastic
descriptions. Finally, through combining all of these ideas, it is shown
that the description of diffusion on a growing domain is non-unique and,
due to these distinct descriptions, diffusion is able to support
patterning without the addition of further kinetics.