  There have been significant recent advances in our understanding of the
potential use and limitations of early-warning signs for predicting drastic
changes, so called critical transitions or tipping points, in dynamical
systems. A focus of mathematical modeling and analysis has been on stochastic
ordinary differential equations, where generic statistical early-warning signs
can be identified near bifurcation-induced tipping points. In this paper, we
outline some basic steps to extend this theory to stochastic partial
differential equations with a focus on analytically characterizing basic
scaling laws for linear SPDEs and comparing the results to numerical
simulations of fully nonlinear problems. In particular, we study stochastic
versions of the Swift-Hohenberg and Ginzburg-Landau equations. We derive a
scaling law of the covariance operator in a regime where linearization is
expected to be a good approximation for the local fluctuations around
deterministic steady states. We compare these results to direct numerical
simulation, and study the influence of noise level, noise color, distance to
bifurcation and domain size on early-warning signs.
