Strong-forcing invariant-measure conjecture for the stochastic Ginzburg–Landau equations

Consider a stochastically forced version of the full Gor'kov–Eliashberg equations

$$\text{the equations }\text{--}.$$

For each $\varepsilon>0$, let the forcing be strong enough that the resulting system admits an invariant measure. Strong-forcing invariant-measure conjecture. If the forcing is strong enough, the system admits a unique invariant measure for all $\varepsilon>0$. As $\varepsilon\to0$, this invariant measure converges weakly to a measure supported on the set of all functions that are limits of global minimizers of the Ginzburg–Landau energy functional. Depending on the relation between $h_{\textit{ex}}$ and $h_{c_1}$, these limiting functions correspond either to the purely superconducting state or to a nucleated state with finitely many vortices. This conjecture proposes a physically realistic stochastic mechanism for vortex nucleation and for tunnelling toward lower-energy states, but the source provides no resolution of the claimed existence, uniqueness, or limiting-measure properties.