Conjecture on infinite-dimensional random attractors under infinite energy mean

Let $p$, $a$, $b$ and $q$ satisfy

$$\ln\left(e^{-a}q+e^{\frac{2(p+2)}{p+4}b}(1-q)\right)>0>-aq+\frac{2(p+2)}{p+4}b(1-q),$$

and let $f$ be a nonlinearity and $g$ a right-hand side satisfying the assumptions of Theorem 5. Let

\mathcal A($\eta) be the associated random attractor in the energy space $E$. **Infinite-dimensional attractor conjecture.** There are a nonlinearity $f$ and \right-hand side $g$ satisfying those assumptions such that

mathcal A($\eta) has infinite Hausdorff and fractal dimensions in$E$.

The parameter condition makes the constructed energy bound have infinite mean. The conjecture proposes that this lack of a first energy moment can lead to attractors with infinite Hausdorff and fractal dimensions; the source gives no resolution.