One-force-one-solution in random Hamilton–Jacobi equations

Let $d\in\mathbb{N}$ and let $F_\omega(t,x)$ have exponential decay of space-time correlations. For every $b\in\mathbb{R}^d$, consider global solutions of the random Hamilton–Jacobi equation of the form

$$\Phi_{b,\omega}(t,x)=b\cdot x+\psi_{b,\omega}(t,x),$$

where $\psi_{b,\omega}$ has sublinear growth, and identify solutions modulo time-dependent additive constants. One-force-one-solution conjecture. 1F1S holds: there is a unique time-stationary global solution of this form for every $b$, these solutions are continuous in $\nu\geq0$, and global solutions are preserved under the zero-viscosity limit. The claim extends established compact and special noncompact results to arbitrary dimension and exponentially mixing forcing; its resolution is not supplied in the source.