Uniform treatment of fluctuations at critical points

A generalized critical point is characterized by the vanishing of certain linear relationships. In particular, the dynamics near such a point are non-linear. In this paper, we study fluctuations at such points of spatially homogeneous systems. We discuss thermodynamic critical points as a special case; but the main emphasis is on stochastic kinetic equations. We show that fluctuations at a critical point cannot be characterized by a Gaussian density, but more complicated densities can be used. The theory is applied to the critical harmonic oscillator.