In this work, we study early warning signs for stochastic partial differential equations (SPDEs), where the linearisation around a steady state is characterised by 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 linearisations of the drift operator 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.

In this paper, the boundedness from below of multiplication operators between $\alpha$-Bloch spaces ${{B}^{\alpha }},\,\alpha \,>\,0$, on the unit disk $D$ is studied completely. For a bounded multiplication operator ${{M}_{u}}\,:\,{{B}^{\alpha }}\,\to \,{{B}^{\beta }}$, defined by ${{M}_{u}}f\,=\,uf$ for $f\,\in \,{{B}^{\alpha }}$, we prove the following result:

(i) If $0<\beta <\alpha ,\,\text{or}\,\text{0}<\alpha \le \text{1}\,\text{and}\,\alpha <\beta \text{,}\,{{M}_{u}}$ is not bounded below;

(ii) if $0\,<\,\alpha \,=\,\beta \,\le \,1,\,{{M}_{u}}$ is bounded below if and only if lim ${{\inf }_{z\to \partial D}}\,\left| u\left( z \right) \right|\,>\,0;$

(iii) if $1\,<\,\alpha \,\le \,\beta ,\,{{M}_{u}}$ is bounded below if and only if there exist a $\delta \,>\,0$ and a positive $r\,<\,1$ such that for every point $z\,\in \,D$ there is a point ${{z}^{'}}\,\in \,D$ with the property $d\left( {{z}^{'}},\,z \right)\,<\,r$ and ${{\left( 1\,-\,{{\left| {{z}^{'}} \right|}^{2}} \right)}^{\beta -\alpha }}\left| u\left( {{z}^{'}} \right) \right|\,\ge \,\delta$, where $d\left( \cdot ,\,\cdot \right)$ denotes the pseudo-distance on $D$.

This paper studies the concept of strongly omnipresent operators that was recently introduced by the first two authors. An operator T on the space H(G) of holomorphic functions on a complex domain G is called strongly omnipresent whenever the set of T-monsters is residual in H(G), and a T-monster is a function f such that Tf exhibits an extremely ‘wild’ behaviour near the boundary. We obtain sufficient conditions under which an operator is strongly omnipresent, in particular, we show that every onto linear operator is strongly omnipresent. Using these criteria we completely characterize strongly omnipresent composition and multiplication operators.