Global infinite-oscillation conjecture for parabolic SPDEs

Let $u$ be the solution of the parabolic stochastic partial differential equation considered in the paper, with coefficient $\sigma$ satisfying $\sigma^{-1}\{0\}=\{0\}$ and bounded, and let $f:\mathds{R}^3\to\mathds{R}_+$ be a correlation function satisfying the Dalang condition.

Global infinite-oscillation conjecture. There exist such correlation functions $f$ for which

$$\mathrm{P}\left\{ \text{Osc}_{u(t)}(x)=\infty\text{ for all $(t,x)\in(0,\infty)\times\mathds{R}^3$}\right\}=1.$$

The preceding theorems establish this behavior pointwise, or under a conditional formulation, but do not establish simultaneous infinite oscillation at every space-time point. The conjecture concerns the stronger global blowup of oscillations for nonlinear SPDEs driven by Gaussian noise.