  A quantitatively reliable theoretical description of the dynamics of
fluctuations in non-equilibrium is indispensable in the experimental search for
the QCD critical point by means of ultra-relativistic heavy-ion collisions. In
this work we consider the fluctuations of the net-baryon density which becomes
the slow, critical mode near the critical point. Due to net-baryon number
conservation the dynamics is described by the fluid dynamical diffusion
equation, which we extend to contain a white noise stochastic current.
Including nonlinear couplings from the 3d Ising model universality class in the
free energy functional, we solve the fully interacting theory in a finite size
system. We observe that purely Gaussian white noise generates non-Gaussian
fluctuations, but finite size effects and exact net-baryon number conservation
lead to significant deviations from the expected behavior in equilibrated
systems. In particular the skewness shows a qualitative deviation from infinite
volume expectations. With this benchmark established we study the real-time
dynamics of the fluctuations. We recover the expected dynamical scaling
behavior and observe retardation effects and the impact of critical slowing
down near the pseudo-critical temperature.
