Quantitative estimates for SPDEs on the full space with transport noise and $L^p$-initial data

For the stochastic linear transport equation with $L^p$-initial data ($1<p<2$) on the full space ${\mathbb{R}}^d$, we provide quantitative estimates, in negative Sobolev norms, between its solutions and that of the deterministic heat equation. Similar results are proved for the stochastic 2D Euler equations with transport noise.