Hausdorff dimension of the zero level set of alpha-Stochastic heat equation

Consider the equation

$latex dz+\nu A^{\alpha}z=C^{1/2}dW, z(0)=0$

where for $latex x:=\sum_{k\in Z^{2}\setminus \{(0,0)\}} x_{k}e_{k}$, with $latex \sum |x_{k}|^{2}<\infty$,

we set $latex A^{\alpha}(x):=\sum_{k\in Z^{2}_{*}}|k|^{\alpha} x_{k}e_{k}$ for $latex \alpha>1$

and $latex C(x):=\sum_{k\in Z^{2}_{*}}\sigma_{k}^{2}x_{k}e_{k}$ such that for $latex \delta\in (1-\alpha,2-\alpha)$ and $latex \frac{c_{2}}{|k|^{\delta}}\leq |\sigma_{k}|\leq \frac{c_{3}}{|k|^{\delta}}$.

RO: The dimension of the zero level set is $latex dim(\{x\in T_{2}: z_{t}(x)=0\})=3-\alpha-\delta$.

Reference: Hausdorff dimension of the level sets of some stochastic PDEs from fluid dynamics