  Consider the linear stochastic biharmonic heat equation on a $d$-dimensional
torus ($d=1,2,3$), driven by a space-time white noise and with periodic
boundary conditions: \begin{equation} \label{0} \left(\frac{\partial}{\partial
t}+(-\Delta)^2\right) v(t,x)= \sigma \dot W(t,x),\ (t,x)\in(0,T]\times
\mathbb{T}^d,\ v(0,x)=v_0(x). \end{equation}
  We find the canonical pseudo-distance corresponding to the random field
solution, therefore the precise description of the anisotropies of the process.
We see that for $d=2$, they include a $z(\log \tfrac{c}{z})^{1/2}$ term.
Consider $D$ independent copies of the random field solution to the SPDEs.
Applying the criteria proved in [4], we establish upper and lower bounds for
the probabilities that the path process hits bounded Borel sets. This yields
results on the polarity of sets and on the Hausdorff dimension of the path
process.
