MHS equilibria in the non-resistive limit to the randomly forced resistive magnetic relaxation equations

We consider randomly forced resistive magnetic relaxation equations (MRE) with resistivity $\kappa >0$ and a force proportional to \sqrtκ\ on the flat $d$-torus ${\mathbb{T}}^d$ for $d\ge 2$. We show the path-wise global well-posedness of the system and the existence of the invariant measures, and construct a random magnetohydrostatic (MHS) equilibrium $B(x)$ in $H^1({\mathbb{T}}^d)$ with law $D(B)=\mu$ as a non-resistive limit $\kappa \to 0$ of statistically stationary solutions $B_{\kappa }(x,t)$. For $d=2$, the measure $\mu$ does not concentrate on any compact sets in $H^1({\mathbb{T}}^2)$ with finite Hausdorff dimension. In particular, all realizations of the random MHS equilibrium $B(x)$ are almost surely not finite Fourier mode solutions.