  We extend a randomisation method, introduced by Shiffman-Zelditch and
developed by Burq-Lebeau on compact manifolds for the Laplace operator, to the
case of $\mathbb{R}^d$ with the harmonic oscillator. We construct measures,
thanks to probability laws which satisfy the concentration of measure property,
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. As an application, we show that
there exists a basis of Hermite functions with good decay properties in
$L^{\infty}(\mathbb{R}^d$)$, when $d\geq 2$.
