  The Gierer-Meinhardt system occurs in morphogenesis, where the development of
an organism from a single cell is modelled. One of the steps in the
development, is the formation of spatial patterns of the cell structure,
starting from an almost homogeneous cell distribution. Turing proposed in his
pioneering work different activator-inhibitor systems with different diffusion
rates, which could trigger the emergence of such cell structures.
Mathematically, one describes these activator-inhibitor systems as a coupled
systems of reaction-diffusion equations with hugely different diffusion
coefficients and highly nonlinear interaction. One famous example of these
systems is the Gierer-Meinhardt system. These systems usually are not of
monotone type, such that one has to apply other techniques. The purpose of this
article is to study the stochastic reaction-diffusion Gierer-Meinhardt system
with homogeneous Neumann boundary condition on a one or two-dimensional bounded
spatial domain. To be more precise, we perturb the original Gierer-Meinhardt
system by an infinite-dimensional Wiener process and show under which
conditions on the Wiener process and the system, a solution exists. In
dimension one, we even show the pathwise uniqueness. In dimension two,
uniqueness is still an open question.
