  In this work, we study early-warning signs for stochastic partial
differential equations (SPDEs), where the linearization around a steady state
has continuous spectrum. The studied warning sign takes the form of qualitative
changes in the variance as a deterministic bifurcation threshold is approached
via parameter variation. Specifically, we focus on the scaling law of the
variance near the transition. Since we are dealing here, in contrast to
previous studies, with the case of continuous spectrum and quantitative scaling
laws, it is natural to start with linearizations that are multiplication
operators defined by analytic functions. For a one-dimensional spatial domain
we obtain precise rates of divergence. In the case of the two- and
three-dimensional domains an upper bound to the rate of the early-warning sign
is proven. These results are cross-validated by numerical simulations. Our
theory can be generically useful for several applications, where stochastic and
spatial aspects are important in combination with continuous spectrum
bifurcations.
