  We study in the present article the Kardar-Parisi-Zhang (KPZ) equation $$
\partial_t h(t,x)=\nu\Delta h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\,
\eta(t,x), \qquad (t,x)\in\mathbb{R}_+\times\mathbb{R}^d $$ in $d\ge 3$
dimensions in the perturbative regime, i.e. for $\lambda>0$ small enough and a
smooth, bounded, integrable initial condition $h_0=h(t=0,\cdot)$. The forcing
term $\eta$ in the right-hand side is a regularized space-time white noise. The
exponential of $h$ -- its so-called Cole-Hopf transform -- is known to satisfy
a linear PDE with multiplicative noise. We prove a large-scale diffusive limit
for the solution, in particular a time-integrated heat-kernel behavior for the
covariance in a parabolic scaling.
  The proof is based on a rigorous implementation of K. Wilson's
renormalization group scheme. A double cluster/momentum-decoupling expansion
allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear
PDE in the small-field region where the noise is not too large, following the
broad lines of Iagolnitzer-Magnen. Standard large deviation estimates for
$\eta$ make it possible to extend the above estimates to the large-field
region. Finally, we show, by resumming all the by-products of the expansion,
that the solution $h$ may be written in the large-scale limit (after a suitable
Galilei transformation) as a small perturbation of the solution of the
underlying linear Edwards-Wilkinson model ($\lambda=0$) with renormalized
coefficients $\nu_{eff}=\nu+O(\lambda^2),D_{eff}=D+O(\lambda^2)$.
