  This paper is a continuation of Poiret-Robert-Thomann (2013) where we studied
a randomisation method based on the Laplacian with harmonic potential. Here we
extend our previous results to the case of any polynomial and confining
potential $V$ on $\mathbb{R}^d$. We construct measures, under concentration
type assumptions, on the support of which we prove optimal weighted Sobolev
estimates on $\mathbb{R}^d$. This construction relies on accurate estimates on
the spectral function in a non-compact configuration space. Then we prove
random quantum ergodicity results without specific assumption on the classical
dynamics. Finally, we prove that almost all basis of Hermite functions is
quantum uniquely ergodic.
