  The experimental search for the QCD critical point by means of relativistic
heavy-ion collisions necessitates the development of dynamical models of
fluctuations. In this work we study the fluctuations of the net-baryon density
near the critical point. Due to net-baryon number conservation the correct
dynamics is given by the fluid dynamical diffusion equation, which we extend by
a white noise stochastic term to include intrinsic fluctuations. We quantify
finite resolution and finite size effects by comparing our numerical results to
analytic expectations for the structure factor and the equal-time correlation
function. In small systems the net-baryon number conservation turns out to be
quantitatively and qualitatively important, as it introduces anticorrelations
at larger distances. Including nonlinear coupling terms in the form of a
Ginzburg-Landau free energy functional we observe non-Gaussian fluctuations
quantified by the excess kurtosis. We study the dynamical properties of the
system close to equilibrium, for a sudden quench in temperature and a
Hubble-like temperature evolution. In the real-time dynamical systems we find
the important dynamical effects of critical slowing down, weakening of the
extremal value and retardation of the fluctuation signal. In this work we
establish a set of general tests, which should be met by any model propagating
fluctuations, including upcoming $3+1$ dimensional fluctuating fluid dynamics.
