Drift-blind Frostman disintegration for Gaussian fibers and a solution of the Balka--Darji--Elekes Brownian-drift problem

Let $G$ be a $d$-dimensional Gaussian process whose coordinates are independent copies of a centered scalar process that is strongly locally nondeterministic with index $H$. Fix a continuous deterministic drift $f$ and an $s$-Frostman measure $\mu$ on $[0,1]$, with $s>Hd$. We construct, simultaneously for every spatial level $z$, a conditional occupation measure $\ell_z$ supported on $(G+f)^{-1}(z)\cap\supp\mu$. In the Fourier representations, the drift contributes only unimodular phases, so all deterministic moment constants are independent of $f$. Almost surely, for every $0<\gamma0$, the set $\{z:\ell_z(J)>0\}$ is nonempty and open, captures full $\mu$-occupation in $J$, and every corresponding fiber has Hausdorff dimension at least $s-Hd$. For one-dimensional Brownian motion, $s=1$ and $H=1/2$; this answers both clauses of Problem 9.1 of Balka, Darji and Elekes, including full occupation for the samplewise largest open good-level set. The probability-one event is chosen after the deterministic drift is fixed.